(25 points) In this problem we will be parameterizing a problem to investigate the question of production pathways. So, consider the case of a prot maximizing rm that is a price taker. The producer receives a price, 6 p, , for output, '1: . The producer faces a total production cost given as C (qt , Rf) : cogr + 1 , and the available R I resource evolves such that R = R q . The rm faces inverse linear demand given as p = a a q + a y , {+1 I I f 0 l f 2 I where y! is income. Finally, there is a backstop technology available at constant marginal cost I) where the backstop is a perfect substitute for the depletable resource. 3. Find the first order conditions for a maximum and solve for the optimal path for price and quantity. b. Now, using Excel, graph the time path of price, quantity, and remaining resource. Assume the following: c0 c1 R0 4 9 100 10,000 Po 1' l3 5 0 1 0909 an 31 a2 1 0 1 0.35 b yo Ygl'l rate 30 50 3.00% c. Now, change the growth rate of the economy, to 1.00%. Compare your answer to what you ygmwm rate 7 found in part b. d. What does this tell you about the effect of income growth on the decision to produce, intertemporal arbitrage, and revenues? e. If we add new discoveries, such that Rr+1 : R, q' + N r where the firm's cost of new discoveries is given as ,uNf , what impact does this have on production in each period? (Note, you do not need to solve this part numerically. Just solve the problem and do the comparative statics.)