Econ 500 Problem Set 3
Economics 500 – Problem Set #3
Consider a firm producing an output Q using the two inputs “labor” (L) and “capital” (K) according to the production function Q = K.5L.5 (i.e., the amount of output produced Q is equal to the square root of K times the square root of L). Suppose that the firm is in a short-run situation in which K is fixed at one unit (K = 1) but labor is variable. Suppose that the user cost of capital, r, is $48 for each unit of K and that the wage rate, w, the cost of one unit of labor, is equal to $3.
a. Determine this firm’s TVC, TC, AVC, AC, and MC for the output levels Q = 0, 1, 2, 3, 4, 5, 6, 7, and 8 (some costs may not be defined for Q = 0). (8 points)
Q = √K .√L = 1 . √L
Q2=L
r=48 w=3 TVC = …show more content…
c. Determine the Average Product of Labor (i.e., APL = Q/L) for this case (you can obtain a general equation or, alternatively, simply compute APL for several values of L). What happens to APL in this case as L increases? Explain how this is consistent with the nature of AVC for this case. (3 points)
Q
0
1
2
3
4
5
6
7
8
L
0
1
4
9
16
25
36
49
64
APL
0
1
0.5
0.33
0.25
0.2
0.17
0.14
0.13
The APL decreases as L increases. The AVC for this case increases as L increases since AVC=w/Q/L.
d. Suppose now that capital K increases from one to four units (i.e., capital is now fixed at the level K = 4). Given this increase in K, determine the new values of TVC and TC (call them TVC* and TC*) for Q = 0, 2, 4, 6, and 8. How does TVC* compare with TVC? How does TC* compare with TC? How is MC affected by this increase in K? Explain. (3 points)
Q = √K .√L
Q = √4 . √L
Q = 2. √L
Q2 /4 = L
TVC = w.L
TFC = r.K
TC =TVC+TFC
Q
0
2
4
6
8
TVC
0
12
48
108
192
TC
48
60
96
156
240
MC
0
9
21
33
45
L
0
1
4
9
16
TVC*
0
3
12 …show more content…
Due to this the change in TC decreases leading to a decrease in MC.
e. Suppose that capital is now fixed at its original unit level (i.e., K = 1) but that the user cost of capital r increases from $48 to $192. Explain how (no calculations are required) this change in r affects the firm’s MC curve. Since the values of TFC (= rK) for this new situation and for part ‘d’ are the same, why is it that MC is affected in one case but not in the other? (3 points)
TFC = r.K TC = TFC + TVC
MC=∆TC / ∆Q
The changes in r does not affect the firm’s MC curve since the difference in TC remains constant even though TFC increases.
In problem d the change in Q is larger hence a decrease in MC is
Consider a firm producing an output Q using the two inputs “labor” (L) and “capital” (K) according to the production function Q = K.5L.5 (i.e., the amount of output produced Q is equal to the square root of K times the square root of L). Suppose that the firm is in a short-run situation in which K is fixed at one unit (K = 1) but labor is variable. Suppose that the user cost of capital, r, is $48 for each unit of K and that the wage rate, w, the cost of one unit of labor, is equal to $3.
a. Determine this firm’s TVC, TC, AVC, AC, and MC for the output levels Q = 0, 1, 2, 3, 4, 5, 6, 7, and 8 (some costs may not be defined for Q = 0). (8 points)
Q = √K .√L = 1 . √L
Q2=L
r=48 w=3 TVC = …show more content…
c. Determine the Average Product of Labor (i.e., APL = Q/L) for this case (you can obtain a general equation or, alternatively, simply compute APL for several values of L). What happens to APL in this case as L increases? Explain how this is consistent with the nature of AVC for this case. (3 points)
Q
0
1
2
3
4
5
6
7
8
L
0
1
4
9
16
25
36
49
64
APL
0
1
0.5
0.33
0.25
0.2
0.17
0.14
0.13
The APL decreases as L increases. The AVC for this case increases as L increases since AVC=w/Q/L.
d. Suppose now that capital K increases from one to four units (i.e., capital is now fixed at the level K = 4). Given this increase in K, determine the new values of TVC and TC (call them TVC* and TC*) for Q = 0, 2, 4, 6, and 8. How does TVC* compare with TVC? How does TC* compare with TC? How is MC affected by this increase in K? Explain. (3 points)
Q = √K .√L
Q = √4 . √L
Q = 2. √L
Q2 /4 = L
TVC = w.L
TFC = r.K
TC =TVC+TFC
Q
0
2
4
6
8
TVC
0
12
48
108
192
TC
48
60
96
156
240
MC
0
9
21
33
45
L
0
1
4
9
16
TVC*
0
3
12 …show more content…
Due to this the change in TC decreases leading to a decrease in MC.
e. Suppose that capital is now fixed at its original unit level (i.e., K = 1) but that the user cost of capital r increases from $48 to $192. Explain how (no calculations are required) this change in r affects the firm’s MC curve. Since the values of TFC (= rK) for this new situation and for part ‘d’ are the same, why is it that MC is affected in one case but not in the other? (3 points)
TFC = r.K TC = TFC + TVC
MC=∆TC / ∆Q
The changes in r does not affect the firm’s MC curve since the difference in TC remains constant even though TFC increases.
In problem d the change in Q is larger hence a decrease in MC is