ECON0013: MICROECONOMICS
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) 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 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代 写ECON0013、Java/Python
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.
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