ECON 8010
TEST #2
FALL 2015
Instructions: All questions must be answered on this examination paper. No additional sheets of
paper are permitted; use the backs of the pages if necessary. For every question, show all of your
work in arriving at your answers. Time limit: 75 minutes.
(20) 1. Calculate the missing parameter values from the restrictions imposed by the theory of consumer
behavior.
a. Suppose that a consumer has the following uncompensated demand function:
X * ( Px , Py , M )  aPx0.4 Pyb M 0.7
Px and Py are per-unit prices of goods X and Y, respectively, M is money income, and a and
b are fixed parameters. The value of the parameter b is _______. Why?
b. Suppose that the consumer has the following expenditure function:
M(PX, PY, U) = (PX + 2Px1/2Pyc + Py) ∙ U/2
Px and Py are per-unit prices of goods X and Y, respectively, U is utility, and c is a fixed
parameter. The value of the parameter c is _______. Why?
c. Suppose the consumer has the following indirect utility function:
V(Px, Py, M) = ¼ Px 1Py d M 2
Px and Py are per-unit prices of goods X and Y, respectively, M is money income, and a is
a fixed parameter. The value of the parameter d is _______. Why?
d. If a consumer spends 1/3 of her income on good X and 2/3 of her income on good Y, and
the income elasticity of demand for good X is 0.6, then the income elasticity of demand
for good Y is _______. Why?
(20) 2. TRUE or FALSE and EXPLAIN: Label the statement TRUE or FALSE and briefly explain
how you arrived at your answer.
a. __________ Suppose that a utility function U(X,Y) is strongly (“additively”) separable so that
Uxy = 0 = Uyx. Then the uncompensated demand functions for goods X and Y exhibit zero
cross-price effects; that is, ∂X*/∂Py = 0 = ∂Y*/∂Px.
b. __________ If X and Y are Edgeworth-Fisher-Pareto complements (Uxy > 0) and X and Y
are each subject to diminishing marginal utility, then the utility function U(X,Y) is quasiconcave.
c. __________ If the utility function U(X,Y) is additively separable and both X
and Y exhibit diminishing marginal utility, then both X and Y are normal goods."
d. _________ In a model with three goods (X, Y, and Z), if Y and Z are net complements
then X and Z are net substitutes.
(20) 3. Suppose that an individual maximizes the utility function
U(X,Y) = Y – X -1
subject to the budget constraint M = Px X + Py Y.
a. Derive the uncompensated demand functions for X and Y.
b. Is either X or Y an inferior good? Justify your answer rigorously.
(10) 4. Assume that an individual's preferences are given by the indirect utility function
V ( Px , Py , M )  [( Px1/ 2  Py1/ 2 ) 2 ] / M
Use Roy's Identity to derive the uncompensated demand functions for X and Y.
(15) 5. Suppose an individual’s preferences are given by the expenditure function
M*(Px, Py, U) = 2Px1/2Py1/2 + PyU
a.
Derive the compensated demand functions for X and Y.
b. What is the (direct) utility function?
(15) 6.
Suppose that an individual’s preferences are represented by the utility function
U( x, ) = x       , where x denotes units of a consumption good,  is time spent at
leisure, and  > 0 is a fixed parameter. She works h = T –  hours per week at an hourly
wage of w, and the per-unit price of x is normalized to p = 1. Her total weekly income is the
sum of labor income (w·h) and non-labor income I.
a. Derive her labor-supply function h*(w,I;  ,T).
b. Rigorously analyze the effect of an increase in the wage rate on her labor supply.
~ ) such that
c. An individual’s reservation wage is defined as the value of w (denoted by w
~ (I; ; T).
h*(w, I; , T) = 0. Derive her reservation-wage function w