A company with this model has 50000 available to spend on

Question

Question: Much of the pioneering work in the field of economic production models occurred in the 1920s when Paul Douglas and Charles Cobb proposed that the amount P of output of a company can be expressed in terms of one unit labor L and one unit of equipment K by an equation of the form

P = cL^α K^β,

where c is a constant and alpha and beta are constants such that 0 < alpha < 1 and 0 < beta < 1. Consider the Cobb-Douglas production model

P = 100 L^0.75 K^0.25

a. A company with this model has $50,000 available to spend on labor and equipment. Use the method of Lagrange multipliers to find the maximum output value of P if labor costs $25 per unit and equipment costs $100 per unit. (Your constraint equation is the budget allotment for the company.).

b. How should the $50,000 be divided between labor costs and equipment costs to achieve this maximum?

c. Economists call the Lagrange multiplier/obtained the marginal productivity of money. For each additional dollar spent on production, /additional units of the product can be produced. How much additional spending will be required to produce 1 additional unit of the product?