ECON 8010
TEST #1 SOLUTIONS
FALL 2015
Instructions: All questions must be answered on this examination paper. No additional
sheets of paper are permitted; use the backs of 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. Time Limit: 75 Minutes.
(10)
1. ASSUMPTIONS
a.
What assumption about the production function x  f ( K , L) ensures that the
short-run demand curve for labor of a profit-maximizing, competitive firm is
downward-sloping? Rigorously explain.
∂L*(p,w;K)/∂w = 1/pfLL < 0 iff production is subject to diminishing marginal
productivity of labor; i.e., fLL < 0.
b. What assumption about the (minimum) cost function C(r,w,x) ensures that the
conditional (constant-output) demand curves for capital and labor of a firm
operating in competitive input markets are downward-sloping? Rigorously
explain.
Concavity of C(r,w,x) in r and w. Specifically, from Shephard’s Lemma we
have ∂C/∂r = K*(r,w,x) > 0 and ∂C/∂w = L*(r,w,x) > 0 . Differentiating again
with respect to r and w, respectively, and invoking concavity of C(r,w,x) in r
and w gives ∂2C/∂r2 = ∂K*/∂r < 0 and ∂2C/∂w2 = ∂L*/∂w < 0.
(10)
2. TRUE or FALSE and EXPLAIN: Label each statement TRUE or FALSE, and
rigorously explain or justify your answer.
For a profit-maximizing firm operating in competitive output and input markets,
a. ___TRUE___ “the short-run supply curve for output is upward-sloping.”
Evaluate the production function x  f ( K , L) at the optimum values of L and
x: x*(p,w) ≡ f [ K , L*(p,w)]. Now, differentiate this identity with respect to p:
∂x*/∂p ≡ fL (∂L*/∂p) ≡ fL [(-fL)/(p fLL)] ≡ - (fL)2/(p fLL) > 0
b.
FALSE
“an increase in product price, ceteris paribus, increases the longrun demand for labor.”
∂L*/∂p ≡ - (fL p fKK + fK p fKL) /p2[(fKK fLL) ̶ (fKL)2] < ≡ > 0
(10)
3. Suppose that a worker’s output x is determined by the production function
x = f (e, h),
where e is the effort level of the worker and h is the fixed level of his human capital
(education, training, work experience, etc.) The marginal product of the worker’s
effort is assumed to be strictly positive and decreasing in effort, so that fe > 0 and fee <
0, respectively. The per-unit price of output is normalized at p = 1. The worker’s
compensation (or “revenue”) is determined by the value of the output he produces and
the amount he is paid per-unit of output (the “piece rate”), given by 0 < b < 1. The
cost of effort is given by the function C(e), and it is assumed that Ce > 0 and Cee > 0.
The worker chooses effort e to maximize the net revenue from working, R[x,b] – C(e).
Given the stated assumptions, rigorously analyze the effect of an increase in the piece
rate b on the worker’s effort e.
Net revenue is given by R(e, h) = b f (e, h) ̶ C(e). The first-order condition for a
maximum of R with respect to e is
Re = b fe ̶ Ce = 0.
Evaluate the first-order condition at the optimal value of e, e*, and totally
differentiate the identity
b fe [e*(b)] ̶ Ce[e*(b)] ≡ 0
with respect to the parameter b:
fe [e*(b)] + b fee (de*/db) ̶ Cee (de*/db) ≡ 0 → fe + (de*/db) (b fee ̶ Cee) ≡ 0
(de*/db) ≡ - fe /(b fee ̶ Cee) > 0
(25)
4. Suppose that the firm’s profit function is
π*(p,w) = p ∙ loge (p/w) – p
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 = loge(p/w) + p(1/p) – 1 = loge(p/w) = x*(p,w)
b. Derive the firm’s labor-demand function L*(p,w).
- ∂π*/∂w = -(-p/w) = p/w = L*(p,w)
c. What is the firm’s production function x = f(L)?
From (a) and (b), respectively, x*(p,w) = loge(p/w) and L*(p,w) = p/w.
Therefore,
x = f(L) = logeL
(20)
5. Suppose that a cost-minimizing firm’s (minimum) cost function is
C * (r , w, x)  ( x  x 1 )  (r  w)1 / 2
where r is the per-unit price of capital, w is the per-unit price of labor, and x is output.
a. Derive the firm’s conditional demand function for labor, and rigorously analyze the
effect of a change in w on the demand for labor.
L*(r,w,x) = ∂C*∕∂w = L *(r , w, x)  (1/ 2)( x  x 1 )  (r / w)1/2
∂L*∕∂w =  (1/ 4)( x  x 1 )  (r )1/2 ( w) 1/2  0
b. Derive the firm’s marginal cost function, and rigorously analyze the effect of an
increase in x on marginal cost.
MC*(r,w,x) = ∂C*∕∂x = (1 ̶ x-2) (r  w)1/2
∂MC*/∂x = 2x-3 (r  w)1/2 > 0
(15) 6. 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 elastic is the demand for the output
it helps to produce.”
TRUE. ∆/∆ = s > 0
c. “The demand for labor is more elastic, the more important it is in the total cost of
production.
UNCERTAIN. ∆/∆s =  +  is indeterminate in sign
(10) 7. MATCHING: Fill in the blank space with the number of the phrase that best matches
the mathematical expression.
a.
______5_____ L *(r , w, x) / r  K (r , w, x) * / w
1. Euler’s Theorem
b.
______1_____ f K  K  f L  L  r  f ( K , L)
2. Hotelling’s Lemma
c.
______4_____ C * w  L* (r , w, x)
3. Reciprocity Relation
d.
______ 2_____  * p  x* (r , w, p)
4. Shephard’s Lemma
e.
______ 3______ ∂x*∕∂w = -∂L*∕∂p
5. Symmetry