Exercise 1 Consider utility maximization problem: Kuhn Tucker Theorem max U (x1, x2) = x1+x2 21,22...
1) Optimization problem 1 Max U(x, y) = x1^0.5 + x2^0.5 s.t. x1 + x2 =16 Find the optimum bundle; check if there is a minimum or a maximum. 2) Give the interpretation of the expenditure function, explain and show its properties. Draw the diagram of the expenditure function. Derive the compensated demand function for x1 and x2 E( p, u) = p(p1. p2)^0,5 and the uncompensated demand function. 3) Derive the expenditure function when the direct utility function...
f. (BONUS Solve the utility maximization problem in general (xỈ + subject to the budget constraint, pM + pr,-I max 2 x1,x2 Again, the marginal utility of good i is MU,-X1ới +均一; the marginal utility of good 2 is MU,-X2(xf + xj)--. Find the quantities of good i and good 2, x1 and X2, in terms of prices, p, and p2, and income I.
Question 2: Identify which of Cases (1)--(4) apply to the following LP problem. max z = 2x1 – X2 s. t. X1 – X2 < 1 2x1 + x2 > 6 X1, X2 > 0 (1) unbounded LP (2) infeasible LP (3) unique optimal solution (4) multiple optimal solutions
max Xi + 3x2 subject to: -X1 – x2 < -3 -21 + x2 = -1 X1 + 2x2 = 2 X1, x2 > 0 Problems. Solve the following problems using the simplex method in the dictionary form. Note that problems 2, 3, and 4, require you to use the two-phase simplex method. For each iteration, in addition to other calculations, clearly show the following: the dictionary, entering variable, minimum ratio, and the leaving variable. Note that we employ Dantzig's...
U = 8x10.5+ 2x2, where x1 is the quantity of good 1 consumed, and x2 is the quantity of good 2 consumed. (Yes the x is raised) 8x1.5 Suppose that the consumer has a budget of M = $400 to spend and that good 1 has a price of p1= 2, and good 2 has a price of p2= 8. Answer the following questions, and write your answers in the Answer Sheet. Write the person’s budget constraint as an equation,...
7. Let X1, X2, ... be an i.i.d. random variables. (a) Show that max(X1,... , X,n)/n >0 in probability if nP(Xn > n) -» 0. (b) Find a random variable Y satisfying nP(Y > n) ->0 and E(Y) = Oo
3. There are two goods, Xi and X2 with prices pı > 0 and P2 = 1. Assume that a consumer has income I> 0 that she will allocate for the bundle (X1, X2), and has preferences represented by the utility function u(X1, X2) = a ln x1 + x2, for some a > 0. (a.) Derive the marginal utilities and bang-for-bucks for each good. (b.) Find the optimal bundle assuming an interior solution, i.e. x > 0 and x...
1. Consider a utility-maximizing price-taking consumer in a two good world. Denote her budget constraint by p1x1 + p2x2 = w, p1,p2,w > 0,x1,x2 ≥ 0 (1) and suppose her utility function is u(x1,x2) = 2x1/2 1 + x2. (2) Since her budget set is compact and her utility function is continuous, the Extreme Value Theorem tells us there is at least one solution to this optimization problem. In fact, demand functions, xi(p1,p2,w),i = 1,2, exist for this example. (i)...
Now minimize P1x1 + P2x2 such that U(x1; x2) = x 3 4 1 x 1 4 2 u and x1; x2 0 (a) Using optimization techniques, nd the Hicksian Demand (Z(p; u))Now minimize P1x1 + P2x2 such that U(x1; x2) = x 3 4 1 x 1 4 2 u and x1; x2 0 (a) Using optimization techniques, nd the Hicksian Demand (Z(p; u))Now minimize P1x1 + P2x2 such that U(x1; x2) = x 3 4 1 x 1...
2. (25%) Consider a consumer with preferences represented by the utility function: u(x1, x2) = min {axı, bx2} If the income of the consumer is w > 0 and the prices are p1 > 0 and P2 > 0. (a) Derive the Marshallian demands. Be sure to show all your work. (b) Derive the indirect utility function. (c) Does the utility function: û(x1, x2) = axı + bx2 represent the same preferences?