API 111 / Econ 2020a / HBS 4010 Fall 2007 -- Problem Set #4 Due:
Wednesday, October 24, 2007
1. Consumption over time (Adapted from Silberberg, 1990) The
world lasts T > 2 periods. A consumer has income wt in any
period t. The real interest rate is r 0, and the consumer can
borrow and lend as much as he/she wants at this rate. The consumer
consumes ct in each period and has utility function U(c). 1a. Write
down the consumers budget constraint. 1b. Write down the consumers
utility maximization problem. Consider the following two utility
functions: (i) 1 log ct U (c ) = t =0 1 + T T t t
1 a ct for 0 < a < 1. (ii) U (c ) = t =0 1 + 1c. Show that
for the optimal consumption stream, consumption in any time period
after period 0 can be expressed as a constant times the consumption
in the previous period. 1d. Under what conditions on the exogenous
variables will consumption be increasing over time? Under what
conditions will consumption be decreasing over time?
(Problem Set Continues)
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Production2. From MWG, Exercise 5.C.13. Hint: What is the firms
revenue maximization problem? Given that, what is the function that
the econometrician has estimated? Apply the envelope theorem (or a
version of Roys Identity). [See Simon and Blume p. 453-457 or MWG
p. 964966 for the envelope theorem.] 3. Cobb-Douglas Production
Function Consider the following production function:
Assuming that the price of the output is p and the prices of
inputs are w1 and w2 respectively: Part I: Profit Maximization: (a)
State the firms profit maximization problem. (b) Derive the firms
factor demand functions for z1 and z2. (c) Derive the firms supply
function. (d) Derive the firms profit function. (e) Verify
Hotellings lemma for q(w,p), z1(w,p), and z2(w,p). Part II: Cost
Minimization: (f) (g) (h) (i) State the firms cost minimization
problem Derive the firms conditional factor demand functions.
Derive the firms cost function. Verify Shepards Lemma.
y = (z) =1 4 z2 2 . f z1 1
Part III: PMP and CMP: (j) Suppose the firm chooses quantity q
to maximize p q c(w,q). Show that the supply function determined
from this problem is the same as the supply function determined by
solving the profit maximization problem in Part I. (Problem Set
Continues)
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4. The Le Chtelier Principle Part I: Comparative statics when
both inputs are variable. A profit-maximizing firm produces a
single output q using two inputs, z 1 and z2. The price of the
single output is p and the cost of the inputs are w1 and w2
respectively, and these prices are taken as given by the firm. The
firms production function q = f(z1,z2) is strictly concave. The
fact that f(z) is concave implies that its second derivative matrix
is negative definite, which in turn implies that: f11 < 0 f22
< 0 f11f22 (f12)2 > 0
a) State the firms profit maximization problem in terms of
inputs z1 and z2.b) Derive the firms first-order conditions. Denote
the firms factor demand functions by z1(w,p) and z2(w,p). c) Use
the factor demand functions and the first order conditions derived
in b) to produce two identities (i.e., substitute the solution into
the first-order conditions). We are interested in determining the
effect of an increase in w1 on the firms utilization of inputs z1
and z2. These effects are captured by the derivatives dz1(w,p)/dw1
and dz2(w,p)/dw1.
d) Totally differentiate the two identities in c with respect to
w1. This yields two equationsthat are linear in dz1(w,p)/dw1 and
dz2(w,p)/dw1.
e) Solve the equations in part d) for dz1(w,p)/dw1 and
dz2(w,p)/dw1. Your answer should bein terms of the price of output
and the second derivatives of the production function. f) Use the
fact that f(z) is concave to determine the sign of the
dz1(p,w)/dw1. What is that sign? Under what conditions will
dz2(p,w)/dw1 be positive? Negative? What is the economic meaning of
these conditions? Part II: Comparative statics when one of the
inputs is not variable.* Now assume that the level of z2 is fixed
at z 2 . This would be true, for example, if input 2 is the size of
the firms plant, and it is unable to vary its plant size in the
short run. * g) State the firms profit maximization problem when
the level of input 2 is fixed at z 2 .
h) State the firms first order condition. State the firms second
order condition. * Denote the solution to the firms short-run
profit maximization problem as z1S(w1,p, z 2 )* i) Substitute
z1S(w1, p, z 2 ) into the first order condition in h) in order to
obtain an identity. * Totally differentiate this identity with
respect to w1 and solve for dz1S(w1,p, z 2 )/dw1. * What is the
sign of dz1S(w1,p, z 2 )/dw1? j) Compare dz1(w,p)/dw1 to
dz1S(w1,p,z2*). Is the firm more responsive to an increase in
factor prices in the short run (when input 2 is fixed) or in the
long run? k) Explain you answer to part j) intuitively using the
example of input 2 being plant size. This is an example of the Le
Chtelier Principle, a quite general phenomenon relating
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changes in variables when some other factors are fixed to
changes when all factors are variable.
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