Question

The International Calculator Company of China produces handheld calculators in its plant. It tries to keep the number of workers in the plant constant so the only variable factor that can be measured is materials. Over the last seven monthly periods, the data for materials and quantity produced were the following:

Matrial	Quantity
70	450
60	430
80	460
95	490
77	465
100	550
85	490

a. Calculate a Cobb-Douglas production function of the form QaMb.

b. Discuss the important properties of your results.

c. What is the marginal product of materials?

Answer 1

The cobb douglas production function estimate can be found by any online plotter or regression plotter. I have provdided the desmos output.

(a) The Cobb-Douglas will have the functional form as , taking the number of workers as constant, which would be multiplied with the constant term, becomeing a.

The following is the graph and estimate.

The estimates will be as below.

.

(b) Differentiating and double differentiating the production function, we would have the results as or . Also, or . Hence, for all possible M, we have and . Thus, we can say that, as M increases, the quantity increases (since ), but that increment is in a decreasing rate (since ), ie the marginal product of Q with respect to M increases at a diminishing rate.

(c) The marginal product is the change in quantity per unit change in material. We have founded the marginal product as , ie is the marginal product. Note that the MP changes, as it is said to be decreasing with increase in M. Also, this is an estimate. For example, according to the estimate, the marginal product at M=77, . But, according to the table provided, where M increases from 70 to 77, we have .