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EC9011
THE UNIVERSITY OF WARWICK
January Examinations 2013/2014
Economic Analysis: Microeconomics

Time Allowed: 2 Hours
...

Read carefully the instructions on the answer book provided and make sure that the particulars
required are entered on each answer book
...


Consider a consumer with a Cobb-Douglas utility function over three goods of the form
u  ( x1  z1 ) ( x2  z2 ) ( x2  z3 ) with       1 , and zi ,  ,  ,   0
...

(a)

Find the Marshallian demands for goods 1, 2 and 3 for this consumer, using the
Lagrangean method
...
(15 marks)

(c)

Find the indirect utility function of the consumer
...
(15 marks)

(d)

Using your answer in (b), find the expenditure function for the consumer
...
(15 marks)

(e)

Using your answer in (d), determine whether the goods are Hicksian complements
or substitutes
...
Define the
compensating variation of this price increase to be

CV  e( p '1 , p2 , p3 , u)  e( p1 , p2 , p3 , u)
where e( p1 , p2 , p3 , u) is the expenditure function in part (d)
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What is the economic interpretation of CV? (25 marks)

1
(Continued)

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2
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How can
hyperbolic discounting by individuals give rise to self-control problems? Illustrate with
an example
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Consider the following portfolio design problem
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The safe asset pays a return of zero
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The
c1
investor has a utility function , u (c) 
,   0 , where c is the final wealth (and
1
consumption) of the investor
...
(15 marks)

(c)

Using your answer to (b), find conditions on W, p and r such that the investor
will invest a share s strictly between 0 and 1 in the risky asset
...
(15 marks)

(e)

How does this value of s change with W, p, and r ? Give an economic intuition
for your findings
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Derive the expected utility of the investor, V, as a function of s, and the
parameters of the problem, W, p and r
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Explain how this
result generalizes to the case of n risky assets
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In state of the world 1, Alice can grow 1  a
units of rice, and in state of the world 2, her crop fails and she can grow nothing
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Both farmers believe that state of the world 1 will occur with
probability  , 1    0
...

p1
p2

(Question (4) continued overleaf)
2
(Continued)

EC9011

(Question (4) continued)

(a)

What does it mean for there to be complete contingent markets for rice in this
economy? Do we observe complete contingent markets in practice? If not, why
not? (15 marks)

(b)

Assume that there are complete contingent markets for rice
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You may take good 2 as the
numeraire
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e
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(20 marks)

(d)

Find the values of a, b,  such that both farmers are fully insured, i
...
have the
same levels of consumption in both states of the world
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Find the relative price of these two securities in equilibrium
...
(25 marks)

SECTION B
(Figures in brackets indicate the mark allocated to each part of the question out of a total of 100)
5
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(20 marks)

(b)

Derive all the pure-strategy Nash equilibria of the following strategic-form game,
involving two players, Jack and Gill
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Let dG and d J denote the demands of Gill and Jack,
respectively, where 0  dG  10 and 0  d J  10
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However, if
dG  d J  10 , then Gill and Jack fail to agree on how to divide the ten dollars, and
each of them gets a payoff of zero
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The total amount of capital invested
in country 1 is k1  1  t1  0
...
5t1
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(i)
(ii)

6
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(25marks)

Explain the differences between the First-Price and the Second-Price Auctions
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There are two buyers who wish to buy it
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Furthermore, these are drawn from a
uniform distribution
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(c)

7
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Firm A's
constant marginal cost of production is one, and this is common knowledge
between the two firms
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It is either one or two
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These beliefs are common knowledge
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Derive the Bayes-Nash equilibrium to this game
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First the union makes a wage demand, w
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The firm's
1
payoff is   6L2  wL , and the union's payoff is u  (w   ) L , where   0
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(30 marks)

(ii)

Show that the equilibrium level of employment is inefficient
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Consider a market with two firms, A and B , each of whom produces the
homogeneous good at a constant marginal cost of 1
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Assume that, in each period, market
demand is q  5  p
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In each period, the firms simultaneously set quantities (a la
Cournot), after observing the quantities that they set in all preceding periods
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Derive a subgame perfect
equilibrium in which, along the equilibrium path, each firm sets quantity equal to
one-half of the monopoly quantity in each period, and state the condition on the
discount factor under which this equilibrium holds
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(a)

Consider the following simple version of Spence's job signalling model
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The productivity of a worker in a job is 
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The total cost of obtaining education level e is e / 
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(i)

(ii)

(b)

Find the separating perfect Bayesian equilibrium with the lowest
education level
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If the
project is not a success ( s  L ), the firm's revenue is zero
...

Should the firm pay wage w to the worker, the firm's expected payoff is
 f (w, e)   ev  w , while the worker's payoff is  w (w, e)  w  0
...
Assume

  0 , c   2v and that the worker's outside option has value equal to zero
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(10 marks)
(Question (8) continued overleaf)

5
(Continued)

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(Question (8) continued)

(ii)

Suppose the firm does not observe effort e , but that the project's success
is verifiable
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Assuming the worker chooses e to maximise expected utility, show that
setting wH  wL  v implies the worker will choose the jointly efficient
effort level
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(25 marks)

6
(End)



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