The production function Y=K1/3L2/3−G exhibits________ returns to scale in K and L. (Assume G is a positive real number.)

Question

The production function  Y=K1/3L2/3−G  exhibits________ returns to scale in K and L. (Assume G is a positive real number.)

BlackTom AI · Accepted Answer

The production function is Y = K^{1/3} L^{2/3} − G, where G > 0.

First, consider homogeneous components: K^{1/3} L^{2/3} is a Cobb-Douglas function with exponents summing to 1, which by itself would exhibit constant returns to scale if it were the entire production function.

However, the −G term is a fixed amount that does not depend on K or L. This fixed cost term changes how output scales when we scale inputs.

Let us scale inputs by a factor t > 1: f(tK, tL) = (tK)^{1/3} (tL)^{2/3} − G = t^{1/3} t^{2/3} K^{1/3} L^{2/3} − G = t (K^{1/3} L^{2/3}) − G.

Now compare f(tK, tL) to t f(K, L): t f(K, L) = t [K^{1/3} L^{2/3} − G] = t K^{1/3} L^{2/3} − t G.

Subtracting, we get f(tK, tL) − t f(K, L) = [t (K^{1/3} L^{2/3}) − G] − [t K^{1/3} L^{2/3} − t G] = (t − 1) G.

Since G > 0 and t > 1, (t − 1) G > 0, so f(tK, tL) > t f(K, L) for all t > 1. This is the hallmark of increasing returns to scale in K and L.

Therefore, the function exhibits increasing returns to scale in K and L because the fixed cost term G causes the total output to scale more than proportionally when inputs are scaled up. The key takeaway is that the −G term breaks the pure constant-returns-to-scale property of the K^{1/3} L^{2/3} part, leading to increasing returns due to the fixed deduction becoming relatively less significant at larger scales.

The production function Y=K1/3L2/3−G exhibits________ returns to scale in K and L. (Assume G is a positive real number.) Single choice

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