Question

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(2) Suppose there is a production manager who directs the resources of a firm. There are two outputs for her to have made by her unit, X and Y. Let the quantity produced of each be denoted x and y, respectively.There are two unit of inputs at her disposal:Labor and capital. Each unit of X takes 2 units of labor and 1 unit of capital.Each unit of Y takes 1 unit of labor and 2 units of capital to produce The manager has L>O units of labor and K>O units of capitalto allocate between the production of the two outputs.The manager is evaluated by the firm's senior leadership by how much is produced diminishing returns.Thus,let B(x,y)=In(x)+1-0Iny)be the benefit received.The parameter is the relative importance of goodX.Assume 0<<1. (a) Under which circumstances would this manager under-utilize the labor resources available to him?(Be sure to explain.) (b) Consider the environment where the manager chooses to fully utilize the labor and capital resources available to him.What happens to the production of good X and Y if the capital stock experiences a(small)increase?

Answer 1

(a) The manager would under-utilize the labor resources available to him if he decided to allocate a disproportionately large share of the resources to good Y. The manager would prioritize producing output X over output Y because B(X, Y) is a concave function, meaning it reaches its maximum point at a higher quantity of output X as production increases. This means that producing more of X will produce greater benefits than increasing production of Y. (b) If the capital stock increases, the production of good X will increase because it requires more capital than labor, while the production of good Y will decrease because it requires more labor than capital. The marginal products of both labor and capital will increase, but the marginal product of capital will increase more, resulting in an increase in production of good X and a decrease in production of good Y.