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# Technology and Production Functions

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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

https://brainmass.com/economics/technology/technology-production-functions-449195

#### 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.

\$2.19

## Production function, equilibrium output, and resources employed by a high end solid gold bracelet manufacturer.

The technology of a firm making high end, solid gold bracelets in Soho (NYC) is described by the production function:

q = 6.0 L3/4K1/5

where q is the number of bracelets produced per year, L is the number of metallurgist employed by this firm and K is the number of capital units used, measured in square footage of factory floor space. Capital is available at a cost of \$3.745 per square foot per year and this price is guaranteed regardless of the size of the facility the firm requires at any given time.

The product and labor market conditions facing this firm are described by:

Product Market: Labor Market:
Demand: P = 13,200 - 0.8Q w = 95,000 - 2.5 QL
Supply: P = 258.18 + 0.18Q w = 15,000 + 1.5 QL

where P is the price (rounded to the nearest quarter dollar) of bracelets similar to those produced by the firm in question on the national market, Q is the number of similarly styled bracelets produced by all firms in the national market, w is the annual salary, measured in dollars per year, paid to metallurgists who work in this industry, and QL is the number of metallurgists employed in this industry nationally.

The following relate only to the firm whose technology is given above:

1. Initially, the firm leases a facility with 248,832 square feet. With this size facility, determine the output level of the firm, the number of workers it employs and the profit of the firm.

2. Three years later, the firm renegotiates its capital lease so that it can employ exactly the amount of capital it feels necessary to maximize its profits. Determine the output level of the firm, the number of workers it employs and the profit of the firm.

3. Suppose the firm decides to set an output target so that it will be making a 50 per cent return on its capital (i.e., it wants to make a profit equal to 50 per cent of its capital cost). Determine the output level of the firm, the number of workers it employs and the profit of the firm.

4. Suppose the metallurgists working for the firm in question negotiate a 2.5 per cent wage hike over and above the current wage in the labor market. Determine many workers, if any, lose their jobs at the new wage. Then determine the substitution and output effects on labor of the wage

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