Consider a simple model for non-renewable resources, where society only values extraction, so the utility function is U(qt)), where U'(qt) > 0 and U''(qt) < 0. Suppose that the marginal cost of extraction is 0, and the competitive mining industry faces a linear inverse demand curve,Pt=a-bqt . Also, suppose that the shadow price on remaining reserves in period T + 1 is zero so that Lambdat+1=0. This means we can replace our difference equation with the exhaustion condition, R0 qt = 0 . Let u denote the shadow price on the initial reserves, the rate of extraction in period t is denoted by qt). (a) Write out the Lagrangian and the first-order necessary conditions for this problem. (5 points) (b) What is dL/dR0, equal to, and what is its economic interpretation? (3 points) (c) From the first order conditions, derive the relationship between marginal utility for q0, and for the marginal utility for qt)