ECON 8010
TEST #2 SOLUTIONS
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 -0.3. Why?
The uncompensated demand function is H0 in Px , Py , and M.
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 _1/2 . Why?
The expenditure function is H1 in Px and Py.
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 _ -1__. Why?
The indirect utility function is H0 in Px , Py , and M.
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 _1.2_. Why?
The share-weighted income elasticities sum to 1. (Generalized Engel’s
Law.)
(20) 2. TRUE or FALSE and EXPLAIN: Label the statement TRUE or FALSE and briefly 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 cross-partial
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 quasiconcave.
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
(–)
(+) (+) (+)
(–)
c.
TRUE 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."
∂X*/∂M = – (– UXY ·PY + UYY · PX)/H and ∂Y*/∂M = (– UXX ·PY + UYX · PX)/H
where H > 0 is the determinant of the bordered Hessian matrix. Since additive
separability implies UXY = UYX = 0, and diminishing marginal utility implies UXX < 0
and UYY < 0, we have
∂X*/∂M = – (UYY · PX)/H > 0 and ∂Y*/∂M = (– UXX ·PY)/H > 0
d. TRUE. In a model with three goods (X, Y, and Z), if Y and Z are net complements then X
and Z are net substitutes.
S ∙ p = 0 where S is a 3x3 singular, symmetric matrix of own- and cross- substitution
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.
(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.
ʆ = Y – X -1 + (M – Px X – Py Y)
∂ʆ/∂X = 1/X2 –  Px = 0.
∂ʆ/∂Y = 1 –  Py = 0.
MRSy,x = Ux/ Uy = (1/X2)/1 = 1/X2 = Px/Py  X = (Py/Px)1/2
∂ʆ/∂ = M – Px X – Py Y = 0  M = Px X + Py Y
M = Px [Py/Px)1/2]+ Py Y  Y* = [M – (Px Py) ½]/Py
X* = (Py/Px)1/2
b. Is either X or Y an inferior good? Justify your answer rigorously.
No. ∂Y*/∂M = 1/Py > 0
∂X*/∂M = 0
(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.
∂V/∂Px = ( ̶ 2/M) ( Px1/2  Py1/2 ) (1/2 Px ½)
∂V/∂Px = ( ̶ 2/M) ( Px1/2  Py1/2 ) (1/2 Py ½)
∂V/∂M = (1/M 2) ( Px1/2  Py1/2 ) 2
X*(Px , Py , M) = ̶ ∂V/∂Px / ∂V/∂M = M / [ ( Px1/2  Py1/2 ) Px1/ 2 ] = M / [ Px + (Px Py) ½]
X*(Px , Py , M) = ̶ ∂V/∂Py / ∂V/∂M = M / [ ( Px1/2  Py1/2 ) Py1/2 ]= M / [ Py + (Px Py) ½]
(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.
Xc*(Px,Py,U) = ∂M*/∂Px = Px - ½ Py ½ = (Py / Px) ½
Yc*(Px,Py,U) = ∂M*/∂Py = U + Px ½ Px - ½ = U + (Px / Py) ½
b. What is the (direct) utility function?
U[Xc*(Px, Py, U), Yc*(Px, Py, U)] = Yc* – (Xc*) -1
U(X,Y) = Y – X -1
(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).
Substitute x = w·(T –  ) + I into U( x, ) = x       and solve
max U( ) = [ w·(T – )  I ]    
The F.O.C. is
∂U/∂  = w·T – 2·w·  + I +  = 0
Solving for * (w,I;  ,T), we have
* (w,I;  ,T) = (w·T + I +  ) / 2·w
or
T – * (w,I;  ,T) = h*(w,I;  ,T) = (w·T –  – I) / 2·w = T/2 – [(I +  ) / 2w]
b. Rigorously analyze the effect of an increase in the wage rate on her labor supply.
∂ h*/∂w = [T·2·w – (w·T –  – I)·2] / 4w2
= (I +  ) / 2w2 > 0
~ ) 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
Set h*(w,I;  ,T) = (w·T –  – I) / 2·w = 0 and solve for w:
~ (I;  ,T) = (I +  ) / T
w