Question

Problem 5: Consider the following variation of the aggregate production function. Now firms must use oil M to produce output (in addition to labor and capital). The price of a unit of oil is p (a) Find a first-order condition for the firm's demand for oil. (b) What must be true about the parameters α, β, and γ if this production function exhibits constant returns to scale? (c) If the price of oil p rises, what would you expect to happen to carbon intensity (the ratio of oil per unit output: M/Y) in this economy? What happens to the revenue share of oil (the ratio of total oil payments to output: pM/Y)?

Answer 1

Pi = Ak^alpha L^beta M^gamma - wL - rk - pM a) FOC: - partial differential pi/partial differential M implies Ak^alpha L^beta (gamma) M^gamma - 1 - P = 0 so Ak^alpha L^beta y M^gamma - 1 = P Ans b) Prodn funcn Q(L, K, M) = Ak^alpha L^beta M^gamma Now Q(lambda, lambda, lambda M) = A(lambda k)^alpha (lambda L)^beta (lambda M)^gamma = lambda^alpha + beta + gamma = 1 middot Ak^alpha L^beta M^gamma = lambda^alpha + beta + gamma Q(L, K, M) Thus for CRS, alpha + beta + gamma = 1 Ans c) Now P, so oil become costly, as coleb douglas prodn funcn so M^* = gamma c^bar/(alpha + beta + gamma)p, so use of M, M^* falls, so M/y falls. Now as MP/Y = gamma c^bar/(alpha + beta + gamma)y, so as M becomes costly, y also falls, bcoz M is input of prodn, thus y, so (Mp/y)