Question-1 (Finding optimal bundle utility functions and budget constraints. 1. U(x, y) = lnx + y and 2x + y = 10 Find the optimal bundle using the following
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- 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.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 c1. For each of the following three utility functions below, find the demand function by explicitly solving each utility maximization problem. (a) u(x1,x2) = xx. (b) u(x1, x2) = Inx1+2lnx2. (c) u(x1,x2) = x7x3.
- 2 3) For each utility function, determine the marginal utility with respect to x, marginal utility with respect to y, and the MRS. (3pts each) a) U(x, y)-3x+2y b) U(x, y) 10x¹/5 y c) U(x, y)=x+ y²1. Calculate the marginal rate of substitution for the following utility functions: (a) u(x1, x2) = ax₁ + bx2 (b) u(x1, x2) = 2√√x₁ + x₂ (c) u(x1, x2) = ln(x₁) + x2 (d) u(x1, x2) = x1x2 (e) u(x1, x2) = xx h (f) u(x₁, x₂) = x + x² 2. Marie Curie has a utility function U(x, y) = max{x, 2y}. Note that the max{} function es- sentially says that the value of the function takes the highest of the values in the set (the set is denoted as the things inside the curly braces). (a) Draw a graph, plotting the indifference curve along which U(x, y) = 10. (b) Are Marie's preferences convex?3. Cost of living will become more expensive fuelled by increases in food and electricity bill. Suppose the utility function for a customer in the rural area is estimated to be U=F05E0.5 where Fis the amount of food consumed and E represent electricity usage. If the consumer income is $10,000 per week and the price of food is $500 and the electricity cost $200. a) Calculate the optimal bundles when the electricity rate increases to $250 b) What is the income and substitution effect of the price increase? c) Illustrate the income and substitution effect on a well labeled diagram,
- PROBLEM (6) A consumer's preferences over pizza (x) and steak (y) are given by u(x,y) = x3y (HINT: MUx = 2xy and MUy = x²) and his income is I = $120 and py= $1. %3D (a) Calculate his optimal bundle when px= $8 (call this bundle A) and separately when px= $1 (call this C). (b) Finding the decomposition bundle B, calculate the income and substitution effects on the amount of pizza of a decrease in the price of pizza from px= $8 down to px= $1. %3D (c) Forget about the decomposition bundle and the two effects. In (a), the price of pizza decreases, hence the agent ends dollars. For this, instead of the price drop, suppose the agent is given some money $m and he optimize utility with this additional gift included to his budget. What should m be, so that his optimal utility with his expanded budget is exactly equal to his utility at the bundle C (the bundle he chooses optimally when pizza price drops to px= $1). up better off. Let's quantify how much "better off" the agent becomes after…Question-3 (Finding optimal bundle) utility functions and budget constraints. 1. U(₁, ₂) = 3x₁ +2x2 and 6x₁ + 2x₂ = 24 2. U (1, 2) = ₁ + 2x2 and 3x1 + 6x2 = 48 3. U (11, 12) = min{1, 2} and 41 + 2x2 = 48 1. U(T1, T₂) = min{4x1, 2r2} and 2x₁ + 3x2 = 36 Find the optimal bundle using the followingFor 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
- 4. Minnie enjoys dessert by eating chocolate cake (C) and apple pie (P) according to the utility function U = Min [C, P]. In any given week, Minnie %3D spends £20 to purchase these two products. Assume that Minnie is a utility maximiser and has to pay 50p for a unit of apple pie and £2 for a unit of chocolate cake. (i) Consider the bundle C = 20 and P = 5. Is this the utility maximizing choice subject to Minnie's budget?. %3! (ii) Suppose that Minnie chooses the exact bundle that maximizes her utility. How would the utility maximizing choice of C be affected if price of P went up? (iii) Explain under which conditions the tangency condition does not work. Given that the tangency condition does not work, discuss how we can find out the optimal consumption bundle for a consumer.find optimal bundle from diagramConsider a consumer whose preferences can be represented by the utility function u(x, y) = x + y (a) Originally, px = 1, py = 2 and m = 1. What bundle does the consumer choose, and what is his utility from this bundle? (b) The price of good x then rises to 3. What bundle does the consumer choose after the price change, and what is his utility from this bundle? (c) Calculate the compensating variation. (Hint: at the new price ratio, what good will he spend his income on?) (d) Calculate the equivalent variation.