ECON 5113 Advanced Microeconomics Winter 2011 Test 1 February 4, 2011 Answer ALL Questions Time Allowed: 1 hour 20 min Attention: Please write your answers on the answer book provided. Use the right-side pages for formal an- swers and the left-side pages for your rough work. Re- member to put your name on the front page. 1. Let be a preference relation on R n + . (a) Define the relations and ∼ induced by . (b) Show that ∩∼ = ∅. (c) Show that for every x ∈ R n + , ∼ (x) ∪ (x) ∪≺ (x)= R n + . 2. Let U : R n + → R + be an increasing, twice differen- tiable, and quasiconcave utility function. (a) Define the indirect utility function. Make sure you explain every variable you use. (b) State and prove Roy’s identity. 3. Suppose a consumer’s preference structure is repre- sented by the utility function U (x)= x α 1 x 1−α 2 , 0 < α < 1. (1) (a) Set up the expenditure minimization problem. (b) Derive the Hicksian demand function h(x,u). (c) Find the expenditure function. (d) Find the indirect utility function. 4. Let U : R n + → R + be an increasing, twice differen- tiable, and quasiconcave utility function. (a) Define price elasticity ij . (b) Define income elasticity η i . (c) Prove that for i =1,...,n, n j=1 ij + η i =0. Hint: The ordinary demand function for good i is homogeneous of degree zero in (p,y). 5. Suppose a consumer’s utility function is given by Equation (1) with α =1/2. Define the Laspeyres- Kon¨ us cost of living index as P K (p 0 , p 1 ,u 0 )= E(p 1 ,u 0 ) E(p 0 ,u 0 ) , where E is the expenditure function with the super- scripts 0 and 1 indicating the time periods. (a) The consumer’s income changes from y 0 to y 1 . Show that the consumer will be worse off in period 1 if y 1 /y 0 is less than P K (p 0 , p 1 ,u 0 ). (b) In period 0 the prices are p 0 = (1, 1) and income is y 0 = 12. Suppose in period 1 p 1 = (2, 1). Calculate the price index P K . Happy New Year!
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ECON 5113 Advanced Microeconomics
Winter 2011
Test 1 February 4, 2011Answer ALL Questions Time Allowed: 1 hour 20 min
Attention: Please write your answers on the answer
book provided. Use the right-side pages for formal an-
swers and the left-side pages for your rough work. Re-
member to put your name on the front page.
1. Let � be a preference relation on Rn+.
(a) Define the relations � and ∼ induced by �.
(b) Show that � ∩ ∼ = ∅.
(c) Show that for every x ∈ Rn+,
∼(x) ∪ �(x) ∪ ≺(x) = Rn+.
2. Let U : Rn+ → R+ be an increasing, twice differen-
tiable, and quasiconcave utility function.
(a) Define the indirect utility function. Make sure
you explain every variable you use.
(b) State and prove Roy’s identity.
3. Suppose a consumer’s preference structure is repre-
sented by the utility function
U(x) = xα1 x
1−α2 , 0 < α < 1. (1)
(a) Set up the expenditure minimization problem.
(b) Derive the Hicksian demand function h(x, u).
(c) Find the expenditure function.
(d) Find the indirect utility function.
4. Let U : Rn+ → R+ be an increasing, twice differen-
tiable, and quasiconcave utility function.
(a) Define price elasticity �ij .
(b) Define income elasticity ηi.
(c) Prove that for i = 1, . . . , n,
n�
j=1
�ij + ηi = 0.
Hint: The ordinary demand function for good
i is homogeneous of degree zero in (p, y).
5. Suppose a consumer’s utility function is given by
Equation (1) with α = 1/2. Define the Laspeyres-
Konus cost of living index as
PK(p0,p1, u0) =
E(p1, u0)
E(p0, u0),
where E is the expenditure function with the super-
scripts 0 and 1 indicating the time periods.
(a) The consumer’s income changes from y0 to y1.Show that the consumer will be worse off in
Test 2 March 4, 2011Answer ALL Questions Time Allowed: 1 hour 20 min
Attention: Please write your answers on the answerbook provided. Use the right-side pages for formal an-swers and the left-side pages for your rough work. Re-member to put your name on the front page.
1. Suppose that a consumer’s indirect utility functionis given by
V (p, y) =y
pα1 pβ2
, α,β > 0.
Find the direct utility function.
2. Suppose that the function E : Rn++×R+ → R+ sat-
isfies the seven properties of an expenditure func-tion. Define
U(x) = maxu
�u : pTx ≥ E(p, u) ∀p ∈ Rn
++
�.
Show that U is increasing in x.
3. A consumer chooses consumption bundle x0 whenthe market prices are p0 and x1 when the prices arep1.
(a) State the weak axiom of revealed preference(WARP).
(b) Suppose that
p0 = (2, 2),
x0 = (3, 2),
p1 = (2, 1),
x1 = (3, 2).
Decide whether the observed choices satisfyWARP.
4. Suppose that f : Rn+ → R+ is a strictly increasing,
strictly quasiconcave, and differentiable productionfunction.
(a) Define the elasticity of substitution σij be-tween inputs i and j.
(b) If f is homothetic, show that σij can be ob-tained from the linear homogeneous part of falone.
5. Suppose the production function of a firm is givenby
y = xα1 x
β2 .
(a) Find the profit function given output price pand input prices w.
(b) What restrictions on α and β are required toensure that the profit function is well-defined?Explain.
Test 3 March 25, 2011Answer ALL Questions Time Allowed: 1 hour 20 min
Attention: Please write your answers on the answerbook provided. Use the right-side pages for formal an-swers and the left-side pages for your rough work. Re-member to put your name on the front page.
1. Suppose that in an economy every consumer’s pref-erences can be represented by a linearly homoge-neous utility function. Prove or disprove: the mar-ket demand for a good is independent of incomedistribution.
2. A monopoly producing a single good faces an inversedemand function
p = a− bq, a, b > 0.
The cost function is given by
c(q) = d+ eq2, d, e > 0.
(a) Derive the marginal revenue.
(b) Derive the marginal cost.
(c) Find the quantity produced by the firm andthe market price.
(d) Find the price elasticity of demand � at themarket price.
3. An industry consists of J identical firms each withcost function c(q) = 10q. Each firm faces an iden-tical inverse market demand p = 110 − 2Q, whereQ =
�Jj=1 qj is the total output produced all the
firms.
(a) Find the Cournot equilibrium output of a typ-ical firm.
(b) Find the profit of each firm.
4. A consumer’s demand for a single good is given byx = d(p, y) = y/p. Suppose income is y = $20 andthe price p increases from $1 to $3. Find
(a) the compensating variation,
(b) the equivalent variation, and
(c) the change in consumer surplus
of the price change.
5. In a two-person exchange economy the utility func-tions of Adam and Betty for fish and rice are giveby
Ua(xaf , x
ar) = min{xa
f , xar},
U b(xbf , x
br) = min{xb
f , xbr},
respectively. Adam has 100 fish and Betty has 100unit of rice.
(a) Find the core of this endowment, C(e).
(b) Find the set of Walrasian equilibrium alloca-tions, W (e).
(c) Draw an Edgeworth box showing the endow-ment e, C(e), and W (e).
“Back off. I don’t believe in the Second Welfare Theorem.”