solve the equations please
2(c). In order to limit greenhouse gas emissions, the public authority imposes a fixed cap or quota k on the CO2 emissions of the firm. All inputs may contribute to emissions: more precisely, the firm's emissions are a convex, differentiable function n:91. = 1: (2,.....2) n(2.....2) with nonnegative partial derivatives. 2(c)(1). Is it necessarily true that the production of a larger amount of output requires emissions to increase? Discuss it in the simpler two-dimensional case.Question 2. Capping emissions. A firm that behaves in a perfectly competitive manner in all markets produces one output by using L-1 inputs according to a direct production function f:1 - R:2= (2...2 )Hf(2). assumed to be differentiable with a strictly positive gradient on 1' and concave. Denote by p > 0 the price of the output, and by w= (wi. .... WLI) ER the vector of input prices. Assume that the cost function c:). xR. > R: (w..... w1;q) / c(w...., w:q) is differentiable and convex in q, and that at the profit maximizing solution the quantities of the inputs and of output are positive. 2(a). What is the relation between the output price and the marginal cost at a profit- maximizing solution? Prove your answer.2(b). How does an increase in an input price affect the marginal cost at a profit- maximizing solution? Prove your answer.1(b). The EMIN[p, u] Problem is defined by min, pex subject to u(x) 2 u , with solution function the Hicksian demand function h for consumer goods. Show that S(p, u)p = 0, where S(p, u) is the Slutsky matrix. 1(c). Define the Hicksian demand for marketed good (k, j) by 5 : R XU - R:5, (1,u) = ajh, (p(),"), where U is the relevant domain of utility levels. When can we say that marketed good (m, i) is a (net) complement of marketed good (k, j) at (n, u)? When can we say that marketed good (m, i) is a (net) substitute of marketed good (k, j) at (It, u)? 1(d). Coffee and cream are popular textbook examples of complements. But Paul Samuelson suggested that, when a consumer uses cream both in her coffee and in her tea, cream and coffee may actually behave as substitutes. In order to analyze this somewhat paradoxical result, we specialize the previous model to the case of two home-produced consumer goods: coffee (good 1) and tea (good 2). Coffee requires coffee beans (marketed good (1, 1)) and cream for coffee (marketed good (2, 1)), whereas tea requires tea leaves (marketed good (1, 2)) and cream for tea (marketed good (2, 2)). (The prices of marketed goods (2, 1) and (2, 2) may well be the same, but this does not play any role here.) 1(d)(1). Is marketed good (1, 1) (coffee beans) a complement or a substitute for marketed good (2, 1) (cream for coffee)? Argue your answer.Question 1. Complements and substitutes. A consumer consumes L consumption goods, with quantities denoted (x1, ..., x2), and her preferences are represented by a twice differentiable, strictly quasiconcave and locally nonsatiated utility function #: R - R:x= (x,...,x,) "> u(x). But these consumption goods are not sold in the market. Instead, consumption good / (j = 1...., L) is home-produced by using / marketed goods, with quantities denoted (zij;..., zy.), according to the Leontief technology x = min Z Z21 ENULL where all denominators are positive. (The first subscript indicates the marketed good, and the second one the consumption good.) A double index (k, j) labels the kth marketed good in the list of marketed goods used in the production of consumption good j, j = 1,..., L, k = 1,..., Nj]. Hence, there are altogether N= _"_, M[j] marketed goods, M of which are used in the home production of consumption good j. Denote by my the price of the (k, j) marketed good, j = 1,.... L, k = 1,..., NU]. 1(a). A vector = (; All Day(2125six.."; My[LLL) Of prices of marketed goods induces, via the home production technology, a vector p = (p1...., pz) of (implicit) prices for the consumption goods. For j = 1...., L, define the function p that expresses the price of consumption good j in terms of the prices of the marketed goods
