Write up your solutions to the exercises below. Give explanations and show all work. All work on the assignment is to be your own or with any references clearly cited. Please. follow these guidelines: write legibly in complete sentences; generate graphs with a computer application or draw the graphs neatly on graph. paper; label them carefully: cite any references used. The Least-Cost Combination In this problem, we apply calculus to a problem in economics. We seek to find the amount of each of several inputs which minimizes the cost of producing a certain amount of output. You should concentrate on solving the calculus exercises. You are not required to give economic interpretations of your answers. Suppose that the quantities z₁ of input 1, 12 of input 2,..., of input n can be combined to create a quantity y= g(11.12.....) (1) of output which is given by the function g for any combination of inputs. The function g is called the production function. (We shall assume that g has continuous partial derivatives.) Suppose also that the prices of inputs 1, 2,..., n are fixed at w1, 2,..., wn, respectively. Then the cost of producing y = g(1, 2,...,n) units of output with quantities of inputs. (x1, x2,...,n) would be C=u (2) We fix the quantity of output y and derive necessary conditions for cost minimization. at a given output level using the method of Lagrange multipliers. Let x denote the vector of inputs (1,2,...,xn). For simplicity of notation, we denote the partial derivative of g with respect to z₁ by 91, the partial derivative of g with respect to r2 by 92, etc. The partial derivative g(x) = g(1,2.....) is sometimes called the marginal productivity of input k. We must solve the following system of equations: Solving this system for A gives w₁ = Agi(11,12,1); w2 Ag(1,2,...,n); = w Ag(1.2.....In); y= g(1,2....In). (3) Wi λ= = 91(x) 2 In (x) which means that cost is minimized for a given level of output when the marginal produc- tivity of each input is proportional to the price per unit of the input. We assume that the equations in (3) have exactly one solution x = (1,2,...,) and that the cost of producing y units of output is minimized for this one set of inputs. Each ; is a function of y which we assume is differentiable. (By imposing differentiability and convexity conditions on g, it is possible to guarantee that the unique solution gives a minimum and that each (y) is differentiable.) The minimum cost of producing y units of output is given by C*=x+x++ (4) Let A be the Lagrange multiplier associated with cost minimization for output y. C" and A are also functions of y. The derivative dC"/dy is called the marginal cost. Use the method of Lagrange multipliers to minimize the cost of production for the following examples with two inputs. Exercise 1: Let g(1,2) = 11/212, w₁ = 4, w₂ = 1, and y = 27. a) Find the quantities of inputs and which minimize the cost of producing y=27 units of output. Also find A.. b) Find C, the minimum cost of producing 27 units of output. c) Graph g(1,2)=27 with ₁ on one axis and 2 on the other. On the same graph, sketch and label several level curves of the cost function f(x1,2)= w₁₁+w22. Be sure to include the level curve f(1, 2) point (2) on the graph. C. Locate the d) Describe the features of the graph and their relationship to the minimization problem. Exercise 2. Repeat Exercise 1 with y = 64. Exercise 3: Let g(1,2) = 1}/²±2, w₁ = 4, w2 = 1, as before, but let y remain undetermined (though fixed). a) Find the quantities of inputs 2 and 2; which minimize the cost of producing y units of output. The quantities r and will depend on y. Also find A as a function of y. b) Find C, the minimum cost, as a function of y. Verify that dC* /dy = X". c) Graph the marginal cost dC"/dy as a function of y. 2/3 Exercise 4: Repeat Exercise 3 with g(1,2) = 1/³/³, w₁ = 4, and w₂ = 1.