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Marginal Rate of Technical Substitution

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Problem 7.1 Marginal Rate of Technical Substitution. The following production table provides estimates of the maximum amounts of output possible with different combinations of two input factors, X and Y. (Assume that these are just illustrative points on a spectrum of continuous input combinations.)
A. Do the two inputs exhibit the characteristics of constant, increasing, or decreasing marginal rates of technical substitution? How do you know?
B. Assuming that output sells for $3 per unit, complete the following tables:
X Fixed at 2 Units
Y Fixed at 3 Units
C. Assume that the quantity of X is fixed at 2 units. If output sells for $3 and the cost of Y is $120 per day, how many units of Y will be employed?
D. Assume that the company is currently producing 162 units of output per day using 1 unit of X and 3 units of Y. The daily cost per unit of X is $120 and that of Y is also $120. Would you recommend a change in the present input combination? Why or why not?
What is the nature of the returns to scale for this production system if the optimal input combination requires that X = Y?
Question 7.1 Is use of the least-cost input combinations (technical efficiency) a necessary condition for profit maximization? Is it a sufficient condition? Explain.

Question 7.5 Explain why the MP/P relation is deficient as the sole mechanism for determining the optimal level of resource employment.

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The solutions provides answers to questions that deal with marginal rate of technical substitution, marginal product, Marginal Revenue Product, returns to scale, optimal level of resource employment

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Problem 7.1 Marginal Rate of Technical Substitution. The following production table provides estimates of the maximum amounts of output possible with different combinations of two input factors, X and Y. (Assume that these are just illustrative points on a spectrum of continuous input combinations.)

Units of
Y Used Estimated Output per Day
5 210 305 360 421 470
4 188 272 324 376 421
3 162 234 282 324 360
2 130 188 234 272 305
1 94 130 162 188 210
1 2 3 4 5
Units of X used

A. Do the two inputs exhibit the characteristics of constant, increasing, or decreasing marginal rates of technical substitution? How do you know?

The two inputs exhibit the characteristics of decreasing marginal rates of technical substitution
This can be inferred from the fact that as per calculations below marginal product decreases in the direction of arrow

TABLE 1

Marginal product keeping X constant
Units of
Y Used
5 22 33 36 45 49

4 26 38 42 52 61

3 32 46 48 52 55

2 36 58 72 84 95

1 94 130 162 188 210
1 2 3 4 5
Units of X used

Marginal product decreases in the direction of arrow

TABLE 2

Marginal product keepingY constant
Units of
Y Used
5 210 95 55 61 49

4 188 84 52 52 45

3 162 72 48 42 36

2 130 58 46 38 33

1 94 36 32 26 22
1 2 3 4 5
Units of X used

Marginal product decreases in the direction of arrow

B. Assuming that output sells for $3 per unit, complete the following ...

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