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.

FA25 ECON 302 002 Homework #2 (Model of Production)

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

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FA25 ECON 302 002 Homework #2 (Model of Production)

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

查看解析

登录即可查看完整答案

类似问题

Gale Furniture uses labor, capital, and wood as inputs in its production of office desks. When Gale Furniture doubles the quantity of labor, capital, and wood, the quantity of office desks doubles. Gale Furniture’s production process is characterized by

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Indivisibility of capital causes _________ returns to scale due to ________ in the __________ production model.

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If bread is produced by using a constant returns to scale production function, then if the:

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10. When a firm doubles its inputs and finds that its output has more than doubled, this is known as: A. Economies of scale B. Constant returns to scale C. Diseconomies of scale D. A violation of the law of diminishing returns

更多留学生实用工具

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