5371
DEGREE OF MASTER OF PHILOSOPHY
Master of Philosophy in Economics (Year 1)
Master of Philosophy in Economics (Final Examination)
Probationer Research Student Qualifying Examination in Economics
Probationer Research Student Examination in Management Studies
MACROECONOMICS
TRINITY TERM 2014
Monday 16th June 2014, 09:30 - 12:30
Please start the answer to each question on a separate page.
Answer four short questions from Part A and two long questions from Part B.
Part A attracts 1/2 of the marks in total; part B also attracts 1/2 of the marks.
Candidates may use their own calculators.
Do not turn over until told that you may do so.
1
Part A
1. Suppose a small open economy discovers oil. What impact might that have on its
real exchange rate? You can assume that the oil requires no significant resources to
extract.
2. Assuming that a version of Okun’s law holds, discuss whether you would expect the
level of vacancies posted by firms to be procyclical or countercyclical.
3. What is the empirical evidence about the effects of monetary policy shock on real
activity, inflation and monetary variables? Discuss.
4. In the context of a search and matching model of the labor market, discuss under what
condition the decentralized Nash wage bargaining internalizes the trading externalities.
If such a condition is not fulfilled, can monetary and fiscal policy restore efficiency?
5. Chari, Kehoe and McGrattan (Econometrica, 2007) argue that “...financial frictions
that manifest themselves primarily as investment wedges did not play a primary role in
the Great Depression or postwar recessions.” Explain, possibly by means of a simple
example, how this argument can be invalidated.
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2
Part B
6. There has recently been speculation that the natural (medium term) real interest rate
might have fallen. This question examines the extent to which demographic factors
might account for this.
(a) Derive an expression for the steady state real interest rate in a simple overlap-
pening generations model to investigate how much a decline in population growth
rates might reduce real interest rates. Assume a Cobb Douglas production func-
tion and log utility, that the saving of the young provides the capital stock for
the subsequent period, and that agents only work for the first period. Typical pa-
rameters for the exponent on capital in the production function α and the utility
discount rate β might be 0.4 and 0.5 respectively.
(b) Use this analysis to relate the real interest rate implied by this OLG model to
both the golden rule level, and the interest rate that would be generated in a
Ramsey model. (A discount factor of β would imply a rate of time preference of
(1 − β)/β.) Does a reduction in population growth reduce the chance of dynamic
inefficiency?
(c) Suppose average life expectency rises, but the retirement age does not. Use the
OLG framework to examine what impact this might have on the real interest
rate. Specifically, compare a situation where before the increase in life expectancy
agents worked part of the time when they were old, but after the increase in life
expectancy they only work when young. Does your result make sense intuitively?
If this is a factor behind the fall in global real interest rates, would you expect it
to be permanent?
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7. Consider an economy in which consumption goods can be produced using the following
production technology
yt = Kα,
t
where yt is output, Kt is capital input and α ∈ (0, 1). Capital input is a combination
of private capital kt and public capital κt, which is provided free of charge by the
government:
Kt = kγκθ
t
t
γ, θ > 0 . A social planner maximises the expected present discounted utility of a
representative consumer
∞
X
max E0
βsu(ct+s)
{ct,kt+1}
s=0
by choosing consumption ct and future private capital kt+1. The social planner does
not choose public capital, which is instead set exogenously by the political process.
The utility function satisfies u0 > 0, u00 < 0. There is unit marginal rate of transfor-
mation between output, consumption goods, private capital and public capital so the
law of motion for the private capital stock is
ct + kt+1 + κt+1 = yt + kt + κt
with neither private nor public capital subject to depreciation.
(a) Derive and interpret the first order conditions of the social planner’s problem.
Assume that public capital is constant in all periods so κt = κ for ∀t > 0.
(b) Solve for the steady-state level of private capital.
(c) Show that the steady-steady level of private capital is increasing in κ if and only
if αγ < 1. Interpret this condition.
(d) Under what condition(s) is steady-state output increasing in κ?
Assume that αγ < 1 and the economy is in steady state for κt = κ for t = 0 when
politicians unexpectedly announce a policy to increases the future level of public cap-
ital. The increase is permanent such that κt = κ0 > κ for ∀t > 1.
(e) Describe the impact of the new policy on consumption, investment, private capital
and output. Pay particular attention to the difference between short-run and
long-run effects.
(f) Recently, a number of governments have increased public investment in an at-
tempt to stimulate economic activity. Given your answer to part (e), explain
how the short run effects of increased investment in public capital depends on
how soon the investment becomes productive.
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8. An economy is characterized by final good producers and a continuum of differentiated
intermediate goods producers. The final good market is competitive. Final good
firms use intermediate inputs to produce the final good according to the following
technology:
ε
Z 1
ε−1
ε−1
Yt =
Yt(i) ε di
,
0
where Yt is the final good, Yt(i) is the intermediate good i ∈ [0, 1] and ε is the elasticity
of substitution between intermediate goods in the production function.
(a) Solve the cost minimization problem of the final good producers to find the opti-
mal demand of intermediate goods by the final good producer. Find the expres-
sion for the Lagrange multiplier of the cost minimization problem and give an
economic interpretation of it.
(b) Since the final good market is competitive, final good producers earn zero profit.
Impose the zero profit condition on the final good producers to find the price of
the final good. Why is the price of the final good equal to the Lagrange multiplier
of the previous problem?
(c) Assume that intermediate good producers are subject to the usual Calvo price
stickiness constraint, and they can change price each period only with proba-
bility (1 − θ), which is independent from the time elapsed since last adjust-
ment.
Moreover, they are producing with the following production function:
Yt(i) = AtNt(i)1−α, where α measures the degree of decreasing returns to scale.
Write down the problem of the intermediate goods producers, taking into ac-
count the demand for intermediate good i found above. Solve for the first order
condition for the optimal reset price.
(d) Loglinearize the first order condition to find the New Keynesian Phillips curve
(NKPC). How does the assumption of DRTS affect the slope of the NKPC? Why?
(e) Given that from the household problem, the real wage is equal to ˆ
wt = σ ˆ
yt + ϕˆ
nt,
(where ˆ
wt, ˆ
yt and ˆ
nt represents log-deviations from steady-state of the real wage,
output and hours worked) and given the definition of marginal cost, define the
output gap and re-write the NKPC in terms of output-gap.
(f) Close the model with the following dynamic IS equation and policy rule:
1
˜
yt = Et ˜
yt+1 −
(ˆıt − Etˆ
πt+1)
(1)
σ
ˆıt = φπ ˆ
πt + φyEt ˜
yt+1 + εi, εi ∼ white noise
(2)
t
t
where ˜
y denotes the output gap (i.e. the log-deviation of output from the flexible
price output). Find the solution for ˜
yt and ˆ
πt using the method of undetermined
coefficients, and find the implied volatilities for ˜
yt and ˆ
πt. Explain how these
volatilities depend on the policy parameters (φπ, φy) and on the slope of the
Phillips Curve.
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9. Consider an economy in which aggregate output (y) changes due to deviations of
inflation (π) from expected inflation (πe) and decreases due to taxes (τ ), so that
aggregate supply is: y = a(π − πe) − bτ . The government finances public spending
(g) with taxes (τ ) and seigniorage (i.e. inflation, π), so that the government budget
constraint is: g = µτ + π, where µ is the quality of fiscal institutions. The government
loss function is:
−(1/2) (π − π)2 + λy2 + θ(g − g)2 − γge,
(3)
e
where π is the exogenous inflation target, λ and θ are the weights on output and the
e
deviation of government spending from a positive exogenous target (g) respectively,
e is the level of effort the government may use to improve the quality of institutions,
and γg is the weight assigned to effort.
(a) In the commitment regime the monetary authority sets the target of inflation to
zero (π = 0) and internalizes the effect of its choice of inflation rate over the
e
agents expectation (πe = π). Assuming effort is exogenous, derive the rates of
inflation and taxes that minimize the government loss function (3).
(b) Show the effect of the quality of institutions µ on the optimal tax and inflation
rates and provide an intuition for the results.
(c) Assume that effort improves the quality of the institutions by choosing an ap-
propriate level of effort e. In particular, let µ = µ0eε, where 0 < ε < 1 and µ0
is the initial quality of institutions. Derive the level of effort that minimizes the
government loss function (3).
(d) Assume ε = 1/2, how is optimal effort related to the initial quality of institutions?
If the welfare cost of effort tends to be disproportionally high (i.e γg → ∞) would
the relationship change? Provide an interpratation.
(e) Assume the government acts under a discretionary regime (i.e. πe 6= π) and takes
a positive inflation target as given (π > 0). Derive the rates of inflation and taxes
e
that minimize the government loss function (3) and provide an interpretation.
(f) In equilibrium (i.e. πe = π), is monetary policy able to maintain the inflation
rate equal to the inflation target? If not, show whether the inflation rate is lower
or higher than the inflation target and provide an intuition for the result.
(g) In equilibrium, derive the level of inflation target that enables the discretionary
regime to deliver the same equilibrium of the commitment regime under exogenous
effort you derived in point (a). Provide an economic intuition of your answer.
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10. A continuum of identical households of measure one value consumption ct and housing
services, proportional to the housing stock ht, according to
" ∞
#
X
Vt = Et
βs (ln ct+s + ψ ln ht+s) ,
s=0
where β ∈ (0, 1) and ψ > 0. The budget constraint is
ct + qtht − bt = qtht−1 − (1 + rt−1)bt−1 + yt,
where qt is the price of housing, bt is debt, which carries a real interest rate rt, and yt
is the income endowment, which follows a stationary AR(1) process
yt = ρyt−1 + εt,
with ρ ∈ (0, 1) and εt ∼ i.i.d. N (0, 1).
(a) Derive the first order conditions for the household.
(b) Assume housing (land) is in fixed supply (i.e. ht = h, ∀t) and the economy is
closed (i.e. ct = yt, ∀t). Solve for house prices as a function of the endowment yt.
(c) Suggest a way to calibrate this model.
(d) What is the response of house prices to a 1% innovation in the endowment process?
(e) Suppose now the economy is open but small relative to rest of the world, hence
taking the real interest rate as given. Further assume that now debt is subject to
a borrowing constraint of the form
bt ≤ θqtht,
where θ ∈ (0, 1). Explain the nature of the borrowing constraint and its rationale.
Derive the first order conditions for the household in this case.
(f) Consider an equilibrium in which the real interest rates is constant and the bor-
rowing constraint is binding. Which condition needs to be satisfied for this equi-
librium to exist?
(g) Suppose the above condition is satisfied. Further assume that the endowment
is constant. Solve for the steady state value of house prices and compare with
the case in which the borrowing constraint is not binding. What is the effect
(comparative static) of a one-off increase in θ? Explain intuitively.
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last page
5372
DEGREE OF MASTER OF PHILOSOPHY
Master of Philosophy in Economics (Year 1)
Master of Philosophy in Economics (Final Examination)
Probationer Research Student Qualifying Examination in Economics
Probationer Research Student Examination in Management Studies
MICROECONOMICS
TRINITY TERM 2014
Wednesday 18th June 2014, 09:30 - 12:30
Please start the answer to each question on a separate page.
Answer four short questions from Part A and two long questions from Part B.
Part A attracts 1/2 of the marks in total; part B also attracts 1/2 of the marks.
Candidates may use their own calculators.
Do not turn over until told that you may do so.
1
Part A
1. Prove Roy’s Identity
∂v(p, w)/∂p
x
i
i(p, w) = −
,
∂v(p, w)/∂w
where v is the consumer’s indirect utility function and xi is her demand for good i
when she has wealth w and the vector of prices is p.
Deduce that if v(p, w) is linear in wealth – in the sense that v(p, w) = wV (p) for some
function V (p) – then each demand function is also linear in wealth.
2. Consider the two-player extensive-form game of imperfect information represented in
the following game tree:
X t
A
t
@
(4,8)
L
R
@
@
t
@
@t
Y
A
A
`
r
`
r
A
A
A
A
t
A
A
t
A
t
t
A
(2,3)
(5,4)
(3,5)
(3,7)
The dotted line drawn after the histories L and R indicates an information set.
The payoffs (4, 8), for example, indicate 4 for player X and 8 for player Y.
(a) Write down the payoff matrix of the corresponding normal-form (strategic-form)
game.
(b) Show that R is strictly dominated.
(c) Show that ` is weakly dominated.
(d) What are the pure-strategy Nash equilibria?
(e) Show that there are mixed-strategy Nash equilibria in which player X plays A,
and player Y plays ` with probability p, where p ≥ 1/3.
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3. We denote an agent’s Marshallian demand at price vector p = (p1, p2, . . . , p`) and
wealth w > 0 by ¯
x(p, w) ∈ R` .
++
(a) What does it mean for ¯
x to obey the Law of Demand?
(b) The Cobb-Douglas utility function
`
X
U (x1, x2, . . . , x`) =
αi ln xi ,
i=1
where P`
α
i=1
i = 1, generates the Marshallian demand function
α
¯
x
iw
i(p, w) =
for i = 1, 2, . . . , `.
pi
Show that ¯
x(p, w) obeys the Law of Demand.
(c) State the Milleron-Mitjuschin-Polterovich conditions on a utility function that
guarantee that its Marshallian demand functions obey the Law of Demand.
Show that these conditions hold for the Cobb-Douglas utility function defined in
part (b).
(d) Discuss briefly the use of the Law of Demand property in general equilibrium
theory.
4. (a) Explain briefly when Consumer Surplus is an appropriate measure of welfare.
(b) A government wishes to raise revenue R by taxing two goods. Good i is supplied
in a competitive industry at constant marginal cost ci, i = 1, 2. If the tax rate is
ti, the final price to consumers is pi = ci + ti and demand is qi = ai − bipi, where
ai and bi are constants, i = 1, 2. The government wishes to minimise the total
deadweight loss 1 b
+ 1 b
.
2 1t2
1
2 2t2
2
Show that the optimal tax rates satisfy
ti
a
= µ i − bipi ,
i = 1, 2
pi
bipi
for some µ.
(c) Interpret the formula in the previous part.
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5. A manager and investors play the following game. In period 0, a manager publicly
chooses how to allocate a fixed stock K of capital between a short-term and a long-
term project. If k1 is allocated to the short-term project, its return is S(k1), and if
K − k1 is allocated to the long-term project, its return is θL(K − k1). The functions
S(·) and L(·) are strictly increasing and concave, and θ, which is equally likely be θ
or θ (where θ > θ), is privately observed by the manager before the start of the game.
In period 1, investors update their beliefs about the value of θ, given the manager’s
choice of k1, and the short-term project returns S(k1). In period 2, the long-term
project returns θL(K − k1).
The manager’s payoff is V1 + V2, where V1 equals investors’ posterior expectation
(computed in period 1) of the total return on the two projects, and V2 equals the
actual total return on the two projects (as observed in period 2).
(a) Is this a screening game or a signalling game? Explain. In words, what conditions
must a perfect Bayesian equilibrium (PBE) of this game satisfy?
(b) Assume that in the least-cost separating PBE of this game, the manager distorts
his choice of k1 (compared to what he would do if θ were publicly observed), in
order to influence investors’ beliefs about θ. Which type of manager (i.e. type-θ
or type-θ) will distort k1, and in which direction will he distort it? Explain your
reasoning.
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Part B
6. Suppose a consumer is choosing how much to buy of two goods, and her preferences are
represented by the utility function u(x1, x2) = 2(x1)1/2 + x2, where her consumption
of good i = 1, 2 is xi. The consumer must choose non-negative quantities for both
goods, so that x1 ≥ 0 and x2 ≥ 0. Her wealth is w > 0 and the prices for the two
goods are p1 > 0 and p2 > 0.
(a) Show that the consumer chooses to buy strictly positive quantities of both goods
if and only if
p2
w > 2 .
p1
(b) Derive the consumer’s indirect utility function (i.e. the maximum utility she ob-
tains with wealth w and prices (p1, p2)).
(c) Suppose initially the consumer has wealth w = 12 and the two prices are p1 = 1
and p2 = 3. Suppose that the first price increases to p1 = 3. By how much does
the consumer’s wealth w have to increase to leave her as well off as she was in the
initial situation?
(d) Suppose now that the consumer has wealth w = 12 and faces the certain price
p2 = 3 and the uncertain price p1, where p1 = 1 or 3 with equal probability.
(The consumer chooses her quantities after she discovers the realization of p1.)
Assuming the consumer cares about her expected utility, how much of her wealth,
if any, would she give up to eliminate all risk in p1 (i.e. to face instead the certain
price p1 = 2)?
(e) Discuss your answer to part (d). How is the answer affected if the consumer had
to choose x1 (but not x2) before she learns the realization of p1?
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5
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7. Consider the following first-price sealed-bid auction. Two players i = 1, 2 simultane-
ously make bids for a single good which player i values at vi. Each player’s valuation vi
is drawn independently from the uniform distribution on [0, 1], so that Pr(vi ≤ v) = v
for v ∈ [0, 1].
The players have preferences represented by the following vN-M utility functions:
u1(x) = xα1 and u2(x) = xα2 where x ≥ 0 and 0 < α1, α2 ≤ 1. The player that makes
the higher bid receives the good and pays the price bid, whereas the losing player
receives nothing: so if bi was bid and was higher than bj, then x = vi − bi for player i
and x = 0 for player j.
(a) Define what is meant by a bidding strategy for each player.
(b) Write down the expected utility for player i with valuation vi as a function of the
strategies of both players.
(c) Suppose that player i bids an amount that is proportional to his valuation vi.
What is the other player’s best response strategy?
(d) Show that there is a Bayesian-Nash equilibrium in which each player’s bid is given
v
by b
i
i(vi) =
.
1 + αi
(e) Does this equilibrium yield efficient outcomes?
(f) What do players need to know in order to justify the solution of part (d)?
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8. A two-date financial economy has two assets and, at date 1, three states of the world.
Asset 1 has payoff vector (d11, d12, d13) (so it pays out d1s in state s) and Asset 2 has
payoff vector (d21, d22, d23). The prices of the two assets are q1 and q2 respectively.
An agent in this economy has an endowment (ω0, ω1, ω2, ω3), where ω0 ∈ R+ is his
endowment at date 0 and ωs ∈ R+, for s = 1, 2, 3, is his endowment in state s at
date 1. Suppose this agent’s utility function is
U (x0, x1, x2, x3) = v(x0) + π1v(x1) + π2v(x2) + π3v(x3)
where x0 is his consumption at date 0, (x1, x2, x3) is his state contingent consumption
vector at date 1, πs > 0 for s = 1, 2, 3, and v : R+ → R is a differentiable, strictly
increasing, and concave function.
(a) Let ¯
zi denote the agent’s demand for Asset i (i = 1, 2) at the price vector (q1, q2).
Show that ¯
zi satisfies
v0(¯
x0) q1 = π1v0(¯
x1) d11 + π2v0(¯
x2) d12 + π3v0(¯
x3) d13
and v0(¯
x0) q2 = π1v0(¯
x1) d21 + π2v0(¯
x2) d22 + π3v0(¯
x3) d23
where ¯
x0 = ω0 − q1 ¯
z1 − q2 ¯
z2 and ¯
xs = ωs + d1s ¯
z1 + d2s ¯
z2, for s = 1, 2, 3.
Interpret this result and comment on its generality.
(b) Suppose all agents in the financial economy have the same utility function
U (x0, x1, x2, x3) = ln(x0) + π1 ln(x1) + π2 ln(x2) + π3 ln(x3).
Let A be the set of agents in the economy and suppose that, for every agent a,
ωa− = λa(d
0
11, d12, d13) + µa(d21, d22, d23), i.e. agent a’s endowment at date 1 can be
thought of as consisting of λa units of Asset 1 and µa units of Asset 2. Show that
q1
π
π
π
=
1d11
+
2d12
+
3d13
W a − q1 ˜
za − q
d
+ d
d
+ d
d
+ d
1
2 ˜
za2
11 ˜
za1
21 ˜
za2
12 ˜
za1
22 ˜
za2
13 ˜
za1
23 ˜
za2
q
π
π
π
and
2
=
1d21
+
2d22
+
3d23
W a − q1 ˜
za − q
d
+ d
d
+ d
d
+ d
1
2 ˜
za2
11 ˜
za1
21 ˜
za2
12 ˜
za1
22 ˜
za2
13 ˜
za1
23 ˜
za2
where W a = ωa + q
= λa + ¯
za and ˜
za = µa + ¯
za.
0
1λa + q2µa, ˜
za1
1
2
2
Suppose that for some agent a, (˜
za, ˜
za) = (α, β). Show that (˜
za0, ˜
za0) = k(α, β)
1
2
1
2
for another agent a0, with W a0 = kW a.
(c) Suppose P
λa = λ∗, P
µa = µ∗, and (q
a∈A
a∈A
1, q2) are the equilibrium asset
prices. Show that for agent ¯
a at equilibrium
W ¯a
(˜
z¯a, ˜
z¯a) =
(λ∗, µ∗) .
1
2
P
W a
a∈A
Is the equilibrium allocation of contingent consumption Pareto optimal? Discuss
the generality of your finding.
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9. A principal P and an agent A play the following game. First, A privately chooses an
amount a ≥ 0 of lobbying effort, which is costly both for A and, ultimately, for P as
well. Second, a signal s = a + θ + u is realized and observed by both parties. Third,
P chooses a decision x ∈ (−∞, ∞). Finally, P receives the payoff −(x − θ)2 − ka,
where k > 0 is a constant, and A receives the payoff x − C(a), where C(a) is strictly
increasing and strictly convex. Both P and A maximize their expected payoffs, given
their beliefs about the other’s strategy, and they have common prior beliefs about the
random variables θ and u: θ and u are independent and normally distributed, with
mean 0 and variances Var(θ) and Var(u), respectively.
(a) Suppose P conjectures that A chooses lobbying effort ˆ
a.
Conditional on the
observed value of s, what is P ’s posterior expectation of θ? What is P ’s optimal
choice of x, given his conjecture ˆ
a and the observed s?
(b) What conditions determine the equilibrium value of lobbying effort a? Explain
how this value depends on Var(θ) and Var(u).
In the game analysed above, P cannot commit in advance to how he will respond to
the signal s generated by A. Now suppose that P , before A chooses a, can commit to
using any decision rule of the form x = d + bs, where d and b ≥ 0 are constants.
(c) Given a choice of decision rule by P , what condition determines A’s optimal choice
of a?
(d) Explain the trade-off faced by P in choosing the value of b in his decision rule.
Comparing P ’s optimal decision rule with commitment to P ’s decision rule in
the equilibrium (without commitment) analysed in parts (a) and (b), in which
case will P ’s decision x be more sensitive to the signal s? How would you expect
the value of b in P ’s optimal decision rule to vary with Var(θ), Var(u), and k?
Explain your answers.
10. Are Pigouvian taxes the best way to deal with externalities?
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last page
5376
DEGREE OF MASTER OF PHILOSOPHY
M.Phil. in Economics (Final Examination)
Probationer Research Student Qualifying Examination in Economics
PUBLIC ECONOMICS
TRINITY TERM 2014
Tuesday 17 June 2014, 09:30 - 11:30
Please start the answer to each question on a separate page.
Answer two questions.
Candidates may use their own calculators.
Do not turn over until told that you may do so.
1
1. Discussing both intensive and extensive margins, explain the considerations which
determine the optimal marginal rates of income tax at both ends of the income distri-
bution.
2. Discuss the pros and cons of imposing a system of community rating in an insurance
market subject to adverse selection. Can these pros and cons be evaluated empirically?
3.
a) Explain the conceptual differences between the Marshall, Hicks and Frisch labour
supply elasticities. Which one is considered most appropriate for predicting the
effect of a permanent tax increase where the revenue is redistributed lump sum?
Why?
b) Discuss why labour supply elasticities with respect to after-tax wages are difficult
to estimate empirically. Explain some of the important econometric problems that
arise. Why can’t you just look at actual tax changes and “see what happened”?
c) Assume that workers have the following utility function:
C1+η
h1+γ
U
it
it
it =
− βit
1 + η
1 + γ
where η ≤ 0, γ ≥ 0. Here Cit is consumption of person i at time t, hit is hours
of work, and βit shifts tastes for work. For the case where the wage rate wit is
exogenous, write out the simple MRS condition that states the marginal rate of
substitution between consumption and leisure equals the wage rate.
d) Now consider a learning-by-doing model where hours of work today affect the wage
rate tomorrow, making the wage endogenous. Describe how the MRS condition
changes in this case. (Note: This can be an intuitive explanation rather than a
formal derivation.) Discuss why failure to account for learning-by-doing would
lead to downward bias in estimates of labour supply elasticities.
4. Explain, using theoretical and empirical arguments, how the access to health insurance
influences demand and supply of medical care.
5. Describe and evaluate the Easterlin paradox.
5376
2
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5430
DEGREE OF MASTER OF PHILOSOPHY
Master of Philosophy in Economics (Final Examination)
Probationer Research Student Qualifying Examination in Economics
ADVANCED MACROECONOMICS 1
TRINITY TERM 2014
Wednesday 18th June 2014, 09:30 - 11:30
Please start the answer to each question on a separate page.
Answer TWO questions, ONE from Part A and ONE from Part B.
Candidates may use their own calculators.
Do not turn over until told that you may do so.
1
PART A
1. Answer all four parts below, using models to illustrate your arguments where appropriate.
(i) Explain how agents will have different consumption levels if a social planner assigns
them different Pareto weights in a model with idiosyncratic productivity shocks.
Does the heterogeneity in consumption across agents rule out a representative agent
formulation of the model?
(ii) Discuss whether increasing labor income taxes always leads to a fall in labor supply
when labor supply is endogenous.
(iii) What are the two main problems faced when writing the recursive competitive equi-
librium in the stochastic growth model? Briefly discuss how the problems may be
addressed.
(iv) Explain the behaviour of aggregate demand in a model with uninsurable income
shocks and incomplete markets. Pay particular attention to the condition necessary
for asset holdings to be bounded.
2
5430
2. In a version of the stochastic growth model, the preferences of the representative agent are
defined by u(ct) = log(ct). As utility only depends on consumption ct, labour is supplied
inelastically at its maximum value of 1. The representative agent’s discount rate is β.
Output is produced according to AtKαN 1−α
t
t
, where Kt is the capital stock accumulated
from last period, At is a productivity shock, Nt is total labor employed in production and
0 < α < 1. The key difference compared to the standard stochastic growth model is that
it requires f (x) units of resources to transform x units of the consumption good into x
units of the capital good. The law of motion for capital is therefore given by the usual
equation
Kt+1 = (1 − δ)Kt + xt,
but the resource constraint is
ct + xt + f (xt) = AtKαN 1−α.
t
t
At follows a stationary Markov chain, i.e., At ∈ ( ¯
A1, . . . , ¯
AN ) with ¯
Ai+1 > ¯
Ai. The
transition matrix is Π with Πij = Pr(At+1 = ¯
Aj|At = ¯
Ai). Assume that persistence
PN Π ¯A
j=1
ij
j is increasing in i.
(i) Explain why f (x) is often referred to as a capital adjustment cost. Discuss whether
these costs are likely to be monotone increasing, decreasing or neither.
(ii) Set up the recursive stochastic dynamic programming problem of a benevolent social
planner in this environment, being specific about exogenous and endogenous states
and decision variables. Derive the Euler equation for capital and provide economic
intuition. It can be assumed that f (·) is differentiable.
(iii) Assume now that the adjustment cost term takes the following functional form:
γx if x ≥ 0
f (x) =
0
if x < 0
Interpret these costs and obtain the equation determining the non-stochastic steady
state of the economy. How does the steady state level of capital depend on γ?
(iv) Use the Euler equation to explain how investment and consumption respond to a
positive and negative technology shock. Compare your results to the standard case
with no adjustment costs. For all these comparisons, consider two cases:
a) xt > 0 for all levels of the capital stock and technology in the stationary distri-
bution of the economy.
b) xt > 0 after a high productivity shock and xt < 0 after a low productivity shock.
(v) Assume a recursive competitive equilibrium for this economy and define the problem
of the financial intermediary in competitive equilibrium. Use the fact that
c(A, K)
Q(A′|X) = βΠ(A′|A) c(A′, K′)
to show that the equilibrium will not result in an efficient allocation of capital. What
is the source of inefficiency?
3
5430
TURN OVER
PART B
3. Consider a central bank attempting to control the inflation rate πt by setting the interest
rate it. The central bank faces uncertainty on the exact relationship between πt and it,
but knows it has the following structure:
πt = αt + βtit,
(1)
where αt and βt are unobserved time-varying parameters. Suppose initially that the
parameters evolve according to:
αt = α + εt, εt ∼ i.i.d.N (0, σ2),
(2)
ε
βt = βt−1 + ηt, ηt ∼ i.i.d.N (0, σ2),
(3)
η
with σ2, σ2 and α known.
ε
η
The central bank forms estimates of the unknown persistent process βt by employing the
Kalman filter. Defining Et−1 {βt} ≡ bt|t−1 and vart|t−1 (βt) ≡ Σb
, this implies that:
t|t−1
Σb
i
t|t−1 t
bt|t = bt|t−1 +
πt − α − b
i2
t|t−1it
(4)
t Σb
+ σ2
t|t−1
ε
2
Σb
i2
t|t−1
t
Σb = Σb
−
(5)
t|t
t|t−1
i2tΣb
+ σ2
t|t−1
ε
bt+1|t = bt|t
(6)
Σb
= Σb + σ2
(7)
t+1|t
t|t
η
The central bank’s objective is to keep inflation πt close to its target level π∗, with the
central bank incurring a quadratic loss for any deviations from the target. It furthermore
discounts the future with discount factor δ. Consequently, the objective is to:
(
)
∞
X
min E0
δt (πt − π∗)2
(8)
{it}∞
t=0
t=0
(i) Derive the certainty-equivalent policy rule (iCE).
t
(ii) Derive the policy rule (iPL) if the central bank learns in a passive manner. How does
t
iP L rank relative to iCE?
t
t
(iii) Unfortunately, it is not possible to derive an analytical expression for the active
learning policy rule (iAL). Use intuitive arguments to nevertheless comment on the
t
ranking of iAL relative to iPL. Refer to equation (5) in your answer and explain the
t
t
underlying intuition. Why does (5) only feature i2 while i
t
t itself is absent?
4
5430
Suppose that the stochastic process for βt is replaced by:
βt = ρβt−1 + ηt, ηt ∼ N (0, σ2),
(3’)
η
where ρ is a parameter satisfying 0 < ρ < 1 that is known to the central bank. Call the
resulting policy rule iALρ
t
.
(iv) Argue intuitively how iALρ
t
would rank relative to iAL.
t
Suppose that equations (2) and (3) are now replaced by:
αt = αt−1 + εt, εt ∼ N (0, σ2),
(2”)
ε
βt = β + ηt, ηt ∼ N (0, σ2),
(3”)
η
with β known. In this case the central bank forms an estimate of the persistent component
αt by running the following Kalman filter:
Σat|t−1
at|t = at|t−1 +
π
Σa
+ σ2i2
t − at|t−1 − βit
(4”)
t|t−1
η t
2
Σat|t−1
Σa = Σa
−
(5”)
t|t
t|t−1
Σa
+ σ2i2
t|t−1
η t
at+1|t = at|t
(6”)
Σa
= Σa + σ2
(7”)
t+1|t
t|t
ε
Call the resulting optimal policy rule under active learning iALα.
t
(v) How would iALα rank relative to iAL and why?
t
t
5
5430
TURN OVER
4. Consider the model by Mackowiak and Wiederholt (2009), “Optimal Sticky Prices Un-
der Rational Inattention”, American Economic Review, in which firms sell one particular
product in an environment that is subject to both aggregate and sector-specific idiosyn-
cratic shocks. Each firm has an information processing capacity of κ bits.
(i) Prices in this model are fully flexible. How is it possible that this model still generates
monetary non-neutrality?
(ii) Consider modifying firms in the model so that, instead of selling only one particu-
lar product, they become multi-product firms operating in many different sectors.
Assuming each firm continues to have the same information processing capacity κ,
argue intuitively how this would affect the firm’s optimal attention allocation. What
would it imply for the degree of monetary non-neutrality in the economy?
Consider the rational inattention model of Paciello and Wiederholt (2012), “Exogenous
Information, Endogenous Information, and Optimal Monetary Policy” forthcoming in
Review of Economic Studies.
(iii) What does the model imply for the optimal communication strategy of central banks?
A rationally inattentive agent lives for two periods and has expected lifetime utility:
E − (x1 − y1)2 + βE − (x2 − y2)2 ,
where β is the discount factor and (y1, y2) are the choice variables for periods 1 and 2
respectively. The agent wants to match states x1 and x2, which are independent random
variables distributed according to:
x1 ∼ i.i.d.N (µx , σ2 )
1
x1
x2 ∼ i.i.d.N (µx , σ2 )
2
x2
The agent’s ability to learn is constrained by κ, which represents a lifetime information
processing capacity. The agent can learn by processing information from noisy signals of
the states. These signals are of the form:
s1 = x1 + ε1, ε1 ∼ i.i.d.N (0, σ2 )
ε1
s2 = x2 + ε2, ε2 ∼ i.i.d.N (0, σ2 )
ε2
(iv) What would be the agent’s optimal action (y1, y2) under perfect information? What
would be their optimal action under imperfect information, given noisy signals
(s1, s2)?
(v) Solve for the agent’s attention allocation problem, i.e., solve for the amounts of
attention (κ1, κ2) allocated to periods 1 and 2. Provide intuition for the expression
for κ1. You may neglect the possibility of corner solutions.
6
5430
LAST PAGE
5431
DEGREE OF MASTER OF PHILOSOPHY
Master of Philosophy in Economics (Final Examination)
Probationer Research Student Qualifying Examination in Economics
ADVANCED MACROECONOMICS 2
TRINITY TERM 2014
Saturday 21st June 2014, 09:30 - 11:30
Please start the answer to each question on a separate page.
Answer TWO questions, ONE from Part A and ONE from Part B.
Candidates may use their own calculators.
Do not turn over until told that you may do so.
1
PART A
1. Suppose that it is not known whether the operating procedure for monetary policy is the
overnight inter-bank interest rate or some measure of bank reserves. A researcher wishes
to identify structural monetary policy shocks under the assumption of a recursive vector
autoregression (VAR) in output, the price level and the overnight interest rate.
(i) Explain why in this case the orthogonalized shocks to the overnight interest rate
may not capture the structural shocks to monetary policy.
(ii) Describe the method introduced by Bernanke and Mihov (1998) to test for the
operating procedure of monetary policy and then identify the effects of structural
monetary policy shocks on output and the price level.
(iii) Why do Christiano, Eichenbaum and Evans (1999) reject the use of statistical tests
to address the problem of uncertainty over the operating procedure of monetary
policy and how do they suggest researchers should instead proceed?
2. Consider the identification of structural monetary policy shocks using Factor Augmented
Vectorautoregressions (FAVARs) in Bernanke, Boivin and Eliasz (2005).
(i) Describe the implementation of this procedure by the named researchers. In your
answer refer to (a) the determination of the number of factors to be included in the
model to be used for monetary policy identification and (b) the steps taken to ensure
a recursive ordering between the factors and the measure of monetary policy.
(ii) Critically evaluate the use of FAVARs as a means to overcome the ‘curse of dimen-
sionality’ that arises when a central bank sets monetary policy based on a large
number of leading indicators of future economic conditions.
PART B
3. “The volatility of an economy can increase if borrowers are leveraged”. Set out a model
showing why this can be true, showing clearly why and how leverage can increase the risk
premium. Does it matter to this finding if the borrowers are financial institutions — who
borrow in order to lend - rather than the producers of final goods?
4. Explain how and why the exchange rate can overshoot its final outcome in response to
a shock, in a macroeconomic model of your choosing. Describe another model in which
the exchange rate can undershoot its final outcome in response to a shock. Explain the
reason for the difference between the findings in the two cases.
5431
2
LAST PAGE
5432
DEGREE OF MASTER OF PHILOSOPHY
Master in Philosphy in Economics (Final Examination)
Probationer Research Student Qualifying Examination in Economics
Probationer Research Student Examination in Management Studies
ADVANCED ECONOMETRICS 1
TRINITY TERM 2014
Monday, 16 June 2014, 14.30–16.30
Please start the answer to each question on a separate sheet.
Candidates should answer TWO questions.
Candidates may use their own calculators.
Do not turn over until told that you may do so.
1
1. Consider n IID pairs (Yi; Xi) of random variables. Suppose
Yi = m (Xi) + "i;
(1)
where "i is independent of Xi with mean zero and variance 2. Assume that m ( ) is twice
continuously di¤erentiable with …rst and second derivatives denoted by m0 ( ) and m00 ( ),
respectively.
The Nadaraya-Watson estimator b
mNW (x) of m (x) and the local linear estimator ^
mLL (x)
at a speci…c value x of Xi have bias given respectively by
1
f 0 (x)
E b
mNW (x)
m (x) =
;
(2)
2 2h2
m00 (x) + 2m0 (x) f (x)
1
E b
mLL (x)
m (x) =
2 2h2m00 (x) ,
(3)
where f ( ) denotes the density of Xi, h is the bandwidth and 2 is a positive constant
which depends on the kernel used. Smaller order terms in the bias expressions are ignored
for this question.
(a) Consider the bias of b
mNW (x) in the following situations
(i) Suppose m (x) =
, i.e., a constant, for all values of x. What is the bias of
0
b
mNW (x)?
(ii) Now, suppose m (x) =
+
x for all values of x. What is the bias of
0
1
b
mNW (x)?
(b) Now answer cases (i), (ii) in part (a) for b
mLL (x).
(c) What is the intuition behind these results?
(d) The unconditional median of Yi can be estimated by minimizing w.r.t.
the term
P
n 1
n
i=1 jYi
j. The minimizing value is then an estimate of the median. Now
suppose we are interested in estimating the conditional median of Yi given Xi = x.
Can you suggest a kernel based procedure for estimating this quantity? Provide an
intuitive justi…cation of your suggested procedure.
(e) Suppose we are interested in the mean of Yi for those whose Xi is at most x (rather
than equal to x). For example, suppose Yi is individual income in £ and Xi is
height in inches and we are interested in average income of those who are at most 66
inches tall. How would you estimate this mean based on a random sample? What
are the key di¤erences in the asymptotic properties of this estimator relative to the
Nadaraya-Watson estimator of mean income for those whose height is equal to 66
inches?
5432
2
2. Consider the following data generation process for (yit; xit; wit)
yit =
xit +
+ v
i
it;
(1)
xit = wit +
+
v
v
i
1 i;t 1 +
2 i;t 2 + eit;
(2)
wit =
wi;t 1 + "it;
(3)
in which 0 <
< 1 and the process for wit is covariance stationary. The time-varying
unobserved components vit; ejs and "kr are independent of each other for any combina-
tion of (i; j; k) and (t; s; r); each of these random variables has expectation zero, the
same variance for all cross-section observations (i = 1; : : : ; n) and for all time periods
(t = 1; : : : ; T ), and is both cross-sectionally and serially independent. The time-invariant
unobserved component
has expectation zero, the same variance for all cross-section
i
observations (i), is cross-sectionally independent, and independent of vjt; eks and "lr for
any combination of i; (j; k; l) and (t; s; r).
(a) Derive the following expected values
(i) E(xitvis) for s = t + 1; t; t
1; t
2 and t
3:
(ii) E(yi;t 1vit):
(iii) E(xit ), E(x
) and E( x
):
i
i;t 1 i
it i
(iv) E( yit ):
i
(b) Data on (yit; xit) is available for i = 1; 2; :::; N , where N is large, and for t = 1; 2; 3:
Write down the complete set of linear moment conditions which are available to
estimate the parameter
in equation (1), using (at most) two equations in …rst-
di¤erences and (at most) one equation in levels.
(c) Now suppose it is known that
= 0: Adapt your answer to part (b) to include any
additional moment conditions which are available in this special case. Note which, if
any, of the moment conditions listed in your answer to part (b) are now redundant.
(d) Using your answers to parts (b) and (c), brie‡y explain how the null hypothesis that
= 0 could be tested against the alternative hypothesis that
6= 0:
(e) Now suppose that data on wit is available for the same three time periods. Discuss
how this additional data could be used to obtain a more e¢ cient estimator of the
parameter
in equation (1).
5432
3
Turn Over
3. (a) Consider random ‡ow variables Ti; Xi for i = 1; : : : ; n, where Ti is the observed
minimum of unemployment duration T and a right-censoring time C
i
i, while Xi is a
vector of observed covariates. An indicator variable i has value one if T < C
i
i and
zero otherwise. Suppose the observations fTi; i; Xig are i.i.d. with hazard:
P[t
T
(t) = lim
i < t +
tjTi
t] =
(1)
t!0
t
and
> 0.
(i) Derive the cumulative hazard function
(t), the survivor function S(t), and the
density function f (t) when (t) = .
Now, suppose the hazard is
P[t
Ti < t +
tjTi
t; Xi = xi]
i(t) = lim
= exp(x0
);
(2)
i
0
t!0
t
with Xi and
denoting vectors of dimension (k
1).
0
(ii) What is the shape of the hazard function? What is the e¤ect of covariates on
the hazard function? What is the contribution of an (un-)censored duration to
the log-likelihood function? Derive the log-likelihood function `( ) = log L( ).
Derive the …rst order condition for
(no need to solve!).
(iii) Consider the case without censoring (set i = 1 for all i 2 f1; : : : ; ng). Un-
der some regularity conditions, a fundamental characteristic of the likelihood
function is E[@`( )=@ ] = 0. What does this imply for
0
E[TijXi = xi]?
(iv) Rather than observing random censoring times, you are given the information
that durations with i = 0 fail within the interval [tl; tu], where 0 < tl < tu < 1,
i.e. you do not know the exact failure time for these observations but only that
failure occurred within the …xed interval [tl; tu] (interval censoring). How would
you change the log-likelihood in (ii) to accommodate this information?
(b) Consider i.i.d. observations Ti; Xi for i = 1; : : : ; n where the exit times are discrete
and take values Ti = 1; 2; : : : ; K: There is no censoring. The underlying hazard
model is still speci…ed in continuous time as:
P[t
Ti < t +
tjTi
t; Xi = xi]
i(t) = lim
= 0(t) exp( x0
);
(3)
i
0
t!0
t
(i) Show the steps to get from Equation (3) to:
(t) = x0
+ "
0
i
0
i;
(4)
R
R
where
t
t
"i
log
(t)
log
0
i(s)ds
and 0
0
0(s)ds .
(ii) Brie‡y describe the procedure proposed by Han and Hausman (1990) to estimate
the model in (3). Which (transformed) parts of the model in (3) are estimated
alongside the parameter vector
? Under which speci…cation is this procedure
0
only an approximation of the ‘true’underlying model?
(c) Consider i.i.d. observations Ti; Xi for i = 1; : : : ; n where the exit times are continu-
ous. There is no censoring. The hazard is
i(t) =
0(t) exp(X 0
)V
i
0
i, where Vi is a
multiplicative unobserved heterogeneity term that satis…es certain regularity condi-
tions. Brie‡y explain the problem that arises when introducing Vi into a single-spell
duration model? (no formal derivations required!).
5432
4
Last Page
5433
DEGREE OF MASTER OF PHILOSOPHY
Master in Philosophy in Economics (Final Examination)
Probationer Research Student Examination in Management Studies
ADVANCED ECONOMETRICS 2
TRINITY TERM 2014
Thursday 19 June 2014, 14.30–16.30
Please start the answer to each question on a separate sheet.
Candidates should answer TWO questions.
Candidates may use their own calculators
Do not turn over until told that you may do so.
1
1. Consider the following two stochastic processes
yt = y + yt 1 + uyt
and
xt = x + xt 1 + uxt;
where
y and
x are constant parameters in R, y0 = x0 = 0 and uyt
i:i:d:N (0; 2)
is independent of uxt
i:i:d:N (0; !2). Let Wy (r) be a Brownian motion de…ned on
r 2 [0; 1].
(a) Let
y = 0 and
x 6= 0 and suppose that the following least squares regression is
carried out
XT
XT
1
yt = ^xt + ^
et
where
^ =
ytxt
x2t
t=1
t=1
is the ordinary least squares estimator. Stating all the relevant results being used,
show that:P
d
R
(i) T 5=2
T
y
1 rW
t=1
txt !
x
0
y (r) dr;
P
d
(ii) T 3
T
x2
=3;
t=1
t !
2
x
R
(iii) T 1=2^ d! 3
1
1 rW
x
0
y (r) dr:
(iv) Is ^ consistent?
(b) Under the conditions of part (a), consider the statistic
^
1 XT
2
t
^
=0 =
h
where
s2 =
y
x
;
X
i
T
t
t
T
1
1=2
T
t=1
s2
x2
T
t
t=1
for testing the hypothesis
= 0. Stating all the relevant results being used, show
that
p R 1
1
d
3
rWy (r) dr
t
!
0
:
T 1=2 =0
R
2
1=2
1
R
W 2 (r) dr
3
1 rW
0
y
0
y (r) dr
Is it possible to perform standard inference?
(c) Let
x = 0 and
y 6= 0. Derive the asymptotic distribution of ^ and t =0 stating all
the relevant results being used. Is ^ consistent? Is it possible to perform standard
inference?
5433
2
2. An investigator wishes to estimate, test her theory, then forecast from the model:
yt =
+ zt 1 + ut;
(1)
where it is assumed that ut
IN [0; 2 ], denoting an independent normal distribution
u
with mean zero and variance 2 , and is distributed independently of
u
fzt 1g, with 6= 0.
The data generation process (DGP) for yt over t = 1; : : : ; T is in fact:
k
X
yt = +
S
j
ft tjg +
zt 1 + t;
(2)
j=1
where t
IN [0; 2] is independent of fzt 1g, Sft tjg is a step indicator that is unity till
time tj then zero thereafter, but the number of location shifts k
1 is not known to the
investigator. As she worries about the impact of unanticipated location shifts (where the
previous unconditional means of variables change unexpectedly) on empirical modelling
and forecasting, the investigator wishes to check for shifts in (1) but believes that k = 0.
(a) Carefully explain how to investigate the occurrence of a single location shift, assum-
ing k = 1 in (2), by step indicator saturation, using a ‘split-half’approach where step
indicators for every observation are included in the candidate set with an intercept
and fzt 1g but each half is entered in turn, so the total number of variables included
each time is N = T =2 + 2.
How would you need to modify your proposal for k > 1 location shifts?
(b) When there are in fact no location shifts, what are the properties of your proposed
approach? Explain how to calculate its gauge.
(c) Discuss whether your approach is more or less potent in …nding a location shift
when k = 1 than using impulse-indicator saturation (IIS), where indicators for every
observation, 1ft=tjg, tj = 1; : : : ; T are added to the candidate set?
(d) A …nal step shift in fact occurs in (2) at time T
1, where T is the forecast origin.
Carefully explain what advice you would o¤er the investigator before she uses her
estimated version of the model in (1):
b
yT+1jT = b + bzT
(3)
to forecast yT+1.
5433
3
Turn Over
3. Consider the New Keynesian Phillips curve model
^t = Et (^t+1) + ^
xt + ut;
Et 1 (ut) = 0;
(1)
where ^t =
t
, ^
xt = xt
x;
t denotes in‡ation, xt denotes the output gap,
and x are parameters for the steady state, and ut is an unobserved disturbance. The
structural parameters are
= ( ; )0. Et ( ) denotes (rational) expectations conditional
on information at time t, which includes t, xt and their lags: The variables t and xt are
observed by the econometrician but Et ( t+1) is not observed.
(a) Let ht ( ; c) = t c
t+1
xt, where c = (1
)
x. Prove that E [yt iht ( ; c)] =
0 for any i
1 and yt = ( t; xt)0 ; and hence suggest a generalized instrumental vari-
ables (GIV) estimator of
using a constant, yt 1 and yt 2 as instruments.
(b) Prove that
k
X
^
j
t =
k+1Et (^t+k+1) +
Et (^
xt+j) + ut;
(2)
j=0
for any …xed k
0:
(c) Suppose ^
x
s
t =
^
xt 1+vt; where Et 1 (vt) = 0 and j
j < 1; and that lims!1
Et (^t+s) =
0: Show that the parameters
= ( ; )0 are not identi…ed if ^
xt follows that law of
motion.
(d) Explain the concept of weak identi…cation in the context of this model, and brie‡y
discuss its consequences for GIV estimators of the structural parameters .
(e) The following regressions have been estimated on a sample of quarterly data from
the USA.
M1 :
t+1 = a0 + a1
t 1 + a2
t 2 + a3xt 1 + a4xt 2 + error;
(3)
M2 :
t+1 = b0 + b1
t 1 + b2xt 1 + b3xt 2 + error;
(4)
and below are the p-values corresponding to F (homoskedastic) or Wald (HAC)
tests of the hypothesis that the coe¢ cients on the regressors in each model are
equal to zero (HAC refers to the Newey-West heteroskedasticity and autocorrelation
consistent variance estimator):
p value
null hypothesis
homoskedastic
HAC
a1 = a2 = a3 = a4 = 0
0.37
0.33
b1 = b2 = b3 = 0
0.40
0.31
Using these results, test the null hypothesis H0 :
= 1;
= 0, against the alternative
hypothesis H1 :
6= 1 and/or
6= 0, carefully justifying your choice of test statistic.
(f) Discuss alternative tests of the hypothesis in part (e), and explain how you can
obtain a con…dence set for
that is robust to weak identi…cation.
5433
4
4. Consider the following fractionally integrated model with k regressors:
(1
L)d(yt
x0 ) =
t
t;
t = 1; :::; T;
(1)
where t is identically and independently distributed as t
N [0; 2]. L is the lag opera-
tor, d a real value in ( 1; 0:5) and
is a column vector with k coe¢ cients.
Let y0 = (y1; :::; yT ) and X0 = (x1; :::; xT ), so X is the T
k regressor matrix and
zt = yt
x0 , so
t
z = y
X :
(2)
The log-likelihood of this fractionally integrated model is:
T
1
1
log L d; ; 2 =
log (2 )
log j j
z0
1z;
(3)
2
2
2
with z de…ned above and
the T
T matrix of autocovariances:
0
1
c (0)
c (1)
c (T
1)
B
.
.
C
B
c (1)
c (0)
. .
..
C
= B
C :
(4)
@
..
.
.
.
. .
. .
c (1)
A
c (T
1)
c (1)
c (0)
is symmetric positive de…nite, and assume it can be evaluated for any d 2 ( 1; 0:5).
(a) Switch to autocorrelations:
= 2R, and show how 2 can be concentrated out of
the log-likelihood. With 2 concentrated out, show how
can be concentrated out
next, deriving
b
1
(d) = X0R 1X
X0R 1y:
(5)
(b) Using the Choleski decomposition for R and the QR decomposition for X, show how
to compute:
b
1
= X0R 1X
X0R 1y:
(6)
(c) The …nal concentrated log-likelihood has only one parameter left. Describe the main
di¤erences between using the BFGS method and Newton’s method to maximize the
log-likelihood.
(d) Discuss the di¤erence between numerical di¤erentiation based on central di¤erences
in comparison to forward di¤erences.
5433
5
Last Page
5435
DEGREE OF MASTER OF PHILOSOPHY
Master of Philosophy in Economics (Final Examination)
Probationer Research Student Qualifying Examination in Economics
BEHAVIOURAL ECONOMICS
TRINITY TERM 2014
Friday 20th June 2014, 14:30 - 16:30
Please start the answer to each question on a separate page.
Answer TWO short questions from Part A, and ONE long question from Part B.
Part A attracts 1/2 of the marks in total; part B also attracts 1/2 of the marks.
Candidates may use their own calculators.
Do not turn over until told that you may do so.
1
PART A
1. Explain the crucial difference between the theories of fairness as in Rabin (1993) and those
such as Fehr and Schmidt (1999). Explain how simple dictator and entry games, as considered
by Charness and Rabin (2002), may be used to distinguish between the theories.
5435
2
2. “Our goal was to design a program to help those employees who would like to save more
[for retirement], but lack the willpower to act on this desire. [The Save More Tomorrow] plan
gives workers the option of committing themselves now to increase their savings rate later.
Once employees join, they stay in the plan until they opt out.” –Richard Thaler and Shlomo
Benartzi
Suppose a worker works for two periods and then retires for a third and final period. S/he has
no initial wealth, her/his income is y in each of the two periods in which s/he works, and
borrowing is impossible. There is a single consumption good, whose price is 1 in each period.
Consuming ct in period t yields her/him utility u(ct) at period t, where u(·) is strictly
increasing, strictly concave, and differentiable. The worker is a quasi-hyperbolic discounter,
with 0 <
β ≤ 1 and
δ = 1.
(a) Express the worker’s intertemporal preferences over consumption plans in terms of the
u(ct) and
β, from the point of view of period t for t = 1, 2, and 3. Say what it means for the
worker’s preferences to be time-consistent. For which values of
β are her/his preferences
time-consistent?
(b) What is the optimal plan for a worker with
β = 1? Express it first as a complete contingent
plan, writing the optimal c1, c2, and c3 as functions of wealth w2 = y - c1 and w3 = w2 + y - c2 at
the starts of periods 2 and 3 (recall that w1 = 0). Then express the optimal plan in terms of the
resulting realized values of c1, c2, and c3.
From now on assume
β < 1 is small enough and u(·) is close enough to linear that u
'(ca) >
βu
'(cb) for all feasible ca, cb. A sophisticated worker is one who can predict her/his own future
choices.
(c) What is the optimal plan for a sophisticated worker with
β < 1? Again, express it first as a
complete contingent plan, then in terms of the resulting realized values of c1, c2, and c3.
Assume that a worker who joins a Save More Tomorrow plan at the start of period t can
commit her/himself to save any desired feasible amount in period t+1, after which s/he is free
to opt out.
(d) Would a sophisticated worker with
β = 1 pay any positive amount, however small, to join
a Save More Tomorrow plan in period 1? Would a sophisticated worker with
β < 1? Explain
briefly.
A naïve worker is one who expects her/his own future choices will be optimal not for the true
β, but for
β = 1.
(e) Would a naïve worker with
β < 1 pay any positive amount, however small, to join a Save
More Tomorrow plan in period 1? Explain briefly.
5435
3
turn
over
3. “Son,” the old guy says, “No matter how far you travel, or how smart you get, always
remember this: Someday, somewhere, … a guy is going to come to you and show you a nice
brand-new deck of cards on which the seal is never broken, and this guy is going to offer to
bet you that the jack of spades will jump out of this deck and squirt cider in your ear. But,
son, … do not bet him, for as sure as you do you are going to get an ear full of cider.”
—Obadiah (“The Sky”) Masterson, in Damon Runyon’s story “The Idyll of Miss Sarah
Brown”
Model this situation as a betting game, in which each player (“Son” or “Guy”) can agree to
bet (Y) or not (N), a bet requires mutual agreement, not betting yields either player £0, and
the winner of an agreed bet wins £10, while the loser loses £10. Only Guy has private
information, whether or not he can make the jack of spades jump out and squirt cider in Son’s
ear, which he observes before deciding whether to agree to bet. “Son” assigns prior
probability p to the possibility that Guy can do this, where 0 < p < 1. Guy knows this p, as
common knowledge.
(a) Identify players’ possible pure strategies and the Bayesian Nash equilibrium or equilibria,
for any given p.
(b) It has been observed that many people deviate from equilibrium betting decisions in such
settings, which has been attributed to naivete of players’ inferences about how others will use
their private information. Precisely describe a non-equilibrium model of strategic behavior
that captures such informational naivete, and use it to deduce each player’s strategy, for any
given p. When will both agree to bet?
5435 4
PART B
4.
Data on consumers’ choices of deductibles for home insurance provide evidence that a
surprising level of risk aversion over modest stakes is a reality in the market. Most customers
purchase low deductibles despite costs significantly above the expected value. Fitting these
choices to a standard (expected utility) model yields implausibly large values of risk aversion
parameters. This question invites you to investigate variants of the prospect theory model to
check whether they provide plausible explanations of the phenomenon.
Imagine that the individual faces a choice between one-year insurance contracts, which differ
in their yearly premium and deductible . Assume that, other than a loss to the home, the
individual faces no risk to lifetime wealth . Furthermore, assume for simplicity that the
individual faces at most one loss during the year, which occurs with probability . Finally,
assume that all losses exceed the highest deductible under consideration. Hence, if the
individual takes insurance, her wealth in the loss state is
, and
otherwise.
(a) Consider a basic version of Kahneman and Tversky’s formulation of prospect theory,
with no reweighting of probabilities and linear value functions, which may differ
between the “loss” and “gain” domains. Explain why this theory cannot provide an
explanation for the phenomenon. Explain further why making allowance for
diminishing sensitivity would not change the conclusion.
(b) Next consider a more complete version of Kahneman and Tversky’s formulation that
allows for reweighting of probabilities. Specifically, consider the following value
function .
where denotes the final wealth and the reference level of wealth. And, consider the
following probability weighting functions,
, respectively, for gains
and losses occurring with probability :
1
1
Let
1.25,
0.61,
0.69. Show that a customer, facing a 4% probability of
loss, would be willing to pay only about $48 to lower the deductible from $1000 to
$500. (The data shows that new customers actually paid $95 to lower the deductible
by that amount.)
(c) Finally, consider the version of prospect theory with reference dependence as in the
Kőszegi-Rabin (2006, 2007) framework. Explain how this framework, applying linear
value functions and probability weighting as in part (b), would imply a willingness to
pay (by a customer facing a 4% probability of loss) just over $100 to lower the
deductible from $1000 to $500. (You may assume, as before,
1.25,
0.61,
0.69.)
5435
5
turn
over
5. Choose ONE of the two following quotations, and write an essay explaining (i) why the
issues it raises are not adequately addressed by a standard neoclassical model, and (ii) how
they might be better addressed using ideas from behavioural decision and/or game theory.
(a) From an interview with Eugene Fama by John Cassidy,
The New Yorker, 13 January 2012:
Fama: …I don’t know what a credit bubble means. I don’t even know what a bubble means.
These words have become popular. I don’t think they have any meaning.
Cassidy: I guess most people would define a bubble as an extended period during which asset
prices depart quite significantly from economic fundamentals.
Fama: That’s what I would think it is, but that means that somebody must have made a lot of
money betting on that, if you could identify it….
Cassidy: Are you saying that bubbles can’t exist?
Fama: They have to be predictable phenomena. I don’t think any of this was particularly
predictable.
(b) From an interview with Edmund Phelps by Caroline Baum,
Bloomberg, 11 February
2013:
Phelps: “In the expectations-based framework that I put forward around 1968, we didn’t
pretend we had a correct and complete understanding of how firms or employees formed
expectations about prices or wages elsewhere. We turned to what we thought was a plausible
and convenient hypothesis. For example, if the prices of a company’s competitors were last
reported to be higher than in the past, it might be supposed that the company will expect their
prices to be higher this time, too, but not that much. This is called ‘adaptive expectations’….
Baum: So how did adaptive expectations morph into rational expectations?
Phelps: The “scientists” from Chicago and MIT came along to say, we have a well-established
theory of how prices and wages work. Before, we used a rule of thumb to explain or predict
expectations: Such a rule is picked out of the air. They said, let’s be scientific. In their mind,
the scientific way is to suppose price and wage setters form their expectations with every bit
as much understanding of markets as the expert economist seeking to model, or predict, their
behavior. The rational expectations approach is to suppose that the people in the market form
their expectations in the very same way that the economist studying their behavior forms her
expectations: on the basis of her theoretical model.
5435
6
last
page
5440
DEGREE OF MASTER OF PHILOSOPHY
Master of Philosophy in Economics (Final Examination)
Probationer Research Student Examination in Management Studies
FINANCIAL ECONOMICS I
TRINITY TERM 2014
Monday, 23 June, 2014 — 09:30 am to 11:30 am
Answer two questions, with one from each part.
Start the answer to each question on a separate sheet
Candidates may use their own calculators
Do not turn over until told that you may do so
1
PART A
1. [a] Consider a CAPM economy with just three risky assets, A, B, and C with
expected rates of return of 10%, 15%, and 10%, respectively. Two portfolios, 1 and
2, are known to lie on the minimum variance frontier. They are defined by the
portfolio weights w
1 =
wA, wB, wC
= (0.5, 0.3, 0.2) and w
, wB, wC
=
1
1
1
2 =
wA
2
2
2
(0.9, −0.3, 0.4), respectively. What are the minimum and maximum possible values
for the expected rate of return on the market portfolio?
[b] Consider a different CAPM economy with three risky assets A, B, and C. Their
payoffs are uncorrelated with each other and have the following characteristics.
i
E(ri)
σi
A
7%
7%
B
10%
10%
C
12%
10%
The risk free rate is 5%.
(i) What is the Sharpe Ratio of an efficient portfolio in this economy?
(ii) What is the equation of the Capital Market Line?
(iii) What is the composition of the optimal portfolio for an investor who requires
an expected return of 15%?
[c] Consider the following two assets A and B with normally distributed rates of
return
˜
r
A
0.10
0.003 0.002
∼ N
,
˜
rB
0.15
0.002 0.005
The agent maximizes expected utility, with u(w) = −e−10w, and has initial wealth of
one. The risk free rate is 5%.
(i) What is the agent’s optimal portfolio?
(ii) What is the equation of the Security Market Line?
[d] What assumptions must be made for the CAPM to hold in a one-good GEI
economy? State the CAPM Mutual Fund Theorem.
5440
2
2. [a] Consider a two-period economy with two agents h ∈ H = {α, β}, two possible
states of the world s ∈ S = {1, 2} in the second period and one commodity (L = {1}).
Both agents have the same utility function
u(x1, x2) = ln(x0) + ln(x1) + ln(x2),
where x0, x1 and x2 are consumption in the first period, state 1 and state 2, respec-
tively. The endowments of the agents in the first period, state 1 and state 2 are given
by eα = (1, 2, 0) and eβ = (1, 0, 2). There is one real asset promising payoffs of 1 in
each state of the world in the second period. Default penalties are λh = λ > 0
∀s, h.
s
(i) Discuss agents’ default decisions, i.e., write down and explain the conditions for
an agent to deliver fully, default partially, and default completely.
(ii) Write down the maximization problem for each agent, including the objective
function, the budget constraints, and the market clearing conditions.
(iii) Set λ = 2/3 and solve for the equilibrium allocations, asset trades, the delivery
rates of the asset, and the actual amounts delivered.
(iv) Provide an economic justification for why default penalties in practice are set
at an intermediate level.
[b] Consider a one-period economy with fiat money and two agents (h ∈ H = {α, β}).
Monetary endowments are given by mα and mβ. Trade is subject to the cash-in-
advance constraint, with the central bank offering intra-period loans. The money
supply by the central bank is denoted by M . The amounts of money borrowed by
the agents are µα and µβ.
There are two goods in the economy. One is a consumption good and the other is
a capital good. The consumption good is in zero supply but can be produced using
the capital good. Production occurs instantly, according to the production function
y = k0.3, where y is the output level and k is the level of the capital good.
Agent α is endowed with one unit of the capital good, while agent β is not endowed
with any of the capital good. Both agents have the same production function. Agent
β purchases q2 units of the capital good from agent α. Agent α purchases q1 units
of the consumption good from agent β. The prices of the consumption good and the
capital good are p1 and p2, respectively.
(i) Assuming that the agent maximizes log-consumption, write down the maximiza-
tion problem for each agent, including the objective and all the constraints.
(ii) Demonstrate the non-neutrality of money in this economy.
(iii) Show that the optimal allocation of the capital good occurs when the interest
rate is zero.
5440
TURN OVER
3
PART B
3.
[a] Consider a market with one risky asset, an informed agent and a risk-
neutral uninformed investor. The value of the risky asset is normally distributed,
˜
v ∼ N (0, σ2). The realization of asset value becomes common knowledge only after
v
trade stops. The informed investor knows the realized value v before trading starts
and can trade in two subsequent time periods. All investors submit limit orders.
(i) What are the simplest price conjectures for the first and second rounds of trad-
ing, given that the supply of the risky asset is the same in both rounds of
trading, i.e. z1 = z2 = z, where ˜
zt ∼ N (0, σ2), and this is common knowledge?
z
(ii) How would your answer change if z1 and z2 are i.i.d.?
(iii) Suggest a measure of the informational efficiency of prices. If it were known
that the supply in the second round of trading was correlated with that in the
first round of trading, how would informational efficiency of prices compare with
those in (ii)?
(iv) Would a strategic, informed trader trade most aggressively in the first round of
trading in case (i), (ii) or (iii)? Rank the three cases in terms of the informed
trader’s aggressiveness in first round trading.
[b] Consider the following modification of the Grossman (1976) model. Traders
choose between holding a risk-free asset with gross return rf and a risky asset; the
latter has price p, a payoff ˜
v, where ˜
v ∼ N (rv, σ2), and is in zero net supply. There are
v
two traders, indexed 1 and 2, who maximize expected utility of end-of-period wealth,
˜
Wi, i ∈ {1, 2}. Both agents maximizes expected utility. Their utility functions have
the same functional form, given by U (w) = −e−w. At the beginning of the period,
trader i is endowed with ei and observes a signal ˜
si which is of the form ˜
si = ˜
v + ˜
i,
where ˜
i ∼ N (0, σ2). ˜
s
i is independent of ˜
v. Additionally, the variances of the traders’
signals are the same and E[12] = 0.
(i) Suppose that each trader acts purely as a price taker and does not infer the
other’s information from the equilibrium price. Find the demand functions of
each trader for the risky asset and solve for the equilibrium price.
(ii) Now suppose that each trader uses the the equilibrium price to infer the other’s
information. Find the demand functions of each trader for the risky asset and
solve for the rational expectations equilibrium price.
(iii) In which of the two cases above is the price more informationally efficient?
5440
4
4. A trader must buy ¯
S shares in a stock over t = 1, 2, ..., T periods. She can break
her trade into a set of ‘child’ orders {st : t = 1, 2, ..., T }, subject to the constraint
T
X st = ¯
S.
t=1
The market price at which she trades is Pt. Its dynamics are
Pt = Pt−1 + (γt)st + ηt,
where γ > 0 and {ηt} is an i.i.d. shock with E[ηt] = 0. For simplicity, normalize
P0 = 0.
(i) How might this model describe a situation of deteriorating market liquidity?
(ii) Assume the trader is risk neutral. Write down the formal problem describing
her objective of minimizing execution costs.
(iii) Assuming that the second order conditions are satisfied in this problem, use the
first-order conditions to prove that the optimal quantity to trade at time t = 2
exceeds by ( ¯
S − s1)/3 the optimal quantity to trade at time t = 3.
(Hint: partially differentiate with respect to s2 and s3.)
(iv) Find the optimal value of the ratio s(T −1)/sT .
(v) Suppose we add a temporary price impact term to the definition of Pt. Give a
brief intuitive discussion of how the optimal trading trajectory will change.
5440
LAST PAGE
5
5441
DEGREE OF MASTER OF PHILOSOPHY
Master of Philosophy in Economics (Final Examination)
Probationer Research Student Qualifying Examination in Economics
Probationer Research Student Examination in Management Studies
FINANCIAL ECONOMICS II
TRINITY TERM 2014
Thursday, 19 June, 2014 — 09:30 am to 11:30 am
Answer two questions, with at least one from Part A.
Start the answer to each question on a separate sheet
Candidates may use their own calculators
Do not turn over until told that you may do so
1
PART A
1. In Fulghieri and Lukin (F&L), firms design securities in order to raise funds to
finance projects. Firms know project quality, the investors who finance the project
do not know project quality but they can invest in becoming informed by paying an
information acquisition cost.
(i) Briefly explain why optimal security designs (the designs chosen in equilibrium)
aim to encourage information production by investors in the F&L model.
(ii) In F&L, how does informed trading affect the degree of mispricing of the firm’s
claims?
Consider the following alternative model of security design with informed trade in
secondary security markets.
There are three dates, 0, 1, and 2. As in F&L, all agents are risk neutral and patient
so they simply maximize their expected payoffs. A penniless entrepreneur has an
investment opportunity that requires an investment of I. The funds required for
undertaking the investment must be raised from an outsider, an investor with funds
at date 0. Funding will take the form of selling a single security to the outsider. There
are many uninformed outsiders who might buy the security so that the price of the
security at date 0 is fixed by Bertrand competition.
The investment produces only one cash flow which is realized at date 2. The cash
flow is either H or L, where H > I > L > 0 and H + L > 2 I. At date 1, an
informed speculator is born. The speculator knows whether the value of the firm is H
or L. At date 1, the speculator is able to trade without any short sale or borrowing
restrictions. The price of the security at date 1 is determined in a competitive market
by a risk neutral market-maker who observes aggregate security demand (just as in
F&L) but not the identity of the trader submitting the demand. All agents other than
the speculator have the same prior beliefs. They believe that H and L are equally
likely. At date 1, with probability u, the outsider receives a liquidity shock. This
shock reduces the outsider’s valuation of date 2 consumption to zero. Thus, if the
outsider receives the shock, the outsider will sell the security and consume at date 1.
(iii) In this alternative model, do you think that informed trade increases or reduces
the welfare of the entrepreneur? Briefly justify your conjecture. You do not
need to provide a formal derivation.
5441
2
(iv) Represent the number of units of the security sold by the speculator by n,
i.e., n = 1 represents submitting a buy order for one unit of the security and
n = −1 represents submitting a sell order for one unit of the security. Compute
the optimal trading strategy for the speculator.
(v) If the market-maker observes that total demand for the security at date 1 equals
−1, what price will the market-maker set for the security?
(vi) What is the optimal limited liability security for the entrepreneur to issue at
date 0?
(vii) Assuming that u = 1/2, L = 1, I = 2, H = 5, what will be the expected
payoff to the entrepreneur if the entrepreneur issues the optimal limited liability
security at date 0? Briefly explain your result.
2. In Harris and Raviv (H&R), the liquidation value of assets plays an important role
in determining the outcome of corporate insolvencies. For a given state of the world,
let R1 represent the firm’s current, date 1, cash flow. Let R2 represent the cash flow
at date 2 if no assets are liquidated at date 1 and let L represent the liquidation value
of the firm’s assets at date 1. Denote the ratio R2/L by α. Assume that R2 > R1.
Answer the following questions in the context of the H&R model.
(i) Suppose that the debtor repudiates the debt contract at date 1. What is the
maximum payment that the creditor can extract from the debtor if the debt is
creditor-preferred? What is the maximum payment in the case where the debt
is debtor-preferred?
(ii) How is the debtor’s repayment cost affected by increasing the liquidation value
of the firm’s assets, holding R1 and R2 fixed, in the case where the debt is
creditor-preferred? How is it affected in the case where the debt is debtor-
preferred?
(iii) Suppose that, after the creditor provides financing and after the state has been
observed, the debtor has the opportunity to invest effort in reducing L, holding
R1 and R2 fixed.
If the debt is creditor-preferred, would the debtor have an incentive to invest
effort in reducing L? If the debt is debtor-preferred, would the debtor have an
incentive to invest effort in reducing L? Briefly, explain any contrast between
the two cases.
(iv) Can you think of firm actions that lower asset liquidation value without affecting
the going-concern value of the firm?
5441
TURN OVER
3
PART B
3. The first issue raised in the private equity (PE) literature was whether PE firms
were adding value to their portfolio companies.
(i) Why is this issue important?
(ii) Discuss the findings on this issue. Is there a difference between recent findings
and older ones? If so, what are the causes?
(iii) In your opinion, what are the missing pieces in this literature?
4. “Recent realizations of profits determine both expectations of future profitability
and the availability of internal finance for investment. Consequently, there are no
convincing tests for the impact of financing constraints on corporate investment.”
Discuss this claim with reference to the recent empirical literature on capital market
imperfections and corporate investment.
5. How does corporate tax affect firms’ capital structure? What are the empirical
challenges in identifying the effects of corporate taxes on financial leverage?
5441
LAST PAGE
4
5444
DEGREE OF MASTER OF PHILOSOPHY
Master of Philosophy in Economics (Final Examination)
Probationer Research Student Examination in Economics
INTERNATIONAL TRADE 1
TRINITY TERM 2014
Tuesday 24th June 2014, 09:30 - 11:30
Please start the answer to each question on a separate page.
Answer TWO questions, ONE from Part A and ONE from Part B.
Candidates may use their own calculators.
Do not turn over until told that you may do so.
1
PART A
1. An economy has two sectors each of which uses labour and a sector specific factor, and
produces output according to
x
1
,
x
1
. Employment in sector 1 is
1
1
L K1
2
L2
K 2
L1 and in
sector 2 is
L2, with the full employment condition
L =
L1 +
L2. Let
K1,
K2 denote the specific
factor in each sector, and the production functions have constant returns to scale. World
prices
p1,
p2 are fixed, and good 2 will be used as numeraire,
p2 = 1.
i)
Find the equilibrium level of employment in each sector.
ii)
All consumers have the same preferences,
U
1
. For what parameter
1
C C2
values does the economy import good 1? What happens to trade as
K2 increases?
iii)
The country imposes an import tariff at proportional rate
t. Who gains and who
loses?
iv)
Suppose that due to some exogenous distortion the wage in sector 1 is higher than
that in sector 2 by factor
θ > 1. What is the effect of the import tariff on aggregate
welfare?
2. Consider the constant elasticity of substitution (CES) utility function Q = ( q(w)ρ dw) 1/ρ,
where q(w) denotes quantity consumed of good w and ρ is a parameter.
i)
Show for which values of ρ the goods are substitutes, and for which values they
are complements.
ii)
Discuss properties of the CES utility function when ρ=1 and when ρ → -∞.
iii)
Discuss strengths and weaknesses of the use of CES in general equilibrium models
of international trade.
3. Explain why exporters are larger and more productive in the Melitz (2003) model than
non-exporters. Would this be true if:
i)
variable trade costs were to fall to zero,
ii)
there were no additional fixed costs associated with trade,
iii)
both?
5444 2
PART B
4.
What theory and evidence is there concerning the determinants of bilateral trade flows
between countries? Does the evidence support the theories?
5.
Does trade theory shed any light on the cross-country distribution of per capita
income? Illustrate your answer drawing on at least two models of international trade.
6.
Do increasing returns to scale justify trade restrictions?
5444
3
last
page
5447
DEGREE OF MASTER OF PHILOSOPHY
Master of Philosophy in Economics (Final Examination)
Probationer Research Student Qualifying Examination in Economics
ADVANCED MICROECONOMICS I (A)
TRINITY TERM 2014
Tuesday, 17 June, 2014 — 2:30 pm to 3:30 pm
The answer to each question should begin on a separate page.
Answer TWO questions from different topics.
Topics are indicated in bold at the start of each question.
Candidates may use their own calculators
Do not turn over until told that you may do so
1
1. [Games, Learning and Evolution] Consider the symmetric gold-silver game
with payoff matrix
G
S
G
3,3
0,0
S
0,0
2,2
There is one player at each node of the network below:
Each period one player is chosen at random to update his choice (G or S) and then
plays against each of his neighbours once. She chooses a log-linear (logit) response to
the current choices of her neighbours, where the response parameter is β > 0.
(i) How many states does the Markov process have?
(ii) Identify all of the pure Nash equilibria when β = ∞, that is, when players
choose a best response to their neighbors with no trembles.
(iii) Given 0 < β < ∞, compute the probability of each pure Nash equilibrium in
the stationary distribution (as a function of β).
2. [Games, Learning and Evolution] Consider two disjoint populations of players
– Row and Column. Each period one pair is drawn at random and they play a discrete
version of the Nash demand game in which the only feasible demands are 1/3, 1/2,
or 2/3.
1
Each player’s payoff equals his demand if the demands are compatible; if the demands
are incompatible each gets a payoff of zero.
(i) Write down the payoff matrix of this two-person stage game.
(ii) Consider adaptive play with memory m = 100, sample size s = 12 and error
rate ≥ 0. When = 0 what are the recurrence classes?
(iii) Compute the resistance between each pair of recurrence classes when > 0.
(iv) What states are stochastically stable? Explain why.
5447
2
3. [Incentives, Information, and Organizations] A sender (S) and a receiver
(R) play the following communication game with verifiable disclosure. At time 0, R
and S share a common prior belief that the state of the world, x, takes each of the
three values {0, 1, 2} with probability 1 . At time 1, S may privately receive signals
3
that are informative about x. With probability µA ∈ (0, 1), S receives signal A,
which reveals to S whether x < 0.5 (realization “A−”) or x > 0.5 (realization “A+”).
With probability µB ∈ (0, 1), S receives signal B, which reveals to S whether x < 1.5
(realization “B−”) or x > 1.5 (realization “B+”). The events of receiving signals A
and B are independent. R observes neither the information conveyed by the signals
(if received), nor whether they were received. At time 2, S chooses which (if any)
signals to verifiably disclose to R. After hearing S’s disclosure, R chooses an action,
V , to minimize the expected value of (V − x)2, given his updated beliefs about x.
Regardless of the value of x, S’s payoff from the game is V .
(i) For each possible set of signal realizations S might observe, what will he verifi-
ably disclose in equilibrium?
(ii) How many different choices of V have positive probability, i.e.
are on the
equilibrium path in this game?
(iii) For each choice of V that has positive probability in equilibrium, explain how it
depends on µA and µB and what each approaches as µA and µB both approach
1.
4. [Incentives, Information, and Organizations] An agent (A) privately chooses
a level of lobbying effort a ≥ 0 to influence a public signal that is informative about
the profitability, θ, of a potential investment project. The signal, s, is given by
s = aθ + u, and after observing it, the principal (P ) chooses a decision x ∈ (−∞, ∞),
receiving the payoff −(x − θ)2 − a. The agent receives the payoff x − a. Both P and
A maximize their expected payoffs, and they have common prior beliefs about the
random variables θ and u: θ and u are independent and normally distributed, with
means ¯
θ > 0 and 0 and variances σ2 and σ2, respectively.
θ
u
(i) In words, what conditions must (pure-strategy) equilibrium choices of A’s effort
a and P ’s decision rule x(s) satisfy?
(ii) Explain why there always exists an equilibrium in which A chooses zero effort.
(iii) If there exists a (pure-strategy) equilibrium with strictly positive lobbying effort,
what condition determines the equilibrium value(s)?
5447
TURN OVER
3
5.
[Choice under uncertainty] (i) State the monotonicity assumptions in the
Savage and Anscombe-Aumann frameworks for expected utility.
(ii) Show that, in both frameworks, monotonicity implies the separation of tastes
from beliefs.
(iii) Explain why it is difficult to give a choice theoretic foundation to the notion of
subjective probability without assuming monotonicity.
6.
[Monotone Comparative Statics] An information structure is defined as a
collection of distribution functions H = {H(·; s)}s∈S, where H(·; s) is the distribution
of the signal z ∈ Z, conditional on a state s ∈ S. Let Z be a compact interval of the
real line R and suppose that S is a subset of R.
(i) Given two information structures H = {H(·; s)}s∈S and G = {G(·; s)}s∈S, both of
which have signals in Z, what does it mean for H to be more informative than G in
the sense of Lehmann?
(ii) Suppose that S = {s1, s2}, with s2 > s1. Suppose also that H(·; s1) = G(·; s1)
and H(·; s2) first order stochastically dominates G(·; s2).
(a) Show that {H(·; s)}s∈{s1,s2} is more informative than {G(·; s)}s∈{s1,s2} in Lehmann’s
sense.
(b) An agent has to choose between two actions x or y, where y is superior if s2
is the true state and x is superior if s1 occurs. In formal terms, denoting his
utility by u, we have u(y, s2) > u(x, s2) and u(x, s1) > u(y, s1).
Suppose the agent has the following decision rule: choose action y whenever he
observes z ≥ z∗ ∈ Z and x otherwise. The agent attaches a prior probability
of πi > 0 to state si (for i = 1, 2). Write down an expression for the agent’s ex
ante expected utility, given his decision rule and under information structure G.
Show that the agent’s ex ante expected utility, using the same rule, is higher
under H than under G.
(iii) Is there a generalization of the result in (ii) to a situation with multiple states
and actions? (No proofs are needed here.)
5447
4
7. [Dynamic mechanism design] A risk-neutral monopolist is contracting over
two periods to sell a single good to a risk-neutral buyer. In period 1 the buyer’s
private information is θ, uniformly distributed on [0, 1]. In period 2 the buyer learns
the realization of ε, which is independent of θ and uniform on [−1, 1]. The buyer’s
valuation for the good is θ + ε.
The monopolist commits to a sales contract after the buyer has learnt θ, but before
the realization of ε. A direct mechanism consists of an upfront fee c (payable if the
buyer agrees to the contract) and a strike price p (payable conditional on buying
the good after the realization of ε), both as functions of the buyer’s reported θ. In
period 1 the buyer has the option to acquire the good from a third party at price
r ∈ [0, 1]; this outside option disappears if he accepts the monopolist’s contract.
Assume p(θ) ∈ [0, 1] and participation for all θ throughout, and use the first-order
approach.
(i) Verify that
Z
1
h
i 1
U (θ, b
θ) = −c(b
θ) +
θ + ε − p(b
θ)
dε
2
p(b
θ)−θ
is the buyer’s expected payoff with valuation estimate θ reporting b
θ in a direct
mechanism {c(·), p(·)}.
(ii) Express the slope of the buyer’s indirect expected payoff function, V (θ) ≡
U (θ, θ), in an incentive compatible direct mechanism.
(iii) What is the buyer’s participation constraint?
(iv) Noting that the monopolist’s expected revenue in an incentive compatible direct
mechanism with full participation is
Z
1 Z 1
Π =
(θ + ε) 1 dε − V (θ) dθ,
2
0
p(θ)−θ
formulate the monopolist’s problem (with full participation) as an optimal con-
trol problem. What are the state and control variables?
(v) Using a Hamiltonian and the Maximum Principle (or otherwise) derive p(θ) and
V (θ) in the optimal mechanism when r = 1 . [Hint: The participation constraint
4
is binding for the two extreme types.]
5447
TURN OVER
5
8.
[Dynamic mechanism design] Suppose that in a dynamic principal-agent
problem the agent’s privately-known type θt follows
θt = (1 − λ)Pt
λt−kε
k=0
k for t = 0, . . . , T ,
where εt, i.i.d. uniform on [0, 1] for all t, are the agent’s orthogonalized types. In
period t, having learnt θt (equivalently, εt) the agent takes unobservable action at,
which then generates a contractible public signal st = θt + at.
In what follows, we refer to the history of variables up to period t by superscript t,
e.g., εt = (ε0, . . . , εt) ≡ (ε0, εt− ). The agent’s payoff is
0
T
X
U (θT , aT , P ) = P −
(1 − θt)at,
t=0
where P is the total payment received from the principal.
(i) Suppose the principal offers a contract to the agent in period 0, after the real-
ization of θ0 ≡ (1 − λ)ε0. State the Revelation Principle.
(ii) Assume that a contract induces the agent to take action at(εt, st−1) in equi-
librium in period t after history (εt, st−1). Verify and explain what it means
for
a
ε
ε
, st−1) + (1 − λ)λt(ε
bt(εt, b0, st−1) = at(b0, εt
−0
b0 − ε0)
to be the agent’s lie-masking action at t, and compute
∂ at.
∂ε b
0
(iii) Write down the agent’s incentive compatibility constraint at t = 0 assuming
lie-masking actions for all t and truthful reports of ε1, . . . , εT .
(iv) Let V (ε0) denote the agent’s expected payoff in an incentive compatible mech-
anism with a given action and payment rule. Show that
V (ε0) = V (0) + (λT − 1)ε0
(using the incentive-compatibility constraint from part (iii)).
5447
LAST PAGE
6
5448
DEGREE OF MASTER OF PHILOSOPHY
Master of Philosophy in Economics (Final Examination)
Probationer Research Student Qualifying Examination in Economics
ADVANCED MICROECONOMICS I (B)
TRINITY TERM 2014
Friday, 20 June, 2014 — 09:30 am to 10:30 am
Answer ONE question
Candidates may use their own calculators
Do not turn over until told that you may do so
1
1. Consider the following two-person game G:
A
B
C
A
20,20
10,30
1,20
B
30,10
0,0
0,20
C
20,1
20,0
0,0
(i) Assume that both players are perfectly rational and forward-looking with dis-
count factor δ = 0.9. Construct a subgame perfect equilibrium of G∞ such that
in each period both players get 20. Be sure to specify the strategies after each
and every history.
(ii) Suppose that each player is a naive hypothesis tester, that is,
• at every time t, each player thinks that the opponent is using a stationary
strategy (pure or mixed)
• each player i chooses an action at according to a quantal response func-
i
tion, that is, with probability proportional to eui(at)/γi, where γi > 0 is i’s
response parameter
• every so often i tests his hypothesis against the most recent s plays of the
game
• i then accepts or rejects the current hypothesis
(a) Specify more precisely what criterion is used to accept or reject, and what
happens afterwards.
(b) Under what conditions on the parameters do players learn to play an ap-
proximate equilibrium? Explain what is meant by “learn” and “approximate”.
(c) Which equilibrium of G∞ do they learn in this case?
(iii) Discuss the differences between model (i) and (ii) with respect to the degree of
rationality of the players and the extent of their knowledge.
5448
2
2. A risk-neutral principal (P ) offers a contract to a risk-neutral agent (A). A’s ex
post payoff equals his payment minus his disutility of effort exerted, and his ex ante
reservation utility is zero. A’s disutility of effort e = 0 is 0 and of effort e = 1 is
d > 0. If e = 0, the value to P of A’s output, v, is 0 with probability 1. If e = 1, then
v = H with probability pH, v = L with probability pL, and v = 0 with probability
1 − pH − pL, where 0 < L < H.
(i) Suppose that A’s effort is publicly observable and contractible. What condition
is necessary and sufficient for P ’s optimal contract to have the form: if e = 0, pay A
zero, and if e = 1, pay A the amount d?
Henceforth, suppose that P cannot observe A’s effort and is restricted to offering a
contract specifying a single number k. After A chooses e from {0, 1} and v is observed
by P , P chooses between purchasing A’s output from A at price k and not purchasing,
in which case the output goes to waste. (A himself can make no use of the output.)
Assume that the condition derived in (i) holds.
(ii) What are the two types of inefficiency that might arise under a contract in this
model?
(iii) Under what conditions on the exogenous parameters of this model (H, L, d, pH, pL)
will P ’s optimal contract
(a) induce e = 1 and result in purchase by P when v ∈ {L, H}?
(b) induce e = 1 and result in purchase by P only when v = H?
(c) induce e=0?
For cases (a) and (b), what value of k will P ’s optimal contract specify?
(iv) Now, in addition, suppose that after the realization of v is observed by both
P and A, renegotiation can occur, according to the rule that P names a price, r,
at which he is willing to purchase A’s output. If A accepts, then the new price, r,
applies; if A rejects, then the old price, k, applies. How does the optimal contract
now depend on the exogenous parameters? How does the possibility of renegotiation
affect the efficiency of the outcome under the optimal contract in each of cases (a),
(b), and (c)?
5448
TURN OVER
3
3.
(i) Suppose (X, ≥) is a subcomplete sublattice of the lattice (L, ≥). Given an
increasing function φ : X → X, we define Y = {x ∈ X : x ≥ φ(x)}. Why is Y
nonempty? Let x∗ = inf Y . Show that (a) φ(x∗) is a lower bound of Y and (b)
φ(x∗) ∈ Y . Hence show that x∗ is a fixed point of φ and, in fact, is the smallest fixed
point of φ, i.e., if ˜
x is another fixed point of φ, then ˜
x ≥ x∗.
(ii) Let (X0, ≥) be another subcomplete sublattice of (L, ≥) and ψ : X0 → X0 an
increasing function on X0. Suppose that X0 ≥ X (in the sense of the strong set
order), ψ(x) ≥ φ(x) for all x ∈ X0 ∩ X, and there exists an element ˆ
x ∈ X such that
ˆ
x ≥ ψ(ˆ
x).
It follows from (i) that ψ has a smallest fixed point, which we shall denote by x∗∗;
indeed
x∗∗ = min{x ∈ X0 : x ≥ ψ(x)}.
Show that x∗∗ ∈ X and that x∗∗ ≥ x∗.
(iii) Consider a game with a finite set of players I = {1, 2, ..., m}. Each player i ∈ I
has a strategy set Si = [ai, bi], which is a compact interval of the real line. We denote
a profile of strategies by the vector s, with s = (s
m
1, s2, ..., sm) ∈ R
. The payoff
of player i when he is playing si and the other players are playing s−i is ui(si; s−i).
Assume that ui is a continuous function of si and ui obeys single crossing differences
in (si; s−i).
(a) Show that the map from s−i to min [arg maxs
u
i∈Si
i(si; s−i)] ∈ Si is an increasing
function of s−i.
(b) Use the results in (i) to guarantee that this game has a Nash equilibrium s∗
that is smaller than any other Nash equilibrium in this game.
(c) Suppose that there is a change in the strategy set of Player 1 from [a1, b1] to
[¯
a1, b1], where ¯
a1 > a1. The payoff function of Player 1, and the payoff functions
and strategy sets of the other players remain unchanged. Let s∗∗ be the smallest
Nash equilibrium in this new game. Show that s∗∗ ≥ s∗.
5448
4
4. Assume that we are about to watch a horse race and there are n states, with each
state corresponding to a particular horse winning the race. A bet is a function from
states to monetary outcomes, i.e., x ∈ X ⊂
n
R , with xi denoting the net gain in the
case where horse i wins. Suppose the preference on X is represented as follows:
V (x) = min p · x,
p∈P
where x is a bet, p ∈ ∆n−1 with ∆n−1 denoting the set of probability vectors on
{1, ..., n}, and P is a closed, convex set of such probability vectors.
(i) What does the form of V (x) imply for the individual’s attitude toward risk and
ambiguity?
(ii) Let A1, A2,..., A6 be six axioms on such that the following theorem is true:
The preference on X satisfies A1-A6 if and only if there exists a unique,
non-empty, closed convex set P ⊆∆n−1 such that, ∀x, y ∈ X,
x y ⇔ V (x) ≥ V (y).
(a) Write down such a set of axioms A1, A2,..., A6 and explain, briefly, the
intuitive content of each axiom.
(b) Sketch a proof of the theorem.
5. Explain qualitatively how optimal bidding differs between uniform-price auctions
and discriminatory auctions for multiple units of a homogeneous product, when each
bidder has private values and desires multiple units. Which auction form is likely to
(i) be more robust against collusion, (ii) result in more bidders participating, (iii) be
more efficient. Explain your answers.
5448
LAST PAGE
5
5449
DEGREE OF MASTER OF PHILOSOPHY
Master of Philosophy in Economics (Final Examination)
ADVANCED MICROECONOMICS I AND II (A)
TRINITY TERM 2014
Tuesday, 17 June, 2014 — 2:30 pm to 4:30 pm
The answer to each question should begin on a separate page.
Answer FOUR questions from at least THREE topics.
Topics are indicated in bold at the start of each question.
Candidates may use their own calculators
Do not turn over until told that you may do so
1
1. [Games, Learning and Evolution] Consider the symmetric gold-silver game
with payoff matrix
G
S
G
3,3
0,0
S
0,0
2,2
There is one player at each node of the network below:
Each period one player is chosen at random to update his choice (G or S) and then
plays against each of his neighbours once. She chooses a log-linear (logit) response to
the current choices of her neighbours, where the response parameter is β > 0.
(i) How many states does the Markov process have?
(ii) Identify all of the pure Nash equilibria when β = ∞, that is, when players
choose a best response to their neighbors with no trembles.
(iii) Given 0 < β < ∞, compute the probability of each pure Nash equilibrium in
the stationary distribution (as a function of β).
2. [Games, Learning and Evolution] Consider two disjoint populations of players
– Row and Column. Each period one pair is drawn at random and they play a discrete
version of the Nash demand game in which the only feasible demands are 1/3, 1/2,
or 2/3.
1
Each player’s payoff equals his demand if the demands are compatible; if the demands
are incompatible each gets a payoff of zero.
(i) Write down the payoff matrix of this two-person stage game.
(ii) Consider adaptive play with memory m = 100, sample size s = 12 and error
rate ≥ 0. When = 0 what are the recurrence classes?
(iii) Compute the resistance between each pair of recurrence classes when > 0.
(iv) What states are stochastically stable? Explain why.
5449
2
3. [Incentives, Information, and Organizations] A sender (S) and a receiver
(R) play the following communication game with verifiable disclosure. At time 0, R
and S share a common prior belief that the state of the world, x, takes each of the
three values {0, 1, 2} with probability 1 . At time 1, S may privately receive signals
3
that are informative about x. With probability µA ∈ (0, 1), S receives signal A,
which reveals to S whether x < 0.5 (realization “A−”) or x > 0.5 (realization “A+”).
With probability µB ∈ (0, 1), S receives signal B, which reveals to S whether x < 1.5
(realization “B−”) or x > 1.5 (realization “B+”). The events of receiving signals A
and B are independent. R observes neither the information conveyed by the signals
(if received), nor whether they were received. At time 2, S chooses which (if any)
signals to verifiably disclose to R. After hearing S’s disclosure, R chooses an action,
V , to minimize the expected value of (V − x)2, given his updated beliefs about x.
Regardless of the value of x, S’s payoff from the game is V .
(i) For each possible set of signal realizations S might observe, what will he verifi-
ably disclose in equilibrium?
(ii) How many different choices of V have positive probability, i.e.
are on the
equilibrium path in this game?
(iii) For each choice of V that has positive probability in equilibrium, explain how it
depends on µA and µB and what each approaches as µA and µB both approach
1.
4. [Incentives, Information, and Organizations] An agent (A) privately chooses
a level of lobbying effort a ≥ 0 to influence a public signal that is informative about
the profitability, θ, of a potential investment project. The signal, s, is given by
s = aθ + u, and after observing it, the principal (P ) chooses a decision x ∈ (−∞, ∞),
receiving the payoff −(x − θ)2 − a. The agent receives the payoff x − a. Both P and
A maximize their expected payoffs, and they have common prior beliefs about the
random variables θ and u: θ and u are independent and normally distributed, with
means ¯
θ > 0 and 0 and variances σ2 and σ2, respectively.
θ
u
(i) In words, what conditions must (pure-strategy) equilibrium choices of A’s effort
a and P ’s decision rule x(s) satisfy?
(ii) Explain why there always exists an equilibrium in which A chooses zero effort.
(iii) If there exists a (pure-strategy) equilibrium with strictly positive lobbying effort,
what condition determines the equilibrium value(s)?
5449
TURN OVER
3
5.
[Choice under uncertainty] (i) State the monotonicity assumptions in the
Savage and Anscombe-Aumann frameworks for expected utility.
(ii) Show that, in both frameworks, monotonicity implies the separation of tastes
from beliefs.
(iii) Explain why it is difficult to give a choice theoretic foundation to the notion of
subjective probability without assuming monotonicity.
6.
[Monotone Comparative Statics] An information structure is defined as a
collection of distribution functions H = {H(·; s)}s∈S, where H(·; s) is the distribution
of the signal z ∈ Z, conditional on a state s ∈ S. Let Z be a compact interval of the
real line R and suppose that S is a subset of R.
(i) Given two information structures H = {H(·; s)}s∈S and G = {G(·; s)}s∈S, both of
which have signals in Z, what does it mean for H to be more informative than G in
the sense of Lehmann?
(ii) Suppose that S = {s1, s2}, with s2 > s1. Suppose also that H(·; s1) = G(·; s1)
and H(·; s2) first order stochastically dominates G(·; s2).
(a) Show that {H(·; s)}s∈{s1,s2} is more informative than {G(·; s)}s∈{s1,s2} in Lehmann’s
sense.
(b) An agent has to choose between two actions x or y, where y is superior if s2
is the true state and x is superior if s1 occurs. In formal terms, denoting his
utility by u, we have u(y, s2) > u(x, s2) and u(x, s1) > u(y, s1).
Suppose the agent has the following decision rule: choose action y whenever he
observes z ≥ z∗ ∈ Z and x otherwise. The agent attaches a prior probability
of πi > 0 to state si (for i = 1, 2). Write down an expression for the agent’s ex
ante expected utility, given his decision rule and under information structure G.
Show that the agent’s ex ante expected utility, using the same rule, is higher
under H than under G.
(iii) Is there a generalization of the result in (ii) to a situation with multiple states
and actions? (No proofs are needed here.)
5449
4
7. [Dynamic mechanism design] A risk-neutral monopolist is contracting over
two periods to sell a single good to a risk-neutral buyer. In period 1 the buyer’s
private information is θ, uniformly distributed on [0, 1]. In period 2 the buyer learns
the realization of ε, which is independent of θ and uniform on [−1, 1]. The buyer’s
valuation for the good is θ + ε.
The monopolist commits to a sales contract after the buyer has learnt θ, but before
the realization of ε. A direct mechanism consists of an upfront fee c (payable if the
buyer agrees to the contract) and a strike price p (payable conditional on buying
the good after the realization of ε), both as functions of the buyer’s reported θ. In
period 1 the buyer has the option to acquire the good from a third party at price
r ∈ [0, 1]; this outside option disappears if he accepts the monopolist’s contract.
Assume p(θ) ∈ [0, 1] and participation for all θ throughout, and use the first-order
approach.
(i) Verify that
Z
1
h
i 1
U (θ, b
θ) = −c(b
θ) +
θ + ε − p(b
θ)
dε
2
p(b
θ)−θ
is the buyer’s expected payoff with valuation estimate θ reporting b
θ in a direct
mechanism {c(·), p(·)}.
(ii) Express the slope of the buyer’s indirect expected payoff function, V (θ) ≡
U (θ, θ), in an incentive compatible direct mechanism.
(iii) What is the buyer’s participation constraint?
(iv) Noting that the monopolist’s expected revenue in an incentive compatible direct
mechanism with full participation is
Z
1 Z 1
Π =
(θ + ε) 1 dε − V (θ) dθ,
2
0
p(θ)−θ
formulate the monopolist’s problem (with full participation) as an optimal con-
trol problem. What are the state and control variables?
(v) Using a Hamiltonian and the Maximum Principle (or otherwise) derive p(θ) and
V (θ) in the optimal mechanism when r = 1 . [Hint: The participation constraint
4
is binding for the two extreme types.]
5449
TURN OVER
5
8.
[Dynamic mechanism design] Suppose that in a dynamic principal-agent
problem the agent’s privately-known type θt follows
θt = (1 − λ)Pt
λt−kε
k=0
k for t = 0, . . . , T ,
where εt, i.i.d. uniform on [0, 1] for all t, are the agent’s orthogonalized types. In
period t, having learnt θt (equivalently, εt) the agent takes unobservable action at,
which then generates a contractible public signal st = θt + at.
In what follows, we refer to the history of variables up to period t by superscript t,
e.g., εt = (ε0, . . . , εt) ≡ (ε0, εt− ). The agent’s payoff is
0
T
X
U (θT , aT , P ) = P −
(1 − θt)at,
t=0
where P is the total payment received from the principal.
(i) Suppose the principal offers a contract to the agent in period 0, after the real-
ization of θ0 ≡ (1 − λ)ε0. State the Revelation Principle.
(ii) Assume that a contract induces the agent to take action at(εt, st−1) in equi-
librium in period t after history (εt, st−1). Verify and explain what it means
for
a
ε
ε
, st−1) + (1 − λ)λt(ε
bt(εt, b0, st−1) = at(b0, εt
−0
b0 − ε0)
to be the agent’s lie-masking action at t, and compute
∂ at.
∂ε b
0
(iii) Write down the agent’s incentive compatibility constraint at t = 0 assuming
lie-masking actions for all t and truthful reports of ε1, . . . , εT .
(iv) Let V (ε0) denote the agent’s expected payoff in an incentive compatible mech-
anism with a given action and payment rule. Show that
V (ε0) = V (0) + (λT − 1)ε0
(using the incentive-compatibility constraint from part (iii)).
5449
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6
5450
DEGREE OF MASTER OF PHILOSOPHY
Master of Philosophy in Economics (Final Examination)
ADVANCED MICROECONOMICS I AND II (B)
TRINITY TERM 2014
Friday, 20 June, 2014 — 09:30 am to 11:30 am
Start the answer to each question in a separate booklet.
Answer TWO questions.
Candidates may use their own calculators
Do not turn over until told that you may do so
1
1. Consider the following two-person game G:
A
B
C
A
20,20
10,30
1,20
B
30,10
0,0
0,20
C
20,1
20,0
0,0
(i) Assume that both players are perfectly rational and forward-looking with dis-
count factor δ = 0.9. Construct a subgame perfect equilibrium of G∞ such that
in each period both players get 20. Be sure to specify the strategies after each
and every history.
(ii) Suppose that each player is a naive hypothesis tester, that is,
• at every time t, each player thinks that the opponent is using a stationary
strategy (pure or mixed)
• each player i chooses an action at according to a quantal response func-
i
tion, that is, with probability proportional to eui(at)/γi, where γi > 0 is i’s
response parameter
• every so often i tests his hypothesis against the most recent s plays of the
game
• i then accepts or rejects the current hypothesis
(a) Specify more precisely what criterion is used to accept or reject, and what
happens afterwards.
(b) Under what conditions on the parameters do players learn to play an ap-
proximate equilibrium? Explain what is meant by “learn” and “approximate”.
(c) Which equilibrium of G∞ do they learn in this case?
(iii) Discuss the differences between model (i) and (ii) with respect to the degree of
rationality of the players and the extent of their knowledge.
5450
2
2. A risk-neutral principal (P ) offers a contract to a risk-neutral agent (A). A’s ex
post payoff equals his payment minus his disutility of effort exerted, and his ex ante
reservation utility is zero. A’s disutility of effort e = 0 is 0 and of effort e = 1 is
d > 0. If e = 0, the value to P of A’s output, v, is 0 with probability 1. If e = 1, then
v = H with probability pH, v = L with probability pL, and v = 0 with probability
1 − pH − pL, where 0 < L < H.
(i) Suppose that A’s effort is publicly observable and contractible. What condition
is necessary and sufficient for P ’s optimal contract to have the form: if e = 0, pay A
zero, and if e = 1, pay A the amount d?
Henceforth, suppose that P cannot observe A’s effort and is restricted to offering a
contract specifying a single number k. After A chooses e from {0, 1} and v is observed
by P , P chooses between purchasing A’s output from A at price k and not purchasing,
in which case the output goes to waste. (A himself can make no use of the output.)
Assume that the condition derived in (i) holds.
(ii) What are the two types of inefficiency that might arise under a contract in this
model?
(iii) Under what conditions on the exogenous parameters of this model (H, L, d, pH, pL)
will P ’s optimal contract
(a) induce e = 1 and result in purchase by P when v ∈ {L, H}?
(b) induce e = 1 and result in purchase by P only when v = H?
(c) induce e=0?
For cases (a) and (b), what value of k will P ’s optimal contract specify?
(iv) Now, in addition, suppose that after the realization of v is observed by both
P and A, renegotiation can occur, according to the rule that P names a price, r,
at which he is willing to purchase A’s output. If A accepts, then the new price, r,
applies; if A rejects, then the old price, k, applies. How does the optimal contract
now depend on the exogenous parameters? How does the possibility of renegotiation
affect the efficiency of the outcome under the optimal contract in each of cases (a),
(b), and (c)?
5450
TURN OVER
3
3.
(i) Suppose (X, ≥) is a subcomplete sublattice of the lattice (L, ≥). Given an
increasing function φ : X → X, we define Y = {x ∈ X : x ≥ φ(x)}. Why is Y
nonempty? Let x∗ = inf Y . Show that (a) φ(x∗) is a lower bound of Y and (b)
φ(x∗) ∈ Y . Hence show that x∗ is a fixed point of φ and, in fact, is the smallest fixed
point of φ, i.e., if ˜
x is another fixed point of φ, then ˜
x ≥ x∗.
(ii) Let (X0, ≥) be another subcomplete sublattice of (L, ≥) and ψ : X0 → X0 an
increasing function on X0. Suppose that X0 ≥ X (in the sense of the strong set
order), ψ(x) ≥ φ(x) for all x ∈ X0 ∩ X, and there exists an element ˆ
x ∈ X such that
ˆ
x ≥ ψ(ˆ
x).
It follows from (i) that ψ has a smallest fixed point, which we shall denote by x∗∗;
indeed
x∗∗ = min{x ∈ X0 : x ≥ ψ(x)}.
Show that x∗∗ ∈ X and that x∗∗ ≥ x∗.
(iii) Consider a game with a finite set of players I = {1, 2, ..., m}. Each player i ∈ I
has a strategy set Si = [ai, bi], which is a compact interval of the real line. We denote
a profile of strategies by the vector s, with s = (s
m
1, s2, ..., sm) ∈ R
. The payoff
of player i when he is playing si and the other players are playing s−i is ui(si; s−i).
Assume that ui is a continuous function of si and ui obeys single crossing differences
in (si; s−i).
(a) Show that the map from s−i to min [arg maxs
u
i∈Si
i(si; s−i)] ∈ Si is an increasing
function of s−i.
(b) Use the results in (i) to guarantee that this game has a Nash equilibrium s∗
that is smaller than any other Nash equilibrium in this game.
(c) Suppose that there is a change in the strategy set of Player 1 from [a1, b1] to
[¯
a1, b1], where ¯
a1 > a1. The payoff function of Player 1, and the payoff functions
and strategy sets of the other players remain unchanged. Let s∗∗ be the smallest
Nash equilibrium in this new game. Show that s∗∗ ≥ s∗.
5450
4
4. Assume that we are about to watch a horse race and there are n states, with each
state corresponding to a particular horse winning the race. A bet is a function from
states to monetary outcomes, i.e., x ∈ X ⊂
n
R , with xi denoting the net gain in the
case where horse i wins. Suppose the preference on X is represented as follows:
V (x) = min p · x,
p∈P
where x is a bet, p ∈ ∆n−1 with ∆n−1 denoting the set of probability vectors on
{1, ..., n}, and P is a closed, convex set of such probability vectors.
(i) What does the form of V (x) imply for the individual’s attitude toward risk and
ambiguity?
(ii) Let A1, A2,..., A6 be six axioms on such that the following theorem is true:
The preference on X satisfies A1-A6 if and only if there exists a unique,
non-empty, closed convex set P ⊆∆n−1 such that, ∀x, y ∈ X,
x y ⇔ V (x) ≥ V (y).
(a) Write down such a set of axioms A1, A2,..., A6 and explain, briefly, the
intuitive content of each axiom.
(b) Sketch a proof of the theorem.
5. Explain qualitatively how optimal bidding differs between uniform-price auctions
and discriminatory auctions for multiple units of a homogeneous product, when each
bidder has private values and desires multiple units. Which auction form is likely to
(i) be more robust against collusion, (ii) result in more bidders participating, (iii) be
more efficient. Explain your answers.
5450
LAST PAGE
5
5451
DEGREE OF MASTER OF PHILOSOPHY
Master of Philosophy in Economics (Final Examination)
DEVELOPMENT ECONOMICS 1
TRINITY TERM 2014
Saturday 21st June 2014, 14:30 - 16:30
Please start the answer to each question on a separate page.
Answer TWO questions.
Candidates may use their own calculators.
Do not turn over until told that you may do so.
1
1. In her analysis of technology adoption by Kenyan farmers, Suri (2011) estimates a
model of the following form (where yit is log output by farm i in period t, and hit is a
dummy for hybrid seed adoption by farm i in period t):
yit = δ + βhit + θi + φθihit + γt + uit.
(a) In what sense does this model nest a standard fixed-effects specification? In your
view, what are the main advantages to generalising a fixed-effects specification in
this way? What are the main disadvantages?
(b) Suri estimates φ < 0. What does this imply about sorting and heterogeneous
returns to using hybrid seed? Explain intuitively what factors might drive this
result among small Kenyan farms.
(c) Suppose that a researcher has surveyed 1000 small manufacturing firms in urban
Ghana. She notices that, of the 1000 firms, 250 are using a new mobile phone
‘app’ that is designed to allow firm managers to keep track of firm revenues and
costs. The researcher notices that those 250 adopting firms have significantly
higher average investment and significantly higher average profits that than those
not adopting, and report significantly less pressure from family and friends to
share firm profits. The researcher considers this a perplexing result: average
returns to the new technology appear positive, but most firms have not adopted.
In your view, what are the main competing hypotheses that the researcher should
consider in order to explain this result?
(d) Advise the researcher on a design for a randomised controlled trial, using the
sample of 1000 small firms, to test between your main competing hypotheses.
5451
2
2. Consider the model from Murphy, Shleifer and Vishny (1989). There is a continuum
of sectors producing a quantity of goods x indexed by q ∈ [0, 1]. A representative
consumer’s preferences are represented by the utility function U = R 1 ln x(q) dq, where
0
x(q) is the amount consumed of the good produced by sector q. Each good is produced
in its own sector, and each sector has two types of firms. Cottage producers produce
one unit of output with one unit of labour and operate under perfect competition.
Each sector also has a single potential industrial firm with access to increasing returns
technology, which operates as a monopolist in its sector if it exists. Industrialisation
requires a fixed cost investment of F units of labour and produces α > 1 units of
output with every additional unit of labour. Total labour is fixed at L units.
(a) Explain the role of transportation costs in this model.
(b) In the basic setup of the model, above, is it possible to have multiple equilibria
(i.e. an equilibrium in which all sectors industrialise and an equilibrium in which
no sectors industrialise)? Explain.
(c) Murphy, Shleifer and Vishny consider the case where workers in industrialised
sectors are paid (1 + v) instead of 1 for workers in cottage production. How does
this change the results and why?
(d) In the real world, we observe a high degree of spatial concentration of economic
activity, and there are many other potential reasons beyond demand spillovers.
Interpret this fact using Kremer’s (1993) ‘O-Ring theory’.
(e) Greenstone, Hornbeck and Moretti (2010) use a creative empirical strategy to
estimate agglomeration economies. Discuss why it is difficult to estimate such
effects, the identifying assumption(s) that are required to consider unbiased their
estimates of productivity spillovers, and any concerns about the assumption(s).
5451
3
turn over
3. A household is composed of two adults (A, B) and one child. The total household
expenditure is given by x. In this household, only the two adults make decisions:
we can therefore model expenditure on the child as a public good (Q). Apart from
deriving utility from their child, the two adult members derive utility from consuming
two private goods: qA is the amount of private good j consumed by adult A, and qB
j
j
is the amount of private good j consumed by adult B, with j ∈ {1, 2}. Assume that
A’s individual preferences are given by
uA = α1 ln qA + α
+ α
1
2 ln qA
2
3 ln Q,
and that B’s individual preferences are given by
uB = β1 ln qB + β
+ β
1
2 ln qB
2
3 ln Q,
with α1 + α2 + α3 = 1, β1 + β2 + β3 = 1 and αk, βk > 0 ∀ k. Assume that the household
only makes consumption decisions. For simplicity, assume that the price of every good
is 1.
(a) Assuming efficiency, and letting µ be the Pareto weight of member A, write down
the household program that determines the conditional sharing rule for each adult
member (xA and xB), and the level of child expenditures (Q).
(b) Explain, intuitively, how this optimal conditional sharing rule can be reached by
a two-stage decision process.
(c) Suppose that adult A is the mother, and adult B is the father. What would be the
effect of an increase in mother’s conditional sharing rule on child expenditures?
Why? (Note: You do not need to derive any further results here; focus on the
intuition of the household programme.)
(d) What are the implications of this model for child expenditures regarding income
pooling? How would you test these implications with actual data? Which regres-
sion(s) would you run? What are the challenges to identification?
5451
4
4. Let y denote household income, x a set of consumption goods, p their prices and V (y, p)
consumer indirect utility. Let q be a different set of prices, and yE the equivalent
income, such that V (p, y) = V (q, yE). Finally, let us define a poverty line z such that
anybody with y < z is poor and the equivalent poverty line zE such that anybody
with yE < zE is poor.
(a) Foster et al (1984) have proposed a class of poverty measures:
Z
z z
α
P
E − yE
α =
f (y) dy.
z
0
E
What are the properties of Pα when α = 0 and α = 1 and α > 1?
(b) The government is considering providing a small quantity xs of good i at a subsi-
i
dized price qi < pi to every household. What is the effect on poverty of a marginal
increase in xs? Give a simple interpretation of the effect on poverty when α = 1.
i
(c) Alternatively, the government could implement a subsidy si on every unit of good
i purchased. Let xi denote the consumption of good i and qi its price. What is
the effect on poverty of a marginal increase in the subsidy? (Note: The implicit
function theorem and Roy’s identity imply that xi = − ∂yE .)
∂qi
(d) The government turns to you for advice on the best way to reduce P1 for a given
budget increase ∆B. Which of the two policies would you recommend?
(e) The government chooses to provide a ration xs of the good i at a subsidized
i
price qi < pi and delegates its allocation to local (corruptible) bureaucrats. The
population size is 100. It is composed of 50 poor and 50 rich households. Let yR
denote the income of the rich and yP the income of the poor. The rich can afford
the ration at the market price, but the poor can only afford it at the subsidised
price yR > pixs > y
. Each bureaucrat is given 50 rations of goods with
i
P > qixs
i
the mission to sell one ration to each poor household. Bureaucrats can impose
t units of testing to people who apply for the ration. Testing is pointless, i.e. it
does not actually allow bureaucrats to find out who is poor and who is rich. We
assume that utility is linear. Each unit of testing imposes a non-monetary utility
cost δ to applicants and ν to bureaucrats. What would an honest bureaucrat do?
What would a corrupt bureaucrat do? How does corruption affect the allocation?
5451
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last page
5452
Master of Philosophy in Economics (Final Examination)
Master of Science in Economic and Social History (by coursework)
DEVELOPMENT ECONOMICS 2
TRINITY TERM 2014
Thursday 19 June 9:30am – 11:30am
Please start the answer to each question in a separate booklet.
Candidates should answer
TWO questions.
Do not turn over until told that you may do so.
1
1. In the historical development of today’s rich countries, the manufacturing shares of
employment and GDP first rose with income per capita and then fell, giving rise to a hump-
shaped pattern. What are some possible explanations for this pattern? Do the same
explanations also seem appropriate in describing the patterns of manufacturing that we
observe in today’s developing countries?
2. Who are the likely losers from trade policy liberalization in a developing country? Should
any of them get government assistance, and if so of what sort?
3. How do ethnic institutions such as pre-colonial state centralization of chieftaincies shape
modern development?
4. “The elegance and tractability of the New Keynesian open economy macroeconomic model
increasingly employed by central banks in developing countries draws policymakers’
attention away from the particular challenges of successful macroeconomic stabilization in
low income countries.” Discuss.
5. The transaction-cost approach to institutions, and the Coase Theorem that lies at its core, is
more useful to understand inefficient than efficient institutions. Explain and illustrate.
6. “Secure land tenure is a necessary condition for long-term increases in agricultural
productivity.” Discuss, drawing on both theoretical arguments and empirical evidence.
7. Consider a simple model of ROSCA participation with random allocation of the pot.
a. Explain how credit constraints can be overcome in this model.
b. Under perfect credit markets the Permanent-Income Hypothesis (PIH) provides
predictions for life-time consumption and savings. How would you test the PIH
model using household data?
c. Suppose half the sample had been randomly allocated to a ROSCA: how would you
expect the income coefficients in your specification to be affected?
8. In designing an ideal package of external financing for development, how serious would be
the trade-offs between long-term growth, short-term stability and distributional impact?
5452
LAST PAGE
2
5453
DEGREE OF MASTER OF PHILOSOPHY
Master of Philosophy in Economics (Final Examination)
ECONOMIC HISTORY 1
TRINITY TERM 2014
Wedneday 18th June 2014, 14:30 - 16:30
Please start the answer to each question on a separate page.
Answer TWO questions, ONE from Part A and ONE from Part B.
In the answers, explain more than one interpretation or analysis,
and assess the evidence and methods used.
Candidates may use their own calculators.
Do not turn over until told that you may do so.
1
Part A
1. Was Europe more advanced economically than Asia during the eighteenth century
or did the European lead emerge later? Discuss market performance, income levels,
and any other aspects of economic, political, or social behaviour that you think are
relevant.
2. How has globalization since 1500 shaped institutions? Provide examples from both
Western and non-Western societies.
3. How well does the Malthusian model describe the world before 1700?
Part B
4. Why did the Industrial Revolution happen in Britain rather than elsewhere (for in-
stance, in France, the Netherlands, or China)?
5. Did the slowdown of the growth rate of the USSR post-1970 prove that Big Push
industrialization was not a viable development strategy?
6. “The indirect effects of geography on development persist even after their direct effects
have disappeared.”
How convincing is the empirical evidence for this claim and does it have any theoretical
basis?
5453
2
last page
5455
DEGREE OF MASTER OF PHILOSOPHY
Master of Philosophy in Economics (Final Examination)
INDUSTRIAL ORGANISATION 1
TRINITY TERM 2014
Monday 23rd June 2014, 09:30 - 11:30
Please start the answer to each question on a separate page.
Answer TWO questions.
Candidates may use their own calculators.
Do not turn over until told that you may do so.
1
1. A single-product monopolist has constant marginal cost of production c and faces
demand function q(p). Net consumer surplus at price p is
Z
∞
v(p) =
q(˜
p)d˜
p .
p
The firm chooses its price to maximize its profit q(p)(p − c). For each statement
below, say whether the claim is true or false, justifying your answers by proof or
counterexample.
[You can assume throughout that q(c) > 0 so that the firm can make positive profit.]
(a) If the demand function increases, the firm’s equilibrium profit increases.
(b) If the demand function increases, equilibrium consumer surplus increases.
(c) If the demand function becomes more elastic, the firm’s equilibrium profit de-
creases.
(d) If marginal cost falls, equilibrium total welfare (that is, profit plus consumer
surplus) increases.
(e) If the demand function is linear, equilibrium profit is greater than equilibrium
consumer surplus.
(f) If the demand function is concave, equilibrium profit is greater than equilibrium
consumer surplus.
2. A firm (firm 1) operates on two markets, A and B. On market A, it is a monopolist,
and its marginal cost is constant and equal to cA. On market B, firm 1 competes with
n ≥ 2 other firms whose (common) marginal cost cBn is lower than that of firm 1, cB1.
There is a unit mass of homogeneous consumers, whose willingness to pay is vA > cA
for product A, and vB > cB1 for product B. Let us assume that vA − cA ≥ cB1 − cBn.
The timing is the following:
t = 1: Firm 1 announces whether it bundles products A and B or not.
t = 2: All firms choose a price for their products: pBi for firm i > 1, and, for firm 1,
either pAB if it chooses bundling, or (pA1, pB1) if it does not.
(a) Show that bundling is not a profitable strategy for firm 1.
Now suppose that the n competitive firms on market B must pay a fixed cost F to
enter, where F < cB1 − cBn. The new timing is as follows:
t = 1: Firm 1 announces whether it bundles products A and B or not.
t = 2: Firms i > 1 simultaneously choose whether to enter and pay the fixed cost.
t = 3: All active firms choose a price for their products: pBi for firm i > 1, and, for
firm 1, either pAB if it chooses bundling, or (pA1, pB1) if it does not.
(b) Show that bundling is profitable. Would total welfare increase with a ban on
bundling?
(c) What is the crucial assumption of the model that makes bundling profitable?
Briefly comment. Can you think of real world examples where the assumption is
reasonable, and others where it is not?
5455
2
3. Consider a market for a homogeneous good with one incumbent, I, and one entrant, E.
The entrant’s marginal cost cE is common knowledge. The incumbent’s marginal cost
cI is unknown to the entrant. With probability α, cI = cH > cE, and with probability
1 − α, cI = cL < cE. To enter, E has to incur a fixed cost F ∈ (α(cH − cE), cH − cE).
At each period, there is one (different) consumer whose willingness to pay is 1. The
timing is as follows:
t = 1: The incumbent sets a price p1. The consumer buys if and only if p1 ≤ 1. The
entrant observes p1 and forms beliefs about cI: β(p1) ≡ Pr[cI = cH|p1].
t = 2: The entrant decides whether to enter or not. The incumbent observes the
entrant’s decision.
t = 3: If entry has occurred, both firms observe both marginal costs and compete
in prices (pI2, pE2). If the entrant has not entered, the incumbent acts as a
monopoly.
(a) Suppose first that the entrant observes cI in t = 1. What is the subgame-perfect
equilibrium of the game? Write down the expected total welfare, profits and
consumer surplus.
From now on we assume that E observes cI only at t = 3.
(b) Under which conditions is a pair (pL, pH) a separating equilibrium? (pL (resp.
pH) being the price charged by I in period 1 if its cost is cL (resp. (cH)).
Focus on equilibria with beliefs such that p1 /
∈ {pL, pH} ⇒ β(p1) = 1.
[Hint: show first that pH must be equal to 1.]
(c) In what sense can we talk about ‘limit pricing’ ? Compared to (a), does limit
pricing increase or decrease total welfare/profits/consumer surplus?
(d) Under which conditions is a price pLH a pooling equilibrium?
Focus on equilibria with beliefs such that p1 6= pLH ⇒ β(p1) = 1.
(e) In what sense can we now talk about ‘limit pricing’ ? Compared to (a), does limit
pricing increase or decrease total welfare/profits/consumer surplus?
5455
3
turn over
4. Consider a homogeneous goods market with N identical firms and S identical con-
sumers. The profit of a firm in the market is given by
Π(N ) = SV (N ) − F
where F is fixed costs. V (N ) is the firm’s per-capita variable profit. There is free
entry so that the number of entering firms satisfies
Π(N ) ≥ 0 ≥ Π(N + 1).
(a) Suppose there are M markets (m = 1, . . . , M ) and we observe S and N for
each market. Assume fixed costs can be specified F = β + εm where β is a
parameter and εm is a market-specific fixed cost effect. εm is unobserved to the
econometrician, but its probability distribution is known. Explain how the model
may be estimated using the assumption of free entry.
(b) Let V (N ) be defined as follows:
q(p(N ))
V (N ) = (p(N ) − c)
N
where p(N ) is price, and q(p(N )) is per-capita demand for the product. c is
(constant) marginal cost. Define the ‘threshold’ market size SN as the minimum
market size required to support N firms. Discuss whether the relationship between
SN and N is informative about the nature of price competition.
(c) Discuss the estimation issues that arise if the assumption of symmetric firms is
relaxed.
5455
4
5. Consider a market with J differentiated products (j = 1, . . . , J ) with characteristics
xj, price pj, and quality ξj. Suppose consumer i’s willingness to pay for product j is
given by:
uij = βxj − αpj + ξj + εij
≡ δj + εij
where δj is the deterministic part of utility and εij is random utility. If εij is distributed
according to a Type-1 Extreme Value distribution, then the probability of choosing
product j is
exp(δ
s
j )
j = PJ
exp(δ
k=0
k )
where the consumer is assumed to have an outside option (k = 0) with δ0 = 0.
(a) Provide an interpretation of the parameters β and α.
(b) Suppose ξj and εij are unobserved by the econometrician. Explain whether these
parameters can be estimated using observations of sj, xj, and pj.
(c) Discuss the substitution patterns that are implied by the model when prices
change.
(d) Discuss whether it is possible to generalize the model to allow more flexible cross-
price effects.
6. A competition authority has been asked to consider a merger of two firms operating
chains of outlets in a number of cities. It has access to data for a number of time
periods. This data gives information on the prices at individual outlets for all firms
(including those that are not party to the merger), and the ownership of the outlets.
How should the data be used to assess the merger? Consider the cases where (i)
the merger is a hypothetical one that has not yet happened, and (ii) the merger has
already happened and the data includes observations before and after the merger.
5455
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last page
5456
5446
DEGREE OF MASTER OF PHILOSOPHY
Master of Philosophy in Economics (Final Examination)
Probationer Research Student Qualifying Examination in Economics
LABOUR ECONOMICS
TRINITY TERM 2014
Friday 20th June 2014, 14:30 - 16:30
Please start the answer to each question on a separate page.
Answer TWO questions.
Candidates may use their own calculators.
Do not turn over until told that you may do so.
1
1. Consider a standard labour market matching model as described by Pissarides (2000).
Workers and firms meet in the labour market according to a constant returns matching
function m(u, v) where u and v are the numbers of unemployed workers and vacancies
respectively. All matches produce a constant flow of output p and are destroyed at
exogenous rate δ.
The labour force is of size 1, and there is free entry of firms.
Unemployed workers receive a flow of utility b; a firm with a vacancy incurs a flow
cost c; and the real interest rate is r. The wage, w, is determined by a Nash bargain, in
which workers and firms have bargaining power β and 1 − β. Labour market tightness
is denoted by θ = v/u.
A steady-state equilibrium (w, θ, u) satisfies the following equations:
p − w
c = q(θ)
where
q(θ) = m/v
(1)
r + δ
r + δ + λ(θ)
w = b + (p − b)β
where
λ(θ) = m/u
(2)
r + δ + βλ(θ)
δ(1 − u) = λ(θ)u
(3)
(a) Explain carefully the basis for each of the equations (1) to (3). (There is no need
to derive them formally.)
(b) Show diagrammatically or algebraically how the equilibrium is determined. Is it
necessarily unique?
(c) Suppose that an unexpected shock permanently raises workers’ bargaining power,
β. How does the steady-state equilibrium change? Describe the dynamic be-
haviour of the labour market during the adjustment toward the new steady state.
(d) How would you expect the rise in β to affect aggregate income? Explain.
(e) Briefly assess the usefulness of this model for policy analysis.
2. How far is evidence from the estimation of earnings functions consistent with the
hypothesis that wages are competitively determined?
3. Under what circumstances is promotion in a jobs hierarchy a desirable alternative to
monetary bonuses as an incentive device? To what extent does the optimal promotion
scheme match empirical evidence on internal labour markets?
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4. To what extent can contracting models based on risk sharing, and relationship-specific
investment, explain the following:
(a) rigidity of wages, without complete inflexibility;
(b) sensitivity of wages to market conditions at hiring;
(c) sensitivity to the best market conditions since hiring;
(d) observation of particularly strong rigidity in the case of nominal wage cuts;
(e) a tendency to specify contracts in nominal rather than real terms?
5. How can we explain the use of fixed wage contracts? Do they give rise to inefficient
unemployment?
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DEGREE OF MASTER OF PHILOSOPHY
Master in Philosophy in Economics (Final Examination)
Probationer Research Student Qualifying Examination in Economics
THEORY BASED EMPIRICAL ANALYSIS
TRINITY TERM 2014
Tuesday, 24 June 2014, 9.30–11.30
Please start the answer to each question on a separate sheet.
Candidates should answer TWO questions.
Candidates may use their own calculators.
Do not turn over until told that you may do so.
1
1. Either:
(a) List the elements of a structural model. Give an illustration (do not use a duopoly
example).
(b) We have a linear model with two right hand side variables. One of the variables
is known to be exogenous but the other one is endogenous. We have one excluded
variable (a potential instrument). Derive the exact conditions under which we can
identify the parameters of the model. Relate this to an (auxiliary) regression of the
endogenous variable on the instrument and the exogenous variable.
(c) Consider a model in which an unobserved parameter p (between 0 and 1) gives rise
to an observed outcome, y, for a given environment. The outcome is …rst increasing
in p, then constant and …nally decreasing in p. Draw the mapping from p to y
and outline the identi…cation issues if everyone faces the same environment and has
the same value for p. Then discuss the identi…cation issues that arise if agents are
heterogeneous in their unobserved parameter and we have one cross-section. Finally
discuss how having two waves of a panel changes the identi…cation.
or:
(a) We have a nonlinear model: y = g (x) + u. The object of interest is g (x). We
suspect E (u j x) 6= 0 but we have a variable z which is believed to be independent
of u. Describe how you would consider the identi…cation of g (x) if z and x are both
discrete (that is, take on a …nite number of values).
(b) A set of agents with the same gross real wage face a budget set that includes a
government bene…t for low earners (which is gradually reduced if earnings are above
a certain threshold) and a tax on earnings for high earners. The object of interest is
the distribution of the taste for leisure with respect to net earnings. Draw the budget
set and discuss the identi…cation issues. If the government increases the bene…t and
the tax on high earners, discuss whether we can point identify the proportion of
gainers and losers?
(c) We can always set identify the parameters of a parametric model, so point identi…-
cation is no longer an important issue. Discuss.
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2. Either:
Answer any 3 from the following 5 questions:
(a) State the revealed preference conditions for weak separability of preferences. Provide
a brief intuition for these conditions.
(b) Describe one way in which empirical revealed preference analysis might incorporate
optimisation errors by either …rms or consumers.
(c) What limits the power of the Generalised Axiom of Revealed Preference to detect
non-rational behaviour. In light of this how should a successful empirical test of
GARP be interpreted?
(d) According to Afriat’s Theorem if there exists a non-satiated utility function that
rationalises the data there also exists a nonsatiated, continuous, monotone, and
concave utility function that rationalises the data. Explain why this is the case.
(e) How can empirical revealed preference methods be used to investigate returns to
scale?
or:
How might the Generalised Axiom of Revealed Preference be used as the basis of the
measurement of taste change?
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Turn Over
3. Consider a model where agents have three options in each period: work, school and home.
We denote these by j = 1; 2; 3. Assume that an agent has the following per-period utility
function:
C
U
ijt
ijt =
h
it ijt
itsijt +
"ijt for t = 1; : : : ; T
(1)
Here, Cijt is consumption of person i at time t, conditional on choosing option j. The
< 1 is the risk aversion parameter. The other variables are de…ned as follows: h is an
indicator for work, so hijt = 1 i¤ j = 1, and 0 otherwise. Similarly, s is an indicator for
school, so sijt = 1 i¤ j = 2, and 0 otherwise. The parameters
and
it
it are the disutilties
of work and school, respectively. Finally, the "ijt are type 1 extreme value errors that are
independent both over time and across alternatives, and
is a scaling parameter. Let
denote the discount factor
Next we turn to the budget constraint. Assume that there is no borrowing or saving,
so that consumption is exactly equal to current income in each period. Conditional on
choice of option j, consumption is given by:
Cijt = withijt + Gsijt + B(1
hijt
sijt)
(2)
Here wit is the wage rate, G is a transfer the person receives while in school, and B is an
unemployment bene…t a person receives if they are home.
The last part of the model is the wage function. We assume that observed wages are
determined by the function:
ln (wo ) =
(3)
it
0 +
1Sit +
2Hit +
it
Here wo is the observed wage, S
it
it is total years of school and Hit is total years of work. The
error term
is assumed to be pure measurement error. Thus, true wages are assumed
it
to be given by the deterministic process ln (wit) = 0 + 1Sit + 2Hit, and
is irrelevant
it
to the agent’s decisions.
Let dijt be an indicator function denoting choices, such that dijt = 1 i¤ agent i chooses
option j at time t, and 0 otherwise. The laws of motion for Sit and Hit are:
Hi;t+1 = Hit + di1t
and
Si;t+1 = Sit + di2t
(4)
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(a) What is the complete set of state variables for this problem? Approximately, how
many possible states are there at the start of period T ?
(b) Write out the current period payo¤s associated with each of the three choices.
(c) What is the form of the “Emax” or expected value functions at time T ?
(d) Write out the Bellman equations (not generic but speci…c to this problem) and
explain how to backsolve the dynamic programming (DP) problem.
(e) Is this model set up as a Rust style model or an Eckstein-Keane-Wolpin style model?
Why?
(f) What are the advantages of assuming that the random component of wages is pure
measurement error? What would happen if instead we assumed it was a productivity
shock that agents see each period before they make their choices?
(g) Assume the econometrician cannot observe wage o¤ers unless they are accepted (i.e.,
unless the person chooses to work). Does this cause any di¢ culty or complication in
estimating this model? In particular, can we estimate the wage function consistently
in a …rst stage, using the data on observed wages, schooling and work experience?
(h) How would your answer to (g) change if we consider the alternative model with
productivity shocks to wages?
(i) Given you have a solution to the DP problem in hand, write out the probabilities
for each of the three choice options (conditional on the state).
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