SUMMER TERM 2015
ECON2601: Economics 2 (Combined Studies)
TIME ALLOWANCE: 3 hours
Answer ALL questions from Part A, one from Part B and one from Part C. All answers should be
accompanied by a brief explanation or discussion. Correct but unexplained answers will not receive
high marks.
Questions in Part A carry five per cent of the total mark each and questions in Part B and Part C
carry twenty-five per cent of the total mark each.
In cases where a student answers more questions than requested by the examination rubric, the policy
of the Economics Department is that the student’s first set of answers up to the required number will
be the ones that count (not the best answers). All remaining answers will be ignored.
PART A
Answer ALL questions from this section.
3
1
A1 Suppose that your utility function for two goods is u(x1 , x2 ) = x14 x24 . The prices of the two
goods are p1 = 1 and p2 = 1 (in £ per unit) with exogenous income m = 100 (in £). What are
your optimal consumption bundle and maximised level of utility? Suppose that consumption of
good 1 beyond the level x1 = 50 is subject to a specific tax of £1 per unit. What is the budget
constraint with this incremental specific tax? What are the optimal consumption bundle and
maximised level of utility with this incremental specific tax?
A2 Define a homothetic function in terms of its relation to a homogenous function of degree one. One
way to verify that a utility function, u(x1 , x2 ), is homothetic is to make sure that its marginal
rate of substitution (MRS) at some arbitrary point (x1 , x2 ), where xi > 0 for i = 1, 2, is the same
as the MRS at point (tx1 , tx2 ), where t is a positive constant. Using this definition, confirm that
1
the constant elasticity of substitution utility function, u(x1 , x2 ) = α1 xδ1 + α2 xδ2 δ , for δ ≤ 1,
δ 6= 1, and αi > 0, is homothetic. Is the utility function u(x1 , x2 ) = ln x1 + x2 homothetic?
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pα−1
A3 A consumer has indirect utility function v(p1 , p2 , m) = m 2pα , where pi is the price of good i,
1
i = 1, 2, m is the exogenous income, and 0 < α < 1 is a constant. Invert the indirect utility
function to obtain the expenditure function, e(p1 , p2 , u), where u is a target utility level. Use
Shephard’s lemma to obtain the Hicksian demand functions, h∗i (p1 , p2 , u), from the expenditure
function. What are the Marshallian demand functions, x∗i (p1 , p2 , m)?
A4 An agent has endowment (ω1 , ω2 ) and is a net consumer of good 1 at current prices. If the price
of good 1 increases, ceteris paribus, and the consumer remains a net consumer of good 1, then
will she be better off? Will she necessarily switch to becoming a net supplier of good 1? Now,
suppose that the price of good 1 decreases instead and the consumer switches to being a net
supplier of good 1. Would she be worse off? Explain your answers using the weak axiom of
revealed preference (WARP).
A5 You enjoy gambling and have a utility-of-wealth function u(Y ) = eY , where Y is your wealth.
Suppose that your initial wealth is Y0 = ln 10. You visit a casino that takes only a single bet
of exactly ln 10 on a game that results in a payoff of either Z1 = − ln 10 or Z2 = ln 10 with
equal probability. What is your terminal wealth, Y0h + Z1 or
i Y0 + Z2 , in each outcome? What
is your expected terminal wealth from gambling, E Y0 + Z̃ ? What is your expected utility of
h i
h
i
wealth, E u Y0 + Z̃ ? What is your utility of expected wealth, u E Y0 + Z̃ ? What is the
minimum amount by which your friends will have to bribe you to prevent you from gambling?
A6 In an exchange economy, can one agent be made strictly better off if we are already at a Paretoefficient allocation? If we are not at a Pareto-efficient allocation, can we make all agents strictly
better off by trading to a Pareto-efficient allocation? Explain your answers.
A
A
A
A7 In a two-agent economy, agent
A has utility function uA xA
1 , x2 = x1 + x2 and agent B has
B
B
B
B
B
utility function u x1 , x2 = max x1 , x2 . Agent A starts with an endowment of (1/3, 1/3)
and Agent B starts with an endowment of (2/3, 2/3). (a) Describe the exchange economy
graphically using an Edgeworth box showing the initial endowment position and the indifference
curves of both agents. (b) Is the agents’ initial allocation Pareto-efficient? Justify your answer.
A8 Consider the following game of one-sided bargaining offers. Player A offers player B a share of 1
dollar and B may either accept the offer (in which case the game ends) or reject it. If player B
rejects then player A makes a second offer in the next time period which player B can also either
accept (in which case the game ends) or reject. If player B rejects both offers the game ends one
period later with exogenous payoffs of s to player A and 1 − s to player B. The discount factor
for both players is δ. Determine the outcome of the game for 0 < δ < 1.
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A9 There are two firms in a Stackelberg duopolistic industry competing in quantity (output) space.
Firm 1 is the market leader and produces output level q1 at constant marginal cost k1 . Firm 2
is the follower and produces output level q2 at constant marginal cost k2 . The market demand
schedule is given by p = a − b(q1 + q2 ), where p is the price per unit and a and b are positive
constants with a > max(k1 , k2 ). What is firm 1’s optimal output in a Stackelberg duopoly if
the firms have equal technologies, i.e. k1 = k2 ? How would firm 1’s strategy change if firm 2’s
technology was superior to firm 1’s, i.e. k2 < k1 ? How would it affect the price per unit?
A10 Consider the duopolistic industry as described in A9, but now where firm 1 is an incumbent
monopoly and firm 2 is a potential entrant that faces a fixed cost F > 0 to enter and compete in
the market via a one-shot simultaneous move game in quantity space. The incumbent monopoly
is allowed to precommit to producing a certain output level before the game is played, thus
incurring a sunk cost which cannot be recovered. Draw a diagram in quantity space illustrating
a case where the optimal strategy for the incumbent is to deter entry. The diagram should
include the best response curves of the two firms, the precommitment level and at least one
isoprofit contour for the incumbent monopoly.
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PART B
Answer ONE question from this section.
B1 Suppose that the preferences of a consumer are additively separable in goods 1 and 2, i.e., her
00
0
utility function is of the form u(x1 , x2 ) = v1 (x1 ) + v2 (x2 ), where vi (xi ) > 0 and vi (xi ) < 0 for
i = 1, 2. The prices of the two goods are p1 and p2 , and the exogenous income is m.
(a) Formulate the Lagrangian for this consumer’s constrained utility-maximisation problem.
Make sure you indicate the three decision variables. Find the three first-order necessary conditions. Denote the decision variables that solve this system as x∗1 (p1 , p2 , m), x∗2 (p1 , p2 , m),
and λ∗ (p1 , p2 , m).
(b) The second-order sufficiency condition for this constrained utility-maximisation problem is
that the bordered-Hessian matrix must have principal minor submatrices with determinants
that alternate in sign starting with negative. The bordered-Hessian matrix is defined as
follows:


Hb = 
∂2u
∂x21
∂2u
∂x2 ∂x1
∂2u
∂x1 ∂x2
∂2u
∂x22
−p1
−p2
−p1


−p2  .
0
Using the utility function for this consumer, find Hb . Verify that the first principal minor
00
00
submatrix has a negative determinant and that |Hb | = −v1 p22 − v2 p21 > 0.
(c) Substitute the optimal solutions from part (a) into the three first-order necessary conditions.
Then, partially differentiate each of the three first-order necessary conditions with respect
∂x∗
∂x∗
to m. Solve for ∂m1 and ∂m2 using Cramer’s rule. Are both goods normal?
(d) Substitute the optimal solutions from part (a) into the three first-order necessary conditions.
Then, partially differentiate each of the three first-order necessary conditions with respect
∂x∗
∂x∗
to p1 . Solve for ∂p11 and ∂p12 using Cramer’s rule. What are the signs of own-price and
cross-price elasticities in this case? You may assume that λ∗ (p1 , p2 , m) ≥ 0.
(e) How would your answers to parts (c) and (d) change if the utility function were not additively separable?
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B2 You own a firm that operates in a perfectly competitive industry with the production function
1
1
f (x1 , x2 ) = x13 x23 , where xi indicates your quantity demanded of input factor i, i = 1, 2. Input
factor i costs wi > 0 per unit. The output from your firm is sold at an exogenous price p > 0.
(a) Formulate your firm’s long-run profit-maximisation problem taking care to indicate the
decision variables. What are the first-order necessary conditions for this problem?
(b) For such an unconstrained maximisation problem, the second-order sufficiency condition
is that the Hessian matrix must be negative definite. It can be verified by checking that
the determinants of the principal minor submatrices of the Hessian matrix alternate in
sign starting with negative. Show that the second-order sufficiency condition holds for this
problem.
(c) Use the first-order necessary conditions from part (a) to solve for the optimal input factor
demands, x∗i (p, w1 , w2 ), where i = 1, 2, and the supply function, y ∗ (p, w1 , w2 ).
(d) Assuming an arbitrary output target of y, set up your firm’s cost-minimisation problem
taking care to indicate the decision variables and the Lagrange multiplier. Take the firstorder necessary conditions for this constrained minimisation problem and solve for the
conditional input factor demands, xci (w1 , w2 , y).
(e) Verify that the conditional input factor demand from part (d) is precisely equal to the
unconditional input factor demand from part (c) for good 1 when the target output level
y is set to exactly y ∗ (p, w1 , w2 ) from part (c). Prove that the unconditional input factor
demand is more price elastic with respect
to its own price than the conditional input factor
∂x∗
∂xc demand, i.e., show that ∂w11 < ∂w11 ∗
.
y=y (p,w1 ,w2 )
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PART C
Answer ONE question from this section.
C1 A monopolist produces a single good x and sells it to two consumer types, A and B, whose
preferences are given by the utility functions
UA (x, y) = 24x − x2 + y,
3
and UB (x, y) = 22x − x2 + y,
2
where x is the quantity consumed of the monopolist’s good and y is the consumer’s disposable
income. There is an equal number (say N ) of both types of consumer within the population and
the monopolist has a constant marginal cost equal to 4.
(a) Determine each consumer type’s demand function for the monopolist’s good based on utility
maximisation.
(b) Suppose that the monopoly can accurately identify customer type and wishes to design
specific “take it or leave it” contracts to perfectly price discriminate and maximise
profits. These contracts take the form of a specific quantity qi sold for an overall price of ri
for each consumer type i = A, B. (i) Write down the profit maximisation problem and the
two constraints faced by the monopolist. (ii) Determine the specific quantity and overall
price for each contract (qA , rA ) and (qB , rB ).
(c) Suppose now that the monopoly cannot accurately identify customer type and so the
contracts must be tailored in such a way that the consumers of each type have the incentive
to self select a given contract. (i) Write down the willingness-to-pay and self-selection constraints for the monopolist’s profit maximisation problem and highlight which constraints
are slack and which constraints bind. (ii) Graphically or otherwise determine the revised
contracts (qA , rA ) and (qB , rB ) for this second-degree price discrimination problem that
maximise the monopolist’s profit.
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C2 Suppose that an economy’s aggregate production function is
Y = K α (AN )1−α ,
for 0 < α < 1,
where Y is aggregate output, K is capital, N is labour and A represents the technology level
of the economy. Suppose further that the capital stock depreciates at a rate δ, that the labour
force is growing at a rate gn and that technological progress grows at a rate ga . Households also
save a proportion s of their disposable income.
(a) Derive the evolution equation for capital per effective worker (K/AN ) and thus determine
the long-run equilibrium growth rates of output per worker (Y /N ) and capital per worker
(K/N ) and the long-run steady capital-to-output (K/Y ) ratio.
(b) Determine the real return to capital (the marginal product of capital) and the real wage.
What are their growth rates?
(c) Suppose the economy is at its long-run equilibrium growth rate. Describe in words and
illustrate graphically the impact both transitionally and in the long-term of a sudden fall
in technological progress from ga to ga0 < ga .
(d) Consider two countries, Country 1 and Country 2, that are completely identical in terms
of all the model parameters, except that they have different savings rates. If α = 1/4, and
the steady-state level of output per worker in Country 1 is twice that of Country 2, what
is the savings rate in Country 1 if the savings rate in Country 2 is 3%?
(e) Given α = 1/4, what is the savings rate sG that maximises consumption in the steady
state? Show all working. Is consumption per head maximised in either country in part
(d)? If not, would setting the savings rate immediately and permanently to sG increase
everyone’s welfare?
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END OF PAPER