Consider an oil industry facing a linear demand curve for aggregate output qt. Let the demand for oil be represented by the following inverse demand curve: pt=abqt. An important characteristic of the linear demand curve is that the implied maximum or "choke-off" price occurs at the intercept where pt=a when qt= 0. In scheduling their production, firms are assumed to know about this backstop/substitute.

Let industry profits in any given period t be denoted as t=pt*qtc*qt, and assume that the industry chooses an extraction path {q0,...,qT} to maximize the net present value of industry profits over T years, subject to the stock constraint S0=qt. Let = 1/(1 +r) indicate the industry's discount factor.

B) Suppose that we have the following parameter values: a= 1; b= 0.1; r= 0.05; c= 0.1;

and S0= 75.

(i) Determine the extraction path for the competitive industry. Plot both the extraction and price paths in two separate graphs. Explain whether they make sense.

ii) Suppose there is a technological breakthrough that reduces the marginal cost of oil

extraction to c= 0.01. What happens to the price path, extraction path, and date of exhaustion? Explain.

iii) Suppose there has been a discovery of more reserves so that S0 is actually equal to 125

(instead of 75). What happens to the price path, extraction path, and date of exhaustion for oil? Explain.

(iv) Now suppose that there has been a technological breakthrough in the "backstop technology" (e.g., wind turbines) that has lowered the demand for oil. Specifically, suppose we now have a= 0.5 so that the demand for oil becomes zero if p >0.5. What happens to the price path, extraction path, and date of exhaustion for oil? Explain.