Download ECON 8010 Test #1 Solutions Fall 2016

ECON 8010
TEST #1 SOLUTIONS
FALL 2016
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. Point values for each question are in
parentheses to the left of the question number as a guide for the allocation of your time.
Time Limit: 75 minutes.
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1. ASSUMPTIONS
a. What assumption about the short-run profit function π*(p,w;K) ensures that the
short-run demand curve for labor is downward-sloping? Explain.
- ∂2π*(p,w;K)/∂w2 = ∂L*(p,w;K))/∂w < 0 iff π*(p,w;K) is convex in w
b. What two assumptions about the production function x  f ( K , L) ensure that longrun demand curve for labor of a profit-maximizing, competitive firm is downwardsloping? Explain.
i)
Diminishing marginal productivity of capital: fKK < 0
ii)
Strict concavity of the production function: [fKK·fLL – (fKL) 2] > 0
Together, these two assumptions imply that the comparative-statics effect on
the demand for labor of a parametric change in the price of labor is
unambiguously negative:
∂L*∕∂w = fKK ∕ {p[fKK·fLL – (fKL2)]} < 0
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2. Suppose that a firm’s technology is represented by the CES production function
x(K,L) = [K + (1 ̶ )L] m/ = 0.5K1/2 + 0.5L1/2. Write down the values of the
following parameters, and show in each case how you arrived at your answer:
a. _____2_____ the elasticity of substitution between K and L.
x(K,L) = [K + (1 ̶ )L] m/
 = 1/(1 ̶ ) = 1/(1 ̶ 1/2) = 2
b. _____1/2_____ the degree of returns to scale.
m/ = 1; From part a,  = 1/2. Therefore, m = 1/2
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3. John is an entrepreneur and the only worker in a shoe-repair business. The
production function for this enterprise is x = f(M,L), where x is the quantity of shoe
repairs performed, M denotes material used in the repair process, and L is the number
of hours John devotes to the business. The production function exhibits the usual
properties of monotonicity (fM > 0, fL > 0) and concavity (fMM < 0, fLL < 0, and
fMMfLL – fML2 > 0). In addition, we assume that fML > 0 so that material and labor are
complements in production. Shoes are repaired at the fixed price p, John pays r per
unit for repair materials, and John’s work hours are determined by the operating hours
of the shoe-repair shop he has chosen, L0. John chooses M to maximize “variable
profit,” defined as the difference between total revenue and the cost of materials.
Rigorously answer the following questions:
a. Determine the effect of an increase in r on M*.
Because there is only one choice variable, M, it is possible to set up and
answer the first two of these three questions by substituting L0 for L directly
into the objective function, and solving the resulting one equation for the
one unknown. The problem, then, is to choose M to maximize
π = pf(M,L0) – rM
The first-order condition is
∂π/∂M = pfM(M,L0) – r = 0.
The second-order, sufficient condition for a profit maximum is pfMM < 0.
The solution function for M is M*(r,p,L0). Substituting M* into the firstorder condition, we have the identity
pfM[M*(r,p,L0),L0] – r ≡ 0.
Differentiating the identity with respect to r, we obtain
(pfMM)(∂M*/∂r) ≡ 1
∂M*/∂r ≡ 1/pfMM < 0
b. Determine the effect of an increase in L0 on M*.
(pfMM)(∂M*/∂L0) + pfML ≡ 0
∂M*/∂L0 = - (pfML)/(pfMM) = - (fML)/(fMM) > 0
c. Determine the effect of an increase in p on M*.
fM + pfMM(∂M*/∂p) ≡ 0
∂M*/∂p ≡ - (fM)/(pfMM) > 0
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4. Suppose that the firm’s profit function is
π*(p,w) = p2/4w
where p is the per-unit price of output x, and w is the per-unit price of labor L.
a. Derive the firm’s output-supply function x*(p,w).
∂π*∕∂p = p∕2w = x*(p,w)
b. Derive the firm’s labor-demand function L*(p,w).
– ∂π*∕∂w = p2∕4w2 = (p∕2w)2 = L*(p,w)
c. What is the firm’s production function x = f(L)?
From (a) and (b), x* = f[L*(p,w)] = p/2w and L*(p,w) = (p/2w)2  x(L) = L1/2
Alternatively,, note that the profit equation is π = px – wL and π*(p,w) =
p2/4w. Substituting L*(p,w) = (p/2w)2 from (b) into the profit equation and
evaluating x at x* and L at L* reveals that x(L) = L1/2.
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5. Label TRUE, FALSE, or UNCERTAIN each of the following statements of these
Marshall-Hicks rules, and rigorously defend your answer.
The Marshall-Hicks formula for the own-wage elasticity of demand for labor is
 = - (1 – s) + s
where  < 0 is the labor-demand elasticity, 0 < s < 1 is the share of the total cost
of production paid to labor,   0 is the elasticity of substitution between capital
and labor, and  < 0 is the own-price elasticity of demand for the output
produced.
a. “The demand for labor is more elastic, the more easily it is substituted for other
inputs.”
TRUE. ∆/∆ = - (1 – s) < 0.
b. “The demand for labor is more elastic, the more important it is in the total cost of
production.
UNCERTAIN. ∆/∆s =  +  is indeterminate in sign
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6. Suppose that a cost-minimizing firm’s (minimum) cost function is
C*(r,w,x) = w[1 + x + loge(r/w)],
where r is the per-unit price of capital, w is the per-unit price of labor, and x is output.
Use Shephard’s Lemma to derive the firm’s conditional (constant-output) demand
function for labor L*(r,w,x), and then rigorously analyze the effects of an increase in
w on L*(r,w,x).
L*(r,w,x) = ∂C*∕∂w = 1 + x + loger – (logew + w/w)
= x + loger – logew
L*(r,w,x) = x + loge(r/w)
∂L*∕∂w = – 1/w < 0
(10) 7.
Define the (minimum) average cost function as AC*(r,w,x) = C*(r,w,x)/x, where r and
w are the per-unit prices of capital and labor, respectively, x is the quantity of output
produced, and C*(r,w,x) is the (minimum) total cost function. Rigorously analyze
the effect of an increase in w on AC*.
AC*(r,w,x) = C*(r,w,x)/x = (1/x)[rK*(r,w,x) + wL*(r,w,x)]
∂AC*/∂w = (1/x)(∂C*/∂w)
From Shephard’s Lemma,
∂C*/∂w = L*(r,w,x) > 0
so
∂AC*/∂w = (1/x) L*(r,w,x) > 0