Question

A student wrote on the last year Econ exam the following: ” Every production function exhibits diminishing returns to scale because professor said that all inputs have diminishing marginal productivities. So when all inputs are doubled, output must be less than double.” How would you grade this answer?

Answer the previous question using two specific production functions as examples:

(a) A fixed-proportions production function.

(b) A Cobb-Douglas production function q = √ KL

Answer 1

Answer: the comment of student is wrong.

a.

This is the production function in which inputs are used at equal proportion in order to get output. Suppose there are two inputs K and L and it has a fixed-proportion as 1: 1, means if K is 10 units then L is also 10 units and by the use of these two inputs the output (Q) becomes 50 units. Now, if K increases to 20 units but L remains the same as 10 units there will be no change in Q – it will be still at 50 units. Q would be just double, 100 units, if K and L both become double (20 units each). Therefore, the answer of student doesn’t stand; output becomes double if inputs are double.

b.

All inputs don’t give diminishing returns.

Suppose in this production function the power of K is 0.5 and the power of L is also 0.5, this happens because K and L are under the square root. If the aggregate of powers becomes exactly 1, the production function has constant returns to scale.

Power of K + Power of L = 0.5 + 0.5 = 1

This means every increase in K and L gives a constant return to q; therefore, there is no question of diminishing returns – this could only be possible if the aggregate of powers of K and L gives the number less than 1.