Question

5. An economist is tasked to estimate the following well-known functions in economic theory. In each case, determine choose an appropriate functional form. Demonstrate your answers graphically and comment on the expected signs of the coefficients.

a. The Cobb-Douglas production function with decreasing return to scale. Assume there is only one input to the production function.

b. The utility function of a risk-lover investor.

Answer 1

a) A Cobb Douglas production function with one input of the form where

For to have decreasing returns to scale, if he input is increased to k times, the output would increase by less than k times.

Consider

Changing x to kx gives

Hence, increasing x to kx increased production by less than k times the original production.

Plotting the production function:

b) A risk seeking investor would invest in a scheme which gives an expected return of x over a scheme which gives a return of x.

An example of such utility function is

To prove this is a utility function of a risk lover investor, consider scheme 1 which gives 10 with probability 0.7 and 0 with probability 0.3. Expected return from this scheme = 7. Consider scheme 2 which gives 7. Expected return from this scheme = 7.

Hence, is an example of a utility function of a risk seeking investor.

Plotting the function: