MICROECONOMIC THEORY I. ECONOMICS 665.
PROFESSOR: Alex Anas. TA: Yiqian Lu.
FALL 2014.
MIDTERM EXAMINATION 1.
October 15, 2014. TIME: 9:30 AM-10:50 AM PLACE: 424 FRONCZAK.
Books, notes or calculators are not allowed in the examination room. Please answer
each question. TOTAL POINTS = 100.
PROBLEM 1(30 Points): A firm has two factories. The first factory can
produce only in period 1 with cost function C1  w1 , w2 , y1   y12 min  w1 , 2w2  . Factory
2 can produce only in period 2 with cost function C2 (w1 , w2 , y2 )  (2w1  4w2 ) y2 .
(a) Derive the first factory’s production function and state what kinds of returns to
scale the technology has? Very briefly explain how you know your answers. (5
points)
ANSWER: The two inputs are perfect substitutes since the minimum cost
function is Leontief in input prices. The elasticity of scale is AC/MC = 1/2. Hence
the technology has diminishing returns. The production function is of the form
1/2
1 

y1   x1  x2  .
2 

(b) Derive the second factory’s production function and state what kind of returns
to scale the technology has? Very briefly explain how you know your answers?
(5 points)
ANSWER: The two inputs are perfect complements since the minimum cost
function is linear in input prices. The elasticity is scale is AC/MC = 1. Hence the
technology has constant returns. The production function is of the form
1 1 
y2  min  x1 , x2  .
2 4 
(c) Suppose that the firm minimizes C1  w1 , w2 , y1  
1
C2  w1 , w2 , y2  where
1 r
r is the interest rate, to produce a total target output amount of 1 unit. Prove
that the firm will produce all of this output in period 1 if r  0. (10 points)
ANSWER: The MC in period 1 is 2 y1 min(w1 , 2w2 ). The MC in period 2 is
2w1  4w2 . For any y1  1, producing in year 1 gives lower marginal cost because
marginal cost in period 1 rises with output and when output is 1
2 min(w1 , 2w2 )  2w1  4w2 . Hence, cost can always be reduced by shifting output
increments from period 2 to period 1.
1
(d) Is there an interest rate such that the firm will produce a target total output
amount of 1 unit some of it in period 1 and some of it in period 2? (10 points).
ANSWER: For this to be possible, the interest rate must be high enough that the
discounted period 2 marginal cost is less than the period 1 marginal cost for
some output units below 1 unit. This happens when r >
w1 + 2w2
(
min w1 ,2w2
derived from discounted MC2=MC1 which is 2 y1 min( w1 , 2 w2 ) 
)
-1. It is
1
 2w1  4w2 
1 r
after setting y1=1.
PROBLEM 2 (45 Points): A profit maximizing firm has a production function
f  x1 , x2 , b  which is strictly increasing and strictly concave in the two inputs x1 , x2 .
The prices of the two inputs are w1 , w2 . The price of the output is p.  is a parameter
beyond the firm’s control.
(a) Suppose that f  x1 , x2 ,   is strictly concave in  . Express this using an
inequality. (5 points)
ANSWER: f  x, tb+ (1- t )b  tf  x, b   (1  t ) f  x, b for any b, b  b and any
0  t  1.
(b) Suppose that f  x1 , x2 ,   is increasing in  . Prove using the Envelop
Theorem that the maximized profit increases in  . (10 points)
ANSWER:  ( p, w, b) = pf  x( p, w, b), b   wx( p, w, b)
Using the E.T.

f
p
0
b
b
(c) Suppose that an increase in  increases the marginal product of input 1
leaving the marginal product of input 2 unchanged. Derive the expressions
dx1 dx2
,
, where x1 , x2 are the profit maximizing input quantities.
d d
Simplify as much as possible the expressions you derived. (15 points)
ANSWER: The first order conditions for an interior profit maximum are:
pf1 ( x1 , x2 ,  )  w1  0
pf 2 ( x1 , x2 ,  )  w2  0
 pf11
 pf
 21
pf12   dx1 / d    pf1 

, f  0, f1  0.
pf 22  dx2 / d   pf 2  2
2
By Cramer’s rule:
 f1 f 22
dx1

d f11 f 22  f12 f 21
f1 f 21
dx2

d f11 f 22  f12 f 21
(d) Explain all you can tell about the signs of
dx1 dx2
,
that you derived in (c)
d d
(15 points)
ANSWER:
 f1 f 22
dx1

d f11 f 22  f12 f 21
f1 f 21
dx2

d f11 f 22  f12 f 21
The denominators are positive because the production function is strictly
concave, hence its Hessian is NSD. Therefore the 2nd minor should be positive.
For the same reason, f 22 is negative because it is the first minor. f1  0 by the
assumption in the problem. The sign of f 21 can be negative, positive or zero.
Hence,
 f1 f 22
dx1

0
d f11 f 22  f12 f 21
f1 f 21
dx2

 or  0.
d
f11 f 22  f12 f 21
PROBLEM 3 (25 Points): Consider the following non-differentiable technology:
f ( x)  x1/2 for 0  x  x0 and f ( x)  x01/2 for x  x0 .
(a) Draw the production function. (5 points)
ANSWER: In the range, 0  x  x0 , the student draws f ( x)  x1/2 which rises
concavely from zero until it reaches the height x01/2 and then stays at that height for
any larger x.
(b) Prove that the firm will never choose x  0 to maximize its profit. (5 points)
ANSWER: At x  0 the marginal product is infinitely large which means that
profit can be increased by producing more as long as the marginal cost of the
input, w, is finite.
(c) Derive and correctly express the indirect profit function,  ( p, w) of this
3
technology. (15 points)
ANSWER: The first order condition for a profit maximum in the range 0  x  x0
is p
1
2 x
0 w
 w x 
p
p2
p
p2
p 2 p2 p2
. If
 x  2   p
w 2 


2w
4w
2w
4w
2w 4w 4w
p
, then the maximum profit occurs at x  x0 . In that case profit is
2 x01/2
p x0  wx0 . The profit function is then expressed by:
p

p
x

wx
w

0
0

2 x01/2

 ( p, w)  
p2
p

w
 4w
2 x01/2
4