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Homework answers / question archive / Suppose the cost of capital is 24, the cost of labor is 8 and the firm has an objective to produce 24 units of output with the following production function: Q = K3/2L1/2 1
Suppose the cost of capital is 24, the cost of labor is 8 and the firm has an objective to produce 24 units of output with the following production function: Q = K3/2L1/2 1. Using a factor of 2, show whether this production function exhibits increasing, decreasing or constant returns to scale and define it. In one paragraph, explain if the firm should scale up, scale down or neither. Also, identify the fixed input you described from part A, and explain how it can become variable in the long run. 2. Find the optimal levels of K and L by using a LaGrange to solve the cost minimization problem. Graph the optimal bundle with capital on the horizontal axis and label it point A. Be sure to label the axes and curves.
Ans. Production function, Q = K^1.5 * L^0.5
Suppose both the inputs are increased by the factor of 2, then new output,
Q' = (2K)^1.5 * (2L)^0.5
=> Q' = 4 * (K^1.5 * L^0.5) = 4*Q
Thus, the increase in production is more than the increase in inputs. So, tbe production function exhibits increasing returns to scale. So, if the firm scales up its operations, the relative increase in output will be more than the increase in scale. So, the average cost will decrease if the operations are scaled up. Thus, the firm should scale up its operations to reap the benefits of the economies of scale
