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Technology and Production Functions

Microeconomics Exercises

1. You might think that when a production function has a diminishing marginal rate of technical substitution of labour for capital, it cannot have increasing marginal products of capital and labour. Show that this is not true, using the production function Q = K2L2

2. A firm produces a quantity Q of breakfast cereal using labor L and material M with the production function Q = 50√(ML) + M+ L.
a) Are the returns to scale increasing, constant, or decreasing for this production function?
b) Is the marginal product of labour ever diminishing for this production function? If so, when? Is it ever negative, and if so, when?

Suppose that the production function is Q = K2L0.5 where
Q  Output produced
L  Labour input used
K  Capital input used

Suppose that the firm faces a wage rate of £10 and rate of rental is £20.
In the short run, K is fixed at 2.

Questions 10-12 are based on the above information

10. The production function exhibits

Decreasing Returns to Scale and Increasing Returns to K
Increasing Returns to Scale and Increasing Returns to L
Decreasing Returns to Scale and Diminishing Returns to L
Increasing Returns to Scale and Diminishing Returns to L

11. How many units of K is employed to produce Q = 4 in the long run?
1
2
3
4

12. How many units of L is employed to produce Q = 4 in the long run?

1
2
3
4

Solution Preview

1. First, the marginal rate of technical substitution of L for K is MPL/MPK.

MPK = dQ/dK = 2KL^2 and MPL = dQ/dL = 2LK^2.

Thus, MRTS of L for K is 2LK^2/2L^2K = K/L.

If this quantity (K/L) is decreasing, then the following can happen,
i) K must be decreasing and L is increasing or constant
ii) L is increasing and K is decreasing or constant
iii) both K and L are increasing but L increases faster

In case iii), both MPL and MPK are increasing.

2. Given Q(M,L) =50(ML)^0.5 + M + ...

Solution Summary

Technology and production functions are examined. The returns on scales increasing are given.

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