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ECON 520: Intermediate Microeconomics
Problem Set 3
Solutions
Professor D. Weisman
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1. Q  KL with  2 and
w  1 . [Note: MPK  L and MPL  K ]
a) 32 units of output are to be produced. Hence,
(1)
32  KL
Also, the firm’s equilibrium condition is given by;
(2)
MPK MPL
L K

   L  2K
r
w
2 1
Hence solve the following equations simultaneously:
(3) 32  KL 
2
  32  K  2 K   2 K
(4) L  2 K 
Hence,
b)
k *  4, L*  8
a) minimum cost : C (32)  4(2)  8(1)  16
c)
K
Iso-cost equation: 16=wL+rK, or
8
K=(16/r) – (w/r)*L
K= 8-(1/2)L
4
Q=32
8
16
2. Let the production function be given by Q  2 K  L , with
L
r  3 and w  1 .
Observations: (1) The production function is perfect substitutes
(2) Capital is 2times more productive than labor per unit, but capital is 3 times the cost of
labor per unit. Hence, conjecture that use only labor in production.
In terms of the firm’s equilibrium condition
1)
MPK  2, MPL  1 and
MPK MPL
2

 1.
since
r
w
3
Hence, the marginal product per dollar of labor everywhere exceeds the marginal product per dollar of
capital. Hence, we have a corner solution in which only labor is used in production. To produce 10 units
of output using only labor (K=0) requires 10 units of labor.
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K *  0, L*  10, Q  L
a) minimum cost : C (10)  10 1  10
b)
Iso-cost line: C  wL  rK , C  1L  3K , solve for K with C=10,
K
10 1
 L
3 3
Note 1: slope of iso-cost line  
slope of isoquant  
1
.
3
1
.
2
Note 2: Whenever slope of isoquant
 slope of iso-cost line, optimal input combination will entail a
corner solution (i.e., use only labor or only capital in production)
K
5
Linear
Isoquant
Corner
solution
-1/2
10/3
-1/3
Q=10
Iso-cost line
L
10
3. Let the production function be given by
Q  min 2K , L with r  2 and w  1 .
Observations: (1) The production function is perfect complements.
(2) Efficient production requires operation at the vertex points o the isoquants, or where
there
is
neither
2K  L or K 
excess
capital
nor
excess
labor.
This
implies
that
1
L.
2
Hence, each unit of output requires 1/2 unit of capital and 1 unit of labor in efficient production.
a) The total cost of producing 20 units of output is
1
2




(1) C (20)  20  r  1w   20 1  1  40
b) Each unit of efficient production requires 1/2 unit of capital and 1 unit of labor. Hence, to produce 20
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units of output efficiently requires
Note:
K *  10, L*  20 .


Q  min 2 K , L  min  20, 20   20
 terms are equal 
C  wL  rK with C  40, r  2 and w  1 .
c) The iso-cost line is given by
40  L  2 K , or K  20 
1
L.
2
K
L-Shaped isoquant
Expansion path
20
10
Q=20
1/2
-1/2
20
40
L
Supplemental question
4. Derive the cost functions associated with the production functions in questions 2 and 3?
a) Q  2 K  L with
r  3 and w  1 .
MPK 2
MPL
 1
 use only L in production.
r
3
w
Each unit of output requires 1 unit of labor, which implies that
(1) C (Q)  w  Q  1 Q .
Note C (10)  110  10 (Same answer as in 2a)
b)
Q  min 2K , L with r  2 and w  1 .
Each unit of output requires
1
K and 1L . This implies that the cost function is given by
2
1

1
 2   11 Q
2

2

(2’) C (Q )  2Q . Note C (20)  2  20  40 (Same answers as in 3a).
(2) C (Q)   r  1w  Q  