Question

Suppose that a firm has a production function ? = K^a ?^b , where

a>0 and b>0. K is capital and L is labor. Assume the firm is a price taker and takes the prices of inputs, (r and w) as given.

1) Write down the firm’s cost minimization problem using a Lagrangean.

2) Solve for the optimal choses of L and K for given factor prices and output Q.

3) Now use these optimal choices in the objective function to find the firms cost

function.

4) What properties does the cost function have? Specifically, what happens if you

double both factor prices? What happens if you increase Q? Suppose a+b=1, what

happens to your cost function?

5) Verify Shephard’s lemma by differentiating the cost function with respect to each

factor price.

Answer 1

1. The cost of production would be and the production function is
-12x = %
. The firm's cost minimization problem would be as below.

Min C = wl+rk

Subject to .

-12x = %

The Lagrangian function for minimization would be . The above minimization problem then reduces with respect to the Lagrangian function as below.

I = wL + K+XQ-KL

.

Min 1 = wL+rK+M(Q- Kºby

2. The FOCs would be as below.

$\frac{\partial l}{\partial L} = 0$ or
(wL +rK+ MQ- KºL")) = 0
or
0= 1-2779-)X + in
or
(-3Ꭲ 2Ꭹ Ꮔ)x = _
or
Y = 1-270.99
.

$\frac{\partial l}{\partial K} = 0$ or
(wL+rK+(Q- KºL')) = 0
or
T+A(-a Ka-1") = 0
or
T = X(ak-
or
Y = 71-
.

$\frac{\partial l}{\partial \lambda} = 0$ or
(WL + rK+XQ-KºL")) = 0
or
9 – К°L) = 0
or
-12x = %
.

Comparing first two FOCs, we have
471-00 - 1-4704
or
_ 971-p 1-27. 1 ID
or
4 - 471- I-1 У
or $\frac{K}{L} = \frac{aw}{br}$ or
K= 2
, which is the required optimal combination of inputs that minimizes the cost.

Putting this in the third FOC, we have $Q = (\frac{aw}{br}L)^{a} L^{b}$ or $Q = (\frac{a^a w^a}{b^a r^a}) L^{a+b}$ or
L = 64/(a+b)/(a+b) qa/(a+b) /(a+b)
, and since
K= 2
, we have $K^* = \frac{aw}{br}L^*$ or
... aw 3/(a+b)/(a+b) K=bre brya/(a+b) /(a+b) /(a+b)
or
K = qb/(a+b) /(a+b) 15/(a+b)/(a+)
. These are the required input demand functions, ie the optimal choices of input given w, r and Q.

3. The cost function would be
C =wl + rK
or
C = w /ta+b)/(a+b) /(a+b) /(a+5Q t ) + ba+) /(a+5) /(a+b), 1/(a+5) 1/(a+b)
or
64/(a+b) /(a+b)/(a+b) q/ +8) /(a+b),,,6/4+5 O/la+) b/ +8) -!/(a+8
or
8/(a+b) qb/(a+b) qa/(a+b) + 2/(a+b)) w/(a+b)/(a+b) 1/(a+b)
or
a +b C = /(a+b) */(a+b)/1/(a+b)) w/la+b),/(a+b)
.

4. If w and r are doubled to be 2w and 2r, we have the cost as
C = 6 ( +b qa/(a+b)]b/(a+b)) (2w)/(a+b)(2r)/(a+b)(1/(a+)
or
C = (20/(a+blya/(a+b)) (104670 oto) (w)/(a+b)(r)a/(a+b)Q1/+5)
or
a+b C = (2) qa/(a+b)/b/(a+b)) (w)/(a+b)(r)/(a+b) 1/(a+)
or
C = 20
. This means that doubling the inputs will also exactly double the cost.

We have
+b)) /(a+) -) /(a+b)/(a+b)/(a+b)-1 /(a+b))
or
26/(a+b)/(a+b)(1--)/(a+b) qa/(a+5) /(a+b)
or
де w/(a+b) /(a+b) qa/(a+b)/(a+b) - (1--6)/(a+b) >
. This means that for an increase in quantity, the cost also increases.

For a+b=1, we have b=1-a, and putting it in the cost function, we have . In this case, the cost function is now linearly related to Q.

( et - Ia) = 2

5. For the given cost function, we have
20 = ( bo a +b +6) w/(a+b)–14/(a+b)/(a+b) /(a+b) /(a+b)
or
qa/(a+b) /(a+b) +) w-a/(a+b)/(a+b)/(a+b)
or
20 a+b) qa/(a+b) ) wa/(a+b)/(a+b) /(a+b) Dw
or
OC / /(a+b) /(a+b) = (a/(a+b)qua/(a+b))
, and since we have
L = 64/(a+b)/(a+b) qa/(a+b) /(a+b)
, we can verify that
+7 De
.

Also, we have
20 = (a/(a+b)) ( ) wb/(a+b)/(a+b)–1Q1/(a+b)
or
= (a/(a+b)/b/(a+b) /(a+b),.-6/(a+b) /(a+b
or
ac 77/ a+b) wb/(a+b), -b/(a+ 55/(a+b)) 1/4+ ar
or
ac q/la+b) /(a+b) b/(a+b)/(a+b) 1/(a+b)
, and since we have
K = qb/(a+b) /(a+b) 15/(a+b)/(a+)
, we can verify that
c
.

Hence, since
+7 De
and
c
, the Shephard's lemma is verified.