Production Problem Set Flashcards
Consider the production function
Q = K + √L
L. Assume W = $1 and R = $50
(a) Verify that the the firm uses no capital to produce Q¯ = 10
- Use the “ Bank for Buck” condition to solve for L
MPL/W = MPK/R
MPL is the marginal product of labor
MPK is the marginal product of capital
1/ 2√L = 1/R
Solve for L
L = (R/2W)^2
- Plug into Q to solve for K
Q = K + √L
Q = K + (R/2W)^2
Solve for K
K = Q - R/2W - However, we need to ensure that
K is non-negative because you can’t have negative capital.
K = Q - R/2W
Q - R/2W <0
Q < R/2W
- If the above holds, use only L since K cannot be negative. Plug K as 0:
Check Back: Q = K + √L
Q = K + √L = 0 + √L
L = Q^2
- Putting this all together:
L(Q-, R, W) = 1/4(R/W)^2 if Q- >=1/2(R/W)
K(Q- , R, W)
= max ( Q- - √L(Q,R,W), 0)
- Checking R/2W = 50/2 = 25 and we see from above that this firm only uses L
For Q = 10, at what R does the firm start to use capital (with W = 1)
Q < R/2W
Use K when :
Q = R/2W
10 = R/2(1)
R = 20
For W = $1 and R = $50, at what Q¯ does the firm start to use capital?
Use K when :
Q = R/2W
Q = 50/2(1) = 25
A good is produced with production function:
F(K, L) = (KL)^1/4 − a, where
a ≥ 0. The price of a unit of labor L is W, the price of a unit of capital K is R.
Compute L(Q ¯ , R, W ) and
K(Q ¯ , R, W ).
Derive C(Q, R, W ¯ ).
- Using the bang for buck condition
MPL/W = MPK/R
Solve this calculating the marginal product of labor and capital
K = W/R * L
L = RK/W
- Now take
Q(L) = F(K, L) = (KL)^1/4 − a
Plug K = W/R* L
3.
Rearranging equation, we get:
L(Q, R, W ¯ ) = (Q¯ + a) + ROOT(R/W)
K(Q, R, W ¯ ) = (Q¯ + a) + ROOT(W/R)
C(Q, R, W) = RK + W L
Plug Values of K in
=2(Q¯ + a)^2*√W R
Suppose the input prices are W = R = 1. At what quantity are the average costs minimized? How does it depend on a?
C(Q, R, W ¯ )
=2(Q¯ + a)^2 * √W R
Average Total Cost is Total Cost/Q.
The previous cost equation is
C(Q, R, W ¯ ) = 2(Q¯ + a)^2 * √W R
W = R = 1
C(Q, R, W ¯ ) = 2(Q¯ + a)^2
Finding Average Total Cost
= 2(Q¯ + a)^2 / Q
To find the quantity where average total cost is minimized we take the derivative of average cost with respect to Q and equal it to 0
D(C(Q ¯,1,1)/Q / dq
= 4(Q ¯,+a) *Q ¯- 2(Q ¯ + a)^2 /Q^2
a = Q ¯
Checking Secound Order Condition
SOC > 0
This verify’s that it is a minimum. AC is minimized when MC = AC
Consider the production function Q(K, L) = K2L. You must produce 8,000 units per day, and face a $400 daily rental rate for each unit of capital and a $200 daily wage rate per worker
Due to a contract with the union representing your workers, you must employ exactly
80 workers per day. You can vary the amount of capital you use. What is the cost of producing 8,000 units per day?
Rearrange Q(K,L) to get K(Q,L)
Q(K, L) = K^2L
K = √Q/L
Plug Q = 8000 and L = 80
K = √Q/L = √8000/80 = 10
C(Q) = RK + WL = R√Q/L + WL
= 400 * 10 + 200 *80 = 20,000
What is the marginal cost of production at Q = 8, 000 given that L is fixed at L = 80?
Background Info:
Consider the production function Q(K, L) = K2L. You must produce 8,000
units per day, and face a $400 daily rental rate for each unit of capital and a $200 daily wage
rate per worker.
C(Q) = RK + WL =. R√Q/L + WL
Taking the derivative of the cost function with respect to quantity
C(Q) = RK + WL =. R√Q/L + WL
dC(Q, L) /dQ
= RQ^-1/2 * L^-1/2 + WL / dQ
= R/2 * 1/ √QL
= 200 / √8000*80 = 1/4
If it were the case that k were fixed at K¯ = 10 but L was variable, what is the marginal
cost of production at at Q = 8,000?
Background Information:
Consider the production function Q(K, L) = K2L. You must produce 8,000 units per day, and face a $400 daily rental rate for each unit of capital and a $200 daily wage rate per worker.
dC(Q,K)/DQ = d(RK+WL)/dQ =
= W/K^2 = 200/100 = 2
At the end of the union contract, you are free to vary both L and K. Based on W, R and the current marginal products, do you think the firm will adjust K and L?
Explain.
Compute the ”bang for the buck”; recall that this compares MPK/R to MPL/W
(note that MPL is marginal product of labour and MPK is marginal product of capital)
MPK/R = 2KL/R = 21080 / 400 = 4
MP
