ECON6003_Mathematical Methods of Economic Analysis_2009 Semester 1_Final Practice

Econ 6003 Final Exam Practice USYD 20091. Solve the problemmax x,y (x1/2 + y) subject to px + y≤I, x≥ 0, and y≥ 0where p > 0 and I > 0 are parameters.[You may use without proof the fact that the function x1/2 + y is quasiconcave.]2. Solve the problemmax x,y x2y2 subject to 2x + y≤ 2, x≥ 0, and y≥ 0.[You may use without proof the fact that x2y2 is quasiconcave for x≥ 0 and y≥ 0.] [Note: questions 1 and 2 would be examples of utility maximization.]3. Let u be a quasiconcave, increasing function of n variables. (Increasing means u'i(x) > 0 for all i.) Let w > 0 be a number and let p = (p1, ..., p n) be a vector with p i > 0 for all i. Consider the problemmax x u(x) subject to ∑i=1n p i x i≤w and x≥ 0.a.Write down the Kuhn-Tucker conditions for this problem.b.Suppose that x* solves the problem. Is there necessarily a value of λ* such that(x*,λ*) satisfies the Kuhn-Tucker conditions?c.Suppose that (x*,λ*) satisfies the Kuhn-Tucker conditions. Does x* necessarilysolve the problem?d.Suppose that x* solves the problem. What can you say about the relationshipbetween p i/p j and the value of u'i(x*)/u'j(x*) wheni.x i* > 0 and x j* > 0?ii.x i* = 0 and x j* > 0?iii.x i* = 0 and x j* = 0?4. Consider the following problem.max x−(x1−c1)2− (x2−c2)2 subject to (x1 + 1)2 + x22≤ 4, x1≥ 0, and x2≥ 0, where c1 and c2 are constants.a.Are the Kuhn-Tucker conditions necessary for a solution of this problem?b.Are the Kuhn-Tucker conditions sufficient for a solution of this problem?c.If possible, use the Kuhn-Tucker conditions to find the solution(s) of the problemfor c1 = c2 = 0.d.If possible, use the Kuhn-Tucker conditions to find the solution(s) of the problemfor c1 = 2 and c2 = 0.5. (Cost Minimization)A firm produces output y with capital K and labor L according to the Cobb-Douglas specification K a L b≥ y, where the parameters a and b satisfy 0 < a < 1, 0 < b < 1, and 0 < a + b < 1 (so that production exhibits decreasing returns to scale and decreasing marginal products).Suppose that the firm chooses capital and labor inputs in order to minimize the total cost rK+wL of producing at least y units of output, where r > 0 is the rental rate for capital and w > 0 is the wage rate for labor. That is, suppose that the firm solves min K,L rK + wL subject to K a L b≥ y, K ≥ 0 and L ≥ 0.a) Define the Lagrangian for this cost minimization problem. Write down the full set of first-order conditions and complementary slackness conditions that, according to the Kuhn-Tucker theorem, must be satisfied by K*, L* and the associated value λ* of the Lagrange multiplier. Here K* and L* denote the values of K and L that solve the optimization problem.b) Next, use these Kuhn-Tucker conditions to find solutions for K*, L* and λ* in terms of the model’s five parameters: r, w, y, a, and b. [Note: these solutions for K* and L* define the firm’s conditional factor demand curves for capital and labor.]c) Now define the value function of the optimization problem as the cost function C(r, w, y). Use your results from above to obtain expressions for the derivatives of the cost function, C1(r, w, y), C2(r, w, y), and C3(r, w, y), in terms of the parameters r, w, y, a, and b.d) What do your results from above tell you about the relationship between the conditional factor demand curves and the derivatives of the minimum cost function? What economic interpretation does your results provide for the Lagrange multiplier in this constrained minimization problem?6. (Profit Maximization) A firm produces output y with capital K and labor L according to the Cobb-Douglas specification K a L b≥ y, where the parameters a and b satisfy 0 < a < 1, 0 < b < 1, and 0 < a + b < 1.Assume that the firm faces the rental rate r > 0 for capital and the wage rate w > 0 for labour. Now, however, suppose that the firm maximizes profits py − rK − wL, where p > 0 denotes its output price, subject to the constraint imposed by its production possibilities. That is, suppose that the firm solvesmax y,k,l py − rK − wL subject to K a L b≥ y, K ≥ 0 and L ≥ 0.a) Define (set up) the Lagrangian for this profit maximization problem. Write down the full set of first-order conditions, and complementary slackness conditions that, according to the Kuhn-Tucker theorem, must be satisfied by y*, K*, L* and the associated value λ* of the Lagrange multiplier. Here y*, K* and L* denote the values of y, K, and L that solve the optimization problem.b) Next, use these Kuhn-Tucker conditions to find solutions for y*, K*, L*, and λ* in terms of the model’s five parameters: p, r, w, a, and b. [Note: this solution for y* defines the firm’s supply function, and these solutions for K* and L* define the firm’s factor demand curves for capital and labor.]c) Now define the value function of the optimization problem as the profit functionΠ(p, r, w). Use your results from above to obtain expressions for the derivatives of the profit function, Π1(p, r, w), Π2(p, r, w), and Π3(p, r, w), in terms of the parameters p, r, w, a, and b.d) What do your results from above tell you about the relationship between the supply and factor demand curves and the derivatives of the profit function?。