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Answer the question in Part A, and ONE question from Part B.I and ONE question from Part B.II.
This assessment accounts for 60 per cent of the marks for the course. Each question carries an equal
percentage of the total mark.
In cases where a student answers more questions than requested by the assessment 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. No credit will be given for
reproducing parts of the course notes. The answer to each part of each question should be on at most one
page (for example A.1 has 6 parts and there should be at most 6 pages of answers to this). Any part of
any answer that violates this will be given zero marks.
ECON0013 1 TURN OVER
PART A
You must answer the question in this section.
A.1 (a) An individual lives for two periods, consuming c when young and c when old. He has assets
0 1
worth A at the beginning of the first period and whatever he has not spent at the end of the
period can be carried forward to the second as saving accruing interest at the real rate r. He
has no other source of income. He has no reason to keep resources beyond the end of the
second period so c = (A?c )(1+r).
1 0
He chooses consumption to maximise lifetime utility
U = ν(c )+βν(c )
0 1
where ν(.) is a within-period utility function and β is a preference parameter.
(i) What properties must the function ν(.) have if the weakly preferred sets in the space of c
0
and c are to be convex? How would you interpret the required properties economically?
1
How would you interpret the parameter β?
(ii) Show that he chooses to consume more in the earlier period if and only if β(1+r) < 1.
Interpret this.
(iii) Suppose that within-period utility has the form ν(c) = ?e?c. Find an expression for
the chosen consumption in each period. (You can ignore corner solutions and therefore
consider only cases where c and c are chosen to both be positive.)
0 1
(iv) Show that the lifetime utility achieved will therefore equal
2+r
U = ? e?A(1+r)/(2+r){β(1+r)}1/(2+r).
1+r
Find therefore an expression for the minimum assets A required to sustain a lifetime utility
of at least U.
(b) A firm produces output Q using skilled labour z and unskilled labour z . The production
0 1
technology is summarised by production function
(cid:20) (cid:21)
1 β
Q = ?ln e?z0 + e?z1
1+β 1+β
for z , z ≥ 0. Labour is hired at skilled wage w and unskilled wage w and the firm takes
0 1 0 1
wages as given.
(i) Without explicitly solving the cost minimisation problem, use analogy with the results of
previous parts to explain
A. why the firm chooses to use more skilled than unskilled labour only if w > βw ,
1 0
ECON0013 2 CONTINUED
B. why the firm’s cost function has the form
w +w w
1 0 1
C(Q,w ,w ) = (w +w )Q + (w +w )ln ?w lnw ?w ln
0 1 1 0 1 0 0 0 1
1+β β
and
C. the form of the conditional demand functions for each type of labour.
(Again, you can ignore corner solutions.)
(ii) Is average cost increasing, decreasing or constant in Q? What does this tell you about
whether there are increasing, decreasing or constant returns to scale?
(You can use here the fact that
w +w w
1 0 1
(w +w )ln ?w lnw ?w ln ≤ 0
1 0 0 0 1
1+β β
for all values of w , w and β.)
0 1
(iii) What is the marginal cost for this technology? Discuss the nature of the firm’s output
supply function.
ECON0013 3 TURN OVER
PART B.I
Answer ONE question from this section.
B.I.1 There is a buyer B and a seller S. The seller produces z units of a good at the cost C(z) = czα
(where α ≥ 1). The buyer gets utility U(z,p) = Bzβ?pz (where β < 1) if she consumes z units of
the good and pays p for each unit she buys.
(a) Considerthe followinggame: First thesellersetsthepricepandundertakes toproduce however
many units the buyer wants at that price. Then the buyer decides how many units to buy.
Describe the subgame perfect equilibrium of this game.
(b) Now consider the different game. First the buyer sets the price p and promises to buy all the
units the seller will produce at that price. Then the seller chooses how many units to produce.
Describe the subgame perfect equilibrium of this game and compare it with your answer above.
(c) If the market for the good were competitive what is the buyer’s demand curve and what is the
seller’s supply curve for the good? What would be the outcome if the competitive price were
then set by an external regulator? Explain how this differs from the outcome in both of the
games above.
(d) Describe a Nash equilibrium of the game where the seller moves first that is not a subgame
perfect equilibrium.
ECON0013 4 CONTINUED
B.I.2 A worker is employed by a firm to produce output. If the worker puts in effort there is: probability
p that they produce two units of output, probability q that they produce one unit, and probability
1?p?q that they produce zero units. If the worker does not put in effort these probabilities are:
r, s, 1?r?s respectively. The manager decides to pay the worker u ≥ 0 if two units are produced
v ≥ 0 if only one unit is produced and w ≥ 0 if no units are produced. The worker has a utility
function x2 ?c if she receives the wage x = w,v,u and puts in effort. If she does not put in effort
she has the utility x2, where x = w,v,u . The worker can earn the utility U from working elsewhere.
The manager can sell each unit of the good that the worker produces for a price R.
(a) Supposethatr < pandconsiderthetwocontracts(u,v,w) = (1,1,1)or(u,v,w) = (1/r,0,0)
which does the worker prefer if she puts in low effort? Which one does the worker prefer when
she puts in high effort? Which contract is cheapest for the firm? Explain your results.
(b) The firm decides that it is content with low effort from the worker. Write down and solve a
constrained optimisation that describes the cheapest way for the firm to achieve this. Interpret
what you find. When does the firm make a profit?
(c) Suppose that p = 2r and the firm decides to pay the worker according to the contract u > 0
and v = w = 0. For what values of c,r,p,U is the worker (a) willing to work for the firm and
provide low effort, (b) willing to work for the firm and provide high effort? If the conditions for
case (a) hold what is the most profitable contract for the firm to offer? If the conditions for
case (b) hold what is the most profitable contract?
(d) Discuss what you think an optimal contract would look like in this case (p = 2r). In particular
consider when the firm is willing to pay for high effort from the worker.
ECON0013 5 TURN OVER
PART B.II
Answer ONE question from this section.
B.II.1 Consider an economy in which K firms use labour Lk to produce corn Qk, k = 1,...,K and H
consumers supply labour lh and consume corn ch, h = 1,...,H.
Firms produce according to the technology
(cid:16) (cid:17)
Qk = Aln 1+Lk
where A is a production parameter.
Consumers are potentially of two types. There are H individuals of Type A who have utilities
A
1 (cid:16) (cid:17)2
Uh = ch ? lh
2
whereas there are H = H ?H individuals of Type B who have utilities
B A
1 (cid:16) (cid:17)3
Uh = ch ? lh .
3
Let the price of corn be p and the nominal wage be w so that the real wage expressed in unit of corn
is W = w/p.
Firms choose production plans to maximise profits πk = pQk?wLk taking prices as given. Profits
are distributed as income to consumers according to production shares θhk (where (cid:80)H θhk =
h=1
1 for each k = i,...,K) and consumers maximise utility subject to budget constraints pch =
(cid:80)K θhkπk +wlh taking prices and firm profits as given.
k=1
(a) Find an expression for the labour demand of each firm given W. Hence find each firm’s profit.
(b) Find expressions for the labour supply of each consumer type given W and firm profits.
(c) Suppose all individuals are of type A, H = H and H = 0, that H = K, and that θhk = 1/H
A B
forallhandallk sothatfirmownershipisequallyspread. FindtheuniqueWalrasianequilibrium
real wage W?.
(d) Illustrate the equilibrium on a Robinson Crusoe diagram for the case H = 1 (and explain why
this also represents the more general case H > 1).
(e) How does the equilibrium real wage change if H > K so that there are more workers than
firms? Discuss.
(f) How does the equilibrium real wage change if θhk (cid:54)= 1/H for some h and k so that ownership
is not equally spread? Discuss.
(g) Now suppose that both H > 0 and H > 0 so that the consumer population consists of
A B
individuals of both types. Is the equilibrium still necessarily unique? Either explain why the
equilibrium remains unique or provide an example where it is not.
ECON0013 6 CONTINUED
B.II.2 Individuals in an economy consume n goods q = (q ,q ,...,q )(cid:48), purchased at the prices p =
1 2 n
(p ,p ,...,p )(cid:48) from budgets y. You decide to model behaviour using preferences represented by
1 2 n
the expenditure function c(υ,p) where υ represents consumer utility.
(a) Explain what an expenditure function is and why
?lnc(υ,p)
= w (υ,p) i = 1,2,...,n
i
?lnp
i
where w (υ,p) is a function giving the budget share of the ith good.
i
Suppose that the expenditure function takes the form
lnc(υ,p) = (cid:88) α ilnp
i
+ υe(cid:80) iβilnpi
i
where α = (α ,α ,...,α )(cid:48) and β = (β ,β ,...,β )(cid:48) are vectors of preference parameters.
1 2 n 1 2 n
(b) What homogeneity property must an expenditure function have? Outline a set of restrictions
on α and β which suffice for c(υ,p) to have that property.
(c) Find an expression for the budget shares under these preferences.
Concern is high that recent inflation, under which the prices have changed from p0 to p1, has
aggravated inequality by hitting poorer individuals harder than the more affluent.
(d) Explain what a true or Konu¨s cost-of-living index
K(cid:0) υ,p0,p1(cid:1)
is and show that under these
preferences
lnK(cid:0) υ,p0,p1(cid:1) = (cid:88) α iln pp 01 i + υ(cid:104) e(cid:80) iβilnp1 i ?e(cid:80) iβilnp0 i(cid:105) .
i i
(e) Explain what a Laspeyres cost-of-living index
L(cid:0) υ0,p0,p1(cid:1)
is and show that under these pref-
erences
(cid:40) (cid:41)
lnL(cid:0) υ0,p0,p1(cid:1) = ln (cid:88) α ip p1 i
0
+ υ0(cid:88) β ip p1 i
0
e(cid:80) iβilnp0 i
i i i i
where υ0 denotes utility in the initial period.
(f) Explainwhytheframeworkwhichyouhaveadoptedformodellingbehaviourisusefulforaddress-
ing the question of how inflation aggravates inequality only if preferences are not homothetic.
What must be true of α and β if preferences are not to be homothetic?
(g) What can be said about comparison of the Laspeyres and true indices if preferences are homo-
thetic? What if they are not homothetic?
(h) What aspect of consumer behaviour do the Laspeyres indices fail to account for? Supposing
that preferences are non-homothetic, discuss how this omission might distort judgement of the
distributional effects of inflation.
Answer the question in Part A, and ONE question from Part B.I and ONE question from Part B.II.
This assessment accounts for 60 per cent of the marks for the course. Each question carries an equal
percentage of the total mark.
In cases where a student answers more questions than requested by the assessment 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. No credit will be given for
reproducing parts of the course notes. The answer to each part of each question should be on at most one
page (for example A.1 has 6 parts and there should be at most 6 pages of answers to this). Any part of
any answer that violates this will be given zero marks.
ECON0013 1 TURN OVER
PART A
You must answer the question in this section.
A.1 (a) An individual lives for two periods, consuming c when young and c when old. He has assets
0 1
worth A at the beginning of the first period and whatever he has not spent at the end of the
period can be carried forward to the second as saving accruing interest at the real rate r. He
has no other source of income. He has no reason to keep resources beyond the end of the
second period so c = (A?c )(1+r).
1 0
He chooses consumption to maximise lifetime utility
U = ν(c )+βν(c )
0 1
where ν(.) is a within-period utility function and β is a preference parameter.
(i) What properties must the function ν(.) have if the weakly preferred sets in the space of c
0
and c are to be convex? How would you interpret the required properties economically?
1
How would you interpret the parameter β?
(ii) Show that he chooses to consume more in the earlier period if and only if β(1+r) < 1.
Interpret this.
(iii) Suppose that within-period utility has the form ν(c) = ?e?c. Find an expression for
the chosen consumption in each period. (You can ignore corner solutions and therefore
consider only cases where c and c are chosen to both be positive.)
0 1
(iv) Show that the lifetime utility achieved will therefore equal
2+r
U = ? e?A(1+r)/(2+r){β(1+r)}1/(2+r).
1+r
Find therefore an expression for the minimum assets A required to sustain a lifetime utility
of at least U.
(b) A firm produces output Q using skilled labour z and unskilled labour z . The production
0 1
technology is summarised by production function
(cid:20) (cid:21)
1 β
Q = ?ln e?z0 + e?z1
1+β 1+β
for z , z ≥ 0. Labour is hired at skilled wage w and unskilled wage w and the firm takes
0 1 0 1
wages as given.
(i) Without explicitly solving the cost minimisation problem, use analogy with the results of
previous parts to explain
A. why the firm chooses to use more skilled than unskilled labour only if w > βw ,
1 0
ECON0013 2 CONTINUED
B. why the firm’s cost function has the form
w +w w
1 0 1
C(Q,w ,w ) = (w +w )Q + (w +w )ln ?w lnw ?w ln
0 1 1 0 1 0 0 0 1
1+β β
and
C. the form of the conditional demand functions for each type of labour.
(Again, you can ignore corner solutions.)
(ii) Is average cost increasing, decreasing or constant in Q? What does this tell you about
whether there are increasing, decreasing or constant returns to scale?
(You can use here the fact that
w +w w
1 0 1
(w +w )ln ?w lnw ?w ln ≤ 0
1 0 0 0 1
1+β β
for all values of w , w and β.)
0 1
(iii) What is the marginal cost for this technology? Discuss the nature of the firm’s output
supply function.
ECON0013 3 TURN OVER
PART B.I
Answer ONE question from this section.
B.I.1 There is a buyer B and a seller S. The seller produces z units of a good at the cost C(z) = czα
(where α ≥ 1). The buyer gets utility U(z,p) = Bzβ?pz (where β < 1) if she consumes z units of
the good and pays p for each unit she buys.
(a) Considerthe followinggame: First thesellersetsthepricepandundertakes toproduce however
many units the buyer wants at that price. Then the buyer decides how many units to buy.
Describe the subgame perfect equilibrium of this game.
(b) Now consider the different game. First the buyer sets the price p and promises to buy all the
units the seller will produce at that price. Then the seller chooses how many units to produce.
Describe the subgame perfect equilibrium of this game and compare it with your answer above.
(c) If the market for the good were competitive what is the buyer’s demand curve and what is the
seller’s supply curve for the good? What would be the outcome if the competitive price were
then set by an external regulator? Explain how this differs from the outcome in both of the
games above.
(d) Describe a Nash equilibrium of the game where the seller moves first that is not a subgame
perfect equilibrium.
ECON0013 4 CONTINUED
B.I.2 A worker is employed by a firm to produce output. If the worker puts in effort there is: probability
p that they produce two units of output, probability q that they produce one unit, and probability
1?p?q that they produce zero units. If the worker does not put in effort these probabilities are:
r, s, 1?r?s respectively. The manager decides to pay the worker u ≥ 0 if two units are produced
v ≥ 0 if only one unit is produced and w ≥ 0 if no units are produced. The worker has a utility
function x2 ?c if she receives the wage x = w,v,u and puts in effort. If she does not put in effort
she has the utility x2, where x = w,v,u . The worker can earn the utility U from working elsewhere.
The manager can sell each unit of the good that the worker produces for a price R.
(a) Supposethatr < pandconsiderthetwocontracts(u,v,w) = (1,1,1)or(u,v,w) = (1/r,0,0)
which does the worker prefer if she puts in low effort? Which one does the worker prefer when
she puts in high effort? Which contract is cheapest for the firm? Explain your results.
(b) The firm decides that it is content with low effort from the worker. Write down and solve a
constrained optimisation that describes the cheapest way for the firm to achieve this. Interpret
what you find. When does the firm make a profit?
(c) Suppose that p = 2r and the firm decides to pay the worker according to the contract u > 0
and v = w = 0. For what values of c,r,p,U is the worker (a) willing to work for the firm and
provide low effort, (b) willing to work for the firm and provide high effort? If the conditions for
case (a) hold what is the most profitable contract for the firm to offer? If the conditions for
case (b) hold what is the most profitable contract?
(d) Discuss what you think an optimal contract would look like in this case (p = 2r). In particular
consider when the firm is willing to pay for high effort from the worker.
ECON0013 5 TURN OVER
PART B.II
Answer ONE question from this section.
B.II.1 Consider an economy in which K firms use labour Lk to produce corn Qk, k = 1,...,K and H
consumers supply labour lh and consume corn ch, h = 1,...,H.
Firms produce according to the technology
(cid:16) (cid:17)
Qk = Aln 1+Lk
where A is a production parameter.
Consumers are potentially of two types. There are H individuals of Type A who have utilities
A
1 (cid:16) (cid:17)2
Uh = ch ? lh
2
whereas there are H = H ?H individuals of Type B who have utilities
B A
1 (cid:16) (cid:17)3
Uh = ch ? lh .
3
Let the price of corn be p and the nominal wage be w so that the real wage expressed in unit of corn
is W = w/p.
Firms choose production plans to maximise profits πk = pQk?wLk taking prices as given. Profits
are distributed as income to consumers according to production shares θhk (where (cid:80)H θhk =
h=1
1 for each k = i,...,K) and consumers maximise utility subject to budget constraints pch =
(cid:80)K θhkπk +wlh taking prices and firm profits as given.
k=1
(a) Find an expression for the labour demand of each firm given W. Hence find each firm’s profit.
(b) Find expressions for the labour supply of each consumer type given W and firm profits.
(c) Suppose all individuals are of type A, H = H and H = 0, that H = K, and that θhk = 1/H
A B
forallhandallk sothatfirmownershipisequallyspread. FindtheuniqueWalrasianequilibrium
real wage W?.
(d) Illustrate the equilibrium on a Robinson Crusoe diagram for the case H = 1 (and explain why
this also represents the more general case H > 1).
(e) How does the equilibrium real wage change if H > K so that there are more workers than
firms? Discuss.
(f) How does the equilibrium real wage change if θhk (cid:54)= 1/H for some h and k so that ownership
is not equally spread? Discuss.
(g) Now suppose that both H > 0 and H > 0 so that the consumer population consists of
A B
individuals of both types. Is the equilibrium still necessarily unique? Either explain why the
equilibrium remains unique or provide an example where it is not.
ECON0013 6 CONTINUED
B.II.2 Individuals in an economy consume n goods q = (q ,q ,...,q )(cid:48), purchased at the prices p =
1 2 n
(p ,p ,...,p )(cid:48) from budgets y. You decide to model behaviour using preferences represented by
1 2 n
the expenditure function c(υ,p) where υ represents consumer utility.
(a) Explain what an expenditure function is and why
?lnc(υ,p)
= w (υ,p) i = 1,2,...,n
i
?lnp
i
where w (υ,p) is a function giving the budget share of the ith good.
i
Suppose that the expenditure function takes the form
lnc(υ,p) = (cid:88) α ilnp
i
+ υe(cid:80) iβilnpi
i
where α = (α ,α ,...,α )(cid:48) and β = (β ,β ,...,β )(cid:48) are vectors of preference parameters.
1 2 n 1 2 n
(b) What homogeneity property must an expenditure function have? Outline a set of restrictions
on α and β which suffice for c(υ,p) to have that property.
(c) Find an expression for the budget shares under these preferences.
Concern is high that recent inflation, under which the prices have changed from p0 to p1, has
aggravated inequality by hitting poorer individuals harder than the more affluent.
(d) Explain what a true or Konu¨s cost-of-living index
K(cid:0) υ,p0,p1(cid:1)
is and show that under these
preferences
lnK(cid:0) υ,p0,p1(cid:1) = (cid:88) α iln pp 01 i + υ(cid:104) e(cid:80) iβilnp1 i ?e(cid:80) iβilnp0 i(cid:105) .
i i
(e) Explain what a Laspeyres cost-of-living index
L(cid:0) υ0,p0,p1(cid:1)
is and show that under these pref-
erences
(cid:40) (cid:41)
lnL(cid:0) υ0,p0,p1(cid:1) = ln (cid:88) α ip p1 i
0
+ υ0(cid:88) β ip p1 i
0
e(cid:80) iβilnp0 i
i i i i
where υ0 denotes utility in the initial period.
(f) Explainwhytheframeworkwhichyouhaveadoptedformodellingbehaviourisusefulforaddress-
ing the question of how inflation aggravates inequality only if preferences are not homothetic.
What must be true of α and β if preferences are not to be homothetic?
(g) What can be said about comparison of the Laspeyres and true indices if preferences are homo-
thetic? What if they are not homothetic?
(h) What aspect of consumer behaviour do the Laspeyres indices fail to account for? Supposing
that preferences are non-homothetic, discuss how this omission might distort judgement of the
distributional effects of inflation.