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Subst/Eff 2. [Efficiency via price coordination I] This problem, and the next two problmes, concerns three different three-input/one-output production functions, f(x1, 12, 13), 9(11, 12, 13), and h(x1, 12, 13). These represent three different ways of producing the same good, denoted y. For input goods 1, 2, and 3, the total quantity available to be used to produce y is X1, X2, and X3, and we let X = (X1, X2, X3) denote the vector of scare good amounts. Society's problem is how to allocate these limited quantities between the three different ways of producing y most efficiently. Let x = (x x,,r;), x = (r, r,, x;), and r = (1, 12, 13) represent the amount of the resources used in production processes f, g, and h respectively. Society's problem is (2) maxx/ ,19,ah f(x ) + 9(x) + h(x")] s.t. x/ +x +x X. Throughout this problem, we assume that FOCs are necessary and sufficient for optima and that all solutions are strictly positive. a. The problem in (2) has 9 decision variables and 3 constraints. As a check on your command of vector notation, write out the problem without using vector notation. b. Give the Lagrangean function L(af, x, x, 1, 12, As) where A; is the mul- tiplier associated with the constraint that no more than X; of input good i is available. c. Give the FOCs for the societal Lagrangean function that you just found. Be sure to include the derivatives with respect to the A's. d. Explain how the FOCs require that the marginal efficiency of the inputs in producing the output be equalized "at the margin," i.e. that MRTS's be equalized