Example A consumer’s utility function is given by Where q1 and q2 are the quantities of the two commodities consumed. If the price q1 is 6 and that of q2 is 3 and the budget is 60. a. Write out a constrained utility maximization problem out of the information given. b. What is the corresponding augmented (composite) objective function? c. Find the levels of q1 and q2 that will satisfy the first order condition for maximum. d. Is the second-order condition satisfied? e. Compute the optimum value of U.

Question 1 Consider a consumer with a utility function of U(X,,X2, ... , Xn) = II-1 X , a; > 0, i = 1,2,3, ...,n. P; is a price of 1 unit of the i-th good. Assuming that the consumer has the total budget of $M, derive his optimal consumption bundles in the following procedure. (a) Set up the consumer's utility maximization program with a Lagrangean multiplier. (b) What is an economic intuition of the Lagrangean multiplier? (c) Assuming an interior solution, get all the FOCS. (d) Find an optimal consumption bundle while specifying each step required to get the solution. (e) Characterize the maximized utility as a function of $M and the parameters. What does it look like? (f) Find an exact value of the Lagrangean multiplier.

Donald likes fishing (X1) and hanging out in his hammock (X2). His utility function for these two activities is u(x1, x2) = 3X12X24. (A) Calculate MU1, the marginal utility of fishing. (B) Calculate MU2, the marginal utility of hanging out in his hammock. (C) Calculate MRS, the rate at which he is willing to substitute hanging out in his hammock for fishing. (D)Last week, Donald fished 2 hours a day, and hung out in his hammock 4 hours a day. Using your formula for MRS from (c) find his MRS last week. (E) This week, Donald is fishing eight hours a day, and hanging out in his ham­ mock two hours a day. Calculate his MRS this week. Has his MRS increased or decreased? Explain why? (F) Is Donald happier or sadder this week compared to last week? Explain.

part B

(A) Suppose an individual who derives utility u(x, y) from consuming a units of the good X and y units of the good Y. Prices of goods X and Y are respectively p = 1 and Py = p. Furthermore, the individual has a budget m. What is the demand for Y if the utility function is: u(x, y) = min{x, ay}. (a) y(p, m) = am (b) y(p, m) = = (c) y(p, m)= (d) y(p, m) = = am a+p m a+p m 1+p (B) Same setup as (A) with u(x, y) = ax + y and ap Pr. The quantity demanded of good X is q. Which of the following is true ? (a) The substitution effect increases q but the income effect decreases q and over- come the substitution effect. Overall q decreases. (b) The substitution effect decreases q but the income effect increases q come the substitution effect. Overall q increases. and over- (c) Both the income and substitution effects increase q. Overall q increases. (d) Both the income and substitution effects decrease q. Overall q decreases. (D) Suppose an inferior good X (not Giffen). Its price increases from…

(5) A consumer has the following utility function: U(x, y) = (x+2)(y+1), where x and y are quantities of two consumption goods whose prices are P, and Py, respectively. The consumer also has a budget of B. (a) Write down the consumer's Lagrangian associated with this primal problem. From the first-order conditions, find expressions for the demand functions x and y. (b) Find the expression for indirect utility function U = U(P, P, B) (namely the maximum value function). (c) Write down the associate dual problem with a target level of utility U. Derive an expression for the expenditure function E = E(P, P, U") of the dual problem. (d) Use Shephard's Lemma to find the Hicksian demands of associated dual problem, i.e., OE/OP, rh, OE/OP, y", and OE/OU= u", where " is the Lagrangian multiplier in this problem.

For each of the following utility functions get the marginal utility of consumption of each of the goods. 1/2 1/2 (a) u(a,b) = a b (b) u(x, y) = x³/4y1/4 (c) u(a, b) = n(a) + In(x) (d) u(x, y) = ln(x) + ln(y) (e) u(a, b) = 2xa +Xb

Assume a consumer with the utility functionU(X,Y)=X²Y²and the typical budget constraintM= PxX+ PyY a. Derive the consumer’s demand for X and Y in terms of the parameters.b. If income of the consumer is 100, price of X is 2 and price of Y is 4, what the quantities of X and Y that gives the consumer maximum satisfaction?c. What is the consumer’s marginal rate of substitution between X and Y

Question 1 Let the utility function be U(q1. q2) = Min{ q1, 292). Let p1 =2 and p2=2. Income is $30 per week. a. Suppose that q1 represents a cup of coffee (250ml) and q2 represents a teaspoon (5ml) of sugar. Interpret the utility function above in words. b. On a large well-labelled diagram, indicate the equilibrium number of coffees with sugar that this individual consumes in a week. c. Suppose that the price of a cup of coffee (with no sugar) doubles to $4. Carefully draw in the new equilibrium on your graph from part b d. What are the income and substitution effects arising from this price increase? Explain.,

Suppose that you go to a restaurant with your friends. a) One of your friends always eat exactly one chicken wrap (C) and drinks two cups of ayran (A). He never changes this combination of food choice. (i) Write the utility function of your friend to represent his preferences. (ii) Draw couple of indifference curves of your friend on graph with appropriate labels. (iii) Determine the demand function of your friend for each product as a function of prices and income. Denote prices as Pc, Pa; income as M. b) Suppose that one of your friends does not care if she consumes spaghetti(S or noodle(N).She wants to consume either two-dish spaghetti or two-dish noodle or any combination of both which adds up to two dishes. (i) Write the utility function of your friend to represent his preferences. (ii) Draw couple of indifference curves of your friend on graph with appropriate labels. (iii) Denote prices as Pn, Ps; income as M. If Pn < Ps, then determine the demand…