Question

(c) Assume that output is governed by a production function which features decreasing returns to capital, decreasing returns to labor, and constant returns to scale. Also assume that everyone in the population works, and is employed. You can maintain these assumptions for the remaining parts of the question. (i) Suppose that both capital and labor increase by 3 percent. By how much will output increase? Suppose that both capital and labor decrease by 5 percent. By how much will output decrease? (ii) Suppose there is a war which reduces the capital stock dramatically. What will happen to output, assuming there is no change in the population? What will happen to output per capita, assuming there is no change in the population? Explain, using a diagram to illustrate each case (iii) Suppose there are equal, successive increases in the capital stock over time with no change in population. What will happen to output with each successive increase in capital? What will happen to output per worker with each successive increase in capital? Explain, using a diagram to illustrate each case

Answer 1

The production function in the Solow growth model is Y=f(K,L) or the output per worker is y=f(k).

Also it is given in the question that there are decreasing returns to both labor and capital.

i) If both capital and labor increase by 3%, the output by production function would fall by 9% assuming that the function is Y= K.L

If both labor and capital fall by 5 % each, the output will increase by 25 units according to the same function.

ii)If we assume there is a war which reduces the capital stock but the population remains unchanged, the output will increase as per the conditions and the nature of the production function.

However, since the capital stock reduces and labor remains same, the capital per labor also falls after the war. Thus if the economy was in steady state before the war, the economy has a capital stock that is lower than the steady state. The capital per worker moves from k₁ to k₂.

As the capital per worker falls, the output per worker also falls from y1 to y2.

iii) With each successive increase in the capital, since there are DRC, the output will fall keeping the population constant over time.

Since there is increase in capital but no change in labor, the capital labor ratio will increase over time and thus the output per worker will also increase successively.  Thus in the picture above, the capital per worker will move to the right of k1 and the output per worker will be more than that the steady state level i.e. above y1.