ECON 8010
TEST #2 ANSWER KEY
FALL 2014
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.
a.
RESTRICTIONS: Calculate the missing parameter values from the restrictions
imposed by the theory of consumer behavior.
Suppose that a consumer has the following uncompensated demand function:


X * ( Px , Py , M )  M – Py1/2 Px1/2 Py a
where Px and Py are the per-unit prices of goods X and Y, respectively, M is money
income, and a is a fixed parameter. The value of the parameter a is -1 . Why?
The uncompensated demand function is H0 in Px and Py
b.
Suppose that the consumer has the following expenditure function:
M*(Px,Py,U) = PyU + 2Px1/2Pyb
where Px and Py are per-unit prices of goods X and Y, respectively, U is utility, and b
is a fixed parameter. The value of the parameter b is ½ . Why?
The expenditure function is H1 in Px and Py
(20)
2.
TRUE or FALSE and EXPLAIN: Label each statement TRUE or FALSE and
explain how you arrived at your answer.
a.
FALSE 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.
∂Y*/∂Px = H12 / H + X*H32 / H = [Py(Px + X* Uxx)] / H ≠ 0 , in general, where Hij
is the determinant of the co-factor of the element in row i and column j, and H is
the determinant of the bordered Hessian matrix of second-order own- and crosspartial derivatives of the utility function.
The utility function in question #3, below, provides a counter-example to the
statement. It is additively separable, but the cross-price effects are non-zero.
b.
TRUE__ 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 quasi-concave.
Quasi-concavity ↔ Ux2 Uyy – 2Ux Uy Uxy + Uy2 Uxx < 0
Assuming complementarity (Uxy > 0) and diminishing marginal utility in both X
and Y (Uxx < 0, Uyy < 0), along with monotonicity (Ux > 0, Uy >0), we have
Ux2 Uyy – 2Ux Uy Uxy + Uy2 Uxx < 0
(–)
(25)
3.
(+) (+) (+)
(–)
Suppose that an individual maximizes the utility function
U(X,Y) = X1/2 + Y
subject to the budget constraint M = Px X + Py Y.
a.
Derive the uncompensated demand functions for X and Y.
The Lagrangean function for this problem is
ʆ = X1/2 + Y + (M – Px X + Py Y)
The first-order conditions (F.O.C.s) for a maximum of ʆ are
∂ʆ / ∂X = ½ X-½ – Px = 0
∂ʆ / ∂Y = 1 – Py = 0
M – Px X + Py Y = 0
The first two F.O.C.s imply the tangency condition
½ Px-1 X-½ =  = Py-1
Solving for X yields
X*(Px, Py, M) = (Py/2Px)2
Substituting X* = (Py/2Px)2 into the budget identity M – Px X* + Py Y* ≡ 0 and
solving for Y* yields
Y*(Px, Py, M) = M/Py – (Py/4Px)
b. Use your answers in part (a) to derive the indirect utility function.
V(Px, Py, M) = U(X*,Y*) = (X*)1/2 + Y*
V(Px, Py, M) = (Py/2Px) + M/Py – (Py/4Px)
V(Px, Py, M) = (Py/4Px) + M/Py
(10)
4. Suppose that an individual’s preferences, defined over the two goods X and Y, are
homothetic, so that the ratio of uncompensated demands for the two goods is a
function only of the price ratio Px/Py and is therefore independent of money income M.
Show that the income elasticities of demand for X and for Y are both equal to 1.
Differentiating the ratio X*/Y*with respect to M, and noting that [∂(X*/Y*)/∂M)
= 0, yields
[(Y*∂X*/∂M – X*∂Y*/∂M)]/Y*2 = 0
(∂X*/∂M)/Y* = (X*/Y*2)(∂Y*/∂M)
(1/X*)(∂X*/∂M) = (1/Y*)(∂Y*/∂M)
Multiplying both sides by M*, we have
(M/X*)(∂X*/∂M) = (M/Y*)(∂Y*/∂M)
(1) XM = YM
Substituting (1) into the Engel Aggregation formula SxXM + SyYM = 1,
SxXM + SyXM = 1
(Sx + Sy)XM = 1
Similarly,
SxYM + SyYM = 1
(Sx + Sy)YM = 1
But since Sx + Sy = 1,
XM = 1 = YM
(20)
5. Suppose that a consumer’s preferences are represented by the expenditure function,
M*(Px, Py, U) = Py [1 + U + loge(Px/Py)],
where Px and Py are the per-unit prices of X and Y, respectively, and U is utility.
Derive the compensated demand functions for X and Y.
∂M/∂Px = Xc(Px, Py, U) = (Py/Px)
∂M/∂Py = Yc(Px, Py, U) = U – loge (Py/Px)
(5)
6. TRUE or FALSE and EXPLAIN: Label the statement TRUE or FALSE and explain
how you arrived at your answer.
“In a model with three goods (X, Y, and Z), if Y and Z are net complements then X
and Z are net substitutes.”
TRUE. S ∙ p = 0 where S is a 3x3 singular, symmetric matrix of own- and crosssubstitution terms, and p > 0 is a 3x1 vector of prices. In the third row of S,
S ∙ p = Szx ∙ px + Szy ∙ py + Szz ∙ pz = 0. Szz < 0 by negativity of the own-substitution
effect and Szy = Syz < 0 by the assumption that Y and Z are net complements, so it
must be true that Szx = Sxz > 0 and, therefore, X and Z are net substitutes.