# In 1928, Cobb and Douglas used the following function to model the production of the entire US...

## Question:

In 1928, Cobb and Douglas used the following function to model the production of the entire US economy in the first quarter of the 20th century:

{eq}P = 1.01K^{a}L^{1-a} {/eq}

Where,

P = total yearly production in $K = total yearly capital investment units. L = total yearly amount of labor-force in person-hours. Suppose {eq}a = 0.25 {/eq}, the labor costs are$20 per hour, the capital costs are $40 per capital investment unit, and the budget is$8,000.

1) Write the equation of the constraint dictated by the budget as a function of K & L.

2) Use the method of Lagrange multipliers to find the optimum number of labor force and the optimum number of units of capital. In addition, determine the value of {eq}\lambda {/eq} correct to 4 decimal places. Show each of the solution steps.

3) By what percent should the capital output K (rounded to whole percent) be increased to maintain the same level of production P, if there is a reduction of 10% in labor force L? Explain the meaning of the changes in L and K.

## Input Demand:

Profit Maximization requires that each firm hire each input up to the point at which its marginal revenue product is equal to its market price, where marginal revenue product is the extra revenue a firm receives when it employs one more unit of an input.

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(a) The equation of the constraint is given by: wL+vK = 8000, or

20L+40K = 8000

(b) The Lagrangian function is given by:

{eq}L =...