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.

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

(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

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.,

2) A consumer’s utility function is: U(x,y) = sqrt(9xy). The price of each good is: (PX,PY) = (15,20). The monthly budget is: M = 600, answer the following:a. What is the optimal bundle (X, Y) that maximizes the consumer’s utility?b. What is the optimal utility given your answer from part a?c. If the desired level of utility the consumer is U = 400, what would be the optimal bundle (X, Y) to minimize consumer expenses?d. Given U = 400 from part c, what would be the new optimal bundle for both goods if PX increases to 25?e. Given the price increase in good X from part d, what would be the optimal bundle for this consumer wishing to maximize its utility with an income, M = 600?f.: Graph the budget line and utility curve from part a i need help on part c

2) A consumer’s utility function is: U(x,y) = sqrt(9xy). The price of each good is: (PX,PY) = (15,20). The monthly budget is: M = 600, answer the following:a. What is the optimal bundle (X, Y) that maximizes the consumer’s utility?b. What is the optimal utility given your answer from part a?c. If the desired level of utility the consumer is U = 400, what would be the optimal bundle (X, Y) to minimize consumer expenses?d. Given U = 400 from part c, what would be the new optimal bundle for both goods if PX increases to 25?e. Given the price increase in good X from part d, what would be the optimal bundle for this consumer wishing to maximize its utility with an income, M = 600?f.: Graph the budget line and utility curve from part a.

Ice cream and cakes are perfect substitutes for a child, and 2 units of ice cream is always worth 3 units of cakes (however many ice creams or cakes she might have, she would be willing to give up 2 ice creams to get 3 more cakes to keep the same utility level) .   (a) Write down a utility function u(x,y) that represents the child's preferences, where x is the number of ice creams and y is the number of cakes she has.   (b) If the prices are px= 8 and py =5, and she has $140 to spend on the two goods this summer, what is her optimal bundle?   (c) If the price of ice creams decreases slightly, down to px = 7, what happens to her optimal bundle in this case? Did it change "slightly" compared to (b)?

Q: A consumer’s preferences for food (F) and clothing (C) are given by U(F,C) = F0.2C0.8. The  price of food is $4, and the price of clothing is $8. The consumer has an income of $3600.  a) What is the utility maximizing choice of food and clothing?  b) How would the utility maximizing choice change if price of clothing increased to $14? c) Given the answers to the previous parts plot a linear approximation to the demand function  for clothing.