1. Use algebra to derive the cost function:
• To solve for K as a function of q and L. Show your work, and verify that you have this solution:
K=
q 2 0.022 L
.
• Write the cost function. Cost is equal to the sum of the expenditures to purchase capital plus the
expenditure
The product function is given as q=0.02 √KL.
Q2 = (0.02)^2 KL
K= (q^2) /((0.02)^2L)
The cost of the firm is given as c=wL +rK
At equilibrium MPl/MPk = w/r
Or can be seen as (k/l) = (w/r)
wL = rK
c=2wl=2rk
L=C/2w = 2rk
L=c/2w, k=c/2r
Q= 0.02√c*c/4wr = 0.01 c/√wr
C=(q√wr)/(0.1) = 100 √wr q
The cost function is given as
C=100√wr q
The cost function shows us the production expenses changing at different output levels. The
average variable cost appears to increase as the level of production increases.
2. Use Excel to create and graph isoquant curves:
30,000
25,000
20,000
15,000
10,000
5,000
0
0
20
40
60
80
100
120
140
Q
Qty of L that must be combined with K=5000
5
10
15
to produce each quantity of output (q)
12
1250
115
3. Consider the short run situation in which K is fixed at 5000. Assume r = .05 and w = 40.
Open a new Excel worksheet for cost information. Note the difference between your
production worksheet, in which the first column stored possible values of L, and this new
cost worksheet in which the first column will store possible values of q. The variable
represented in the first column will be graphed on the horizontal axis of the scatter-plot.
For the isoquant diagram, L is shown on the horizontal axis. The new cost worksheet will
be used to graph cost functions, with quantity of output on the horizontal axis.
800
700
600
500
400
300
MC-EQ
200
100
0
0
2
4
6
8
10
12
14
16
Here, we solve for L as a function of q and K with K = 5000 in the short run: Q =
0.02 K^0.5 x L^0.5
L^0.5 = q/ 0.02K^0.5 (L^0.5)^2 =
(q/0.02 x K^0.5)^2 L = q^2/(0.02)^2
(K^0.5)^2
L = q^2 / 0.0004 K L = 2500
q^2 / K
L = 2500 q^2 / 5000 L =
18
20
0.5q^2
L is the only variable input in the short run so wage (w) is multiplied by L which gives you the total variable
cost.
TVC = WL
TVC = (40) [0.5q^2] TVC =
20 q^2
One estimate of Marginal cost by computing the change in total cost as output increases would be: MC =
2 (20*1)/0.0004 * 5000
MC = 20
A second estimate of Marginal cost as output increases: MC = 2
(40*1)/0.0004*5000
MC = 40
4. Find equilibrium P & Q in the perfectly competitive market.
5. Complete the following table for a firm that is producing the profit-maximizing level of
output.
Revenue
Optimal firm q
Short-run equilibrium P
Revenue = q*P
$$$
8
320
2,560
TFC
TVC (incurred by a firm that
250
1280 ((40*q^2)/
is producing the optimal (Q)
TC=TVC+TFC
(0.02^2(5,000)))
1,530
Profit = Revenue - TC
2,560-1,530 = 1,030
Cost
Profit
6. Generate a graph to show the optimal quantity that will be produced by each competitive firm,
and the resulting profit. This graph will include 4 curves to show:
800
700
600
500
ATC
400
300
200
100
0
0
100
200
300
400
500
600
700
800
900 1,000 1,100 1,200 1,300 1,400 1,500 1,600 1,700 1,800
Yes the graph is consistent is consistent with the profit computation. The graph shows that there
are competitive firms that profit according to the direction of the graph.
7. Assume that potential entrants will have exactly the same cost function as the existing
firms. Will new firms enter the market? Why or why not?
New firms will not enter the market place because there are monopolistic forms within the
market and have fixed prices of commodities. This is assuming that potential entrants will
have the same cost function. Whether greater than or equal 120.
8. You work for a firm that produces an input that is used by these competitive firms. Your
marketing vice president has asked you to provide analysis to support the marketing
department's strategic planning committee. They understand that the industry is not
currently in long-run equilibrium, and they have asked you to help them estimate the
output that will be produced and the number of firms that will exist when the industry
reaches long-run equilibrium. This will require several steps:
720-0.5Q = 40Q/n and Q= nq
Q is replaced by nq
720-0.5nq=40nq/n
720-0.5nq=40q
40q+0.5nq=720
Q(40+0.5n)=720
Q=720(40+0.5n)
720-0.5nq=40q
0.5nq=720-40q
0.5n=720/q-40
n=1440/q-80
Long Run Equilibrium
Output of each individual firm
Industry Output
Number of firms
800
320
100
8,000
7,000
6,000
5,000
4,000
3,000
2,000
1,000
0
0
2
4
6
8
10
12
14
16
18
20