```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
(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
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
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