Problem Set #9
Problem Set #10
Suggested Answers to Problems 11.1, 11.3, 11.5, 11.6 and 11.7
11.1 Digging clams by hand in Sunset Bay requires only labor input. The total number of clams obtained per hour (q) is given by
q = 100(L
where L is labor input per hour.
a. Graph the relationship between q and L.
b. What is the average productivity of labor in Sunset Bay? Graph this relationship and show that APL diminishes for increases in labor input.
APL = 100/(L. APL moves inversely with L See graph below.
c. Show that the marginal productivity of labor in Sunset Bay is given by
MPL = 50/(L
dq/dL = .5(100)L-.5
Graph this relationship and show that MPL< APL for all values of L. Explain why this is so.
See graph on the left. The Marginal “drives” the average. When the marginal is below the
average it pulls it down.
3. Power Goat Lawn Company uses two sizes of mowers to cut lawns. The smaller mowers have a 24-inch blade and are used on lawns with many trees and obstacles. The larger mowers are exactly twice as big as the smaller mowers and are used on open lawns where maneuverability is not so difficult. The two production functions available to Power Goat are:
| |Output per hour |Capital Input |Labor Input |
| |(Square Feet) |(# of 24” mowers) | |
|Large Mowers |8000 |2 |1 |
|Small Mowers |5000 |1 |1 |
a. Graph the q= 40,000 square feet isoquant for the first production function. How much K and L would be used if these factors were combined without waste?
Q=40,000 with 5*2 = 10 units of capital and 5*1 =5 units of labor. However, the production technology allows no substitutability. Thus, the production function is lexographical.
b. Answer part (a) for the second function.
For q=40,000, K =8 and L = 8.
See graph on right.
c. How much K and L would be used without waste if half of the 40,000-square-foot lawn were cut by the method of the first production function and half by the method of the second? How much K and L would be used if three-fourths of the lawn were cut by the first method and one-fourth by the second? What does it mean to speak of fractions of K and L?
|40,000 sq. ft. |LM |SM |
|LM/SM mix |L |K |L |K |
|0/100$% |0 |0 |8 |8 |
|25%/75% |1.25 |2.5 |6 |6 |
|50%/50% |2.5 |5 |4 |4 |
|75%/25% |3.75 |7.5 |2 |2 |
|100%/0 |5 |10 |0 |0 |
Fractions of K and L are fractions of an hour.
d. On the basis of your observations in part (c), draw a q=40,000 isoquant for the combined production functions.
|40,000 sq. ft. | |
|LM/SM mix |L |K |
|0/100$% |8 |8 |
|25%/75% |7.25 |8.5 |
|50%/50% |6.5 |9 |
|75%/25% |5.75 |9.5 |
|100%/0 |5 |10 |
[pic]
Observe that even when two technologies use fixed inputs, if combinations of those technologies can be made, substitutability between the primary inputs be made.
11.5 Suppose that
q = L(K( 0 ................
................
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