Q.1 Use Lagrange multipliers to find the maximum utility for the utility function U = 5xy, when subject to a budget of £30, where the price of each unit of X is £5 and each unit of Y is £1.
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Q: For the utility function U = (Qx0.5+Q,0.5)2 and the budget 122= 8Qx + 8Qy find the CHANGE in optimal…
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Q.1 Use Lagrange multipliers to find the maximum utility for the utility function U = 5xy, when subject to a budget of £30, where the
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- Ruby has the following utility function: U(X, Y) = X^3/4 , Y^1/4, where X is her consumption of food items, with a price of $10, and Y is her consumption of clothing items, with a price of $30. She plans to spend $360 on food and clothing over the next week. Using the Lagrange multiplier technique, determine the number of food and clothing items that will maximize Ruby's utility.Kai spends his income on lime (L) and ginger water (G). Lime is priced at $2, while ginger water costs $1. Suppose Kai has $30 to spend and his utility function can be represented as U(L,G) = L0.5 G0.5 What is the optimal number of lime and ginger water for Kai to purchase? b. How much utility does this combination bring him?Suppose that David and his friend Wilson derive utility from consuming two types of snacks: onion rings (9₁) and chips (9₂). The utility function for each individual is U (9₁, 92) = 9192. Their indifference curves for these two goods are assumed to have the usual (convex) shape. Suppose David has an initial endowment of 35 onion rings and 10 chips, and Wilson's initial endowment consists of 5 onion rings and 20 chips. (1) Draw an Edgeworth box and show the initial allocation of goods, to be labelled e. Indicate the initial quantities of each person's goods on the four axes.
- Given the following utility functions for Consumer A and Consumer B and their budget contraints: UA = X0.5Y0.5 UB = X0.2Y0.8 I= Px (X) + Py(Y) and given that: Px = 3, Py= 6 and I = 100 Fill in the blanks in the following table (round to two decimal places): Part 1: Part 2: Part 3: Part 4: For Consumer A, what is the optimal choice of X? Number For Consumer A, what is the optimal choice of Y? Number For Consumer B, what is the optimal choice of X? Number For Consumer B, what is the optimal choice of Y? Number1.2) Suppose that a consumer’s utility function is U=10 lnx+20 lny. A) Find the marginal utility of x, MU_x, and the marginal utility of y, MU_y.B) Suppose that the consumer has R3600 to spend on x and y, while p_x=200 and p_y=400. C) Use the Lagrange multiplier (LM) method to find the levels of x and y that will maximise the consumer’s utility, subject to her budget constraint. D) Find and interpret the value of the Lagrange multiplier. Show that MRCS=p_x/p_y at the utility maximising levels of x and y.Economics Summer break is approaching! Suppose you derive utility from days spent traveling on vacation domestically, D, and days spent traveling on vacation in a foreign country, F. Your utility function over these two “goods” is: U(D, F) : 4D0.25F 0.75 Let your budget constraint be I(D, F) = pDD + pF F; where I is your income, pD is the price of domestic travel per day, and pF is the price of foreign travel per day. a. Determine the demand functions for domestic travel and foreign travel. Make sure you show your work – show the steps used. b. Suppose that you’ve saved $800 for your summer travel, the price of domestic travel per day is $25 and the price of foreign travel per day is $100. How many days of each type of travel will you embark on? c. Illustrate the indifference curve, budget constraint, and the utility maximizing bundle associated with (b). Make sure you show the level of utility, the budget constraint intercepts, and the optimizing equilibrium. Your answer
- Two students go out to lunch and decide to split the bill evenly between them. Each student has a quasi-linear utility function given by ui(fi , xi) = φi(fi) + xi , where φi(·) is strictly concave, fi is the amount of food consumed by student i, and xi is a composite numeraire good. Each student has a fixed budget of mi . EVALUATE THIS CLAIM: Both students eat too much!Which of the following statement is TRUE based on this question : A utility function with 2 goods (X,Y) is given by U = X^1/2Y. If there’s 3 bundles, bundle A = (9,3); bundle B = (4,1); bundle C = (16, 4)Assume, as in Exercise 22.1, that a consumer has utility function F or fruit and chocolate. Determine the consumer's demand functions q1(P1, P2, M) and q2(P1, P2, M). Determine also It* in terms of P1, P2 and M. Find the indirect utility function and show that It* = 8Vj8M. Suppose, as before, that fruit costs $1 per unit and chocolate $2 per unit. If the income is raised from $36 to $36.5, determine the precise value of the resulting change in the indirect utility function. Show that this is approximately equal to (O.5)λ*, where λ* is evaluated at P1 = 1,P2 = 2 and M = 36. Exercise 22.1 A consumer purchases quantities of two commodities, fruit and chocolate, each month. The consumer's utility function is For a bundle (X1, X2) of X1 units of fruit and X2 units of chocolate. The consumer has a total of $49 to spend on fruit and chocolate each month. Fruit cost $1 per unit and chocolate costs $2 per unit. How many units of each should the consumer buy…
- 4. Two individuals, Amir and Budi, consume two goods, clothes (X) and shoes (Y). The utility functions for the two individuals are given as: Utility function of Amir, UA = 15X0.25Y0.75Utility function of Budi, UB = 25X0.5Y0.5 The current price for clothes (Px) is Rp 100,000 and the current price for shoes (PY) is Rp 150,000 a. Determine marginal rate of substitution (MRSXY)between clothes (X) and shoes (Y) for Amir and Budi! Please explain. b. Amir is currently consuming 5 units of clothes (X) and 10 units of shoes (Y), whereas Budi is consuming 12 units of clothes (X) and 8 units of shoes (Y). At this current consumption, have Amir and Budi reached the efficient allocation of clothes and shoes? If they have, explain why. If they have not, calculate the optimal allocation and explain. c. Considering the relative price between of clothes and shoes, at the current consumption, have Amir and Budi reached exchange equilibrium? Please explain d. Use the Edgeworth Box to illustrate the…Utility is a type of function that occurs in economics. When a consumer receives x units of a certain product, a certain amount of pleasure, or utility U, is derived. The graph represents a typical utility function. a) Find the average rate of change of U as x changes from 0 to 1; from 1 to 2; from 2 to 3; from 3 to 4. b) Why do the average rates of change decrease as x increases? a) The average rate of change of U as x changes from 0 to 1 is. (Type an integer or a decimal.) C Utility (pleasure units) AU 200- 150- 100- 50- 0 (1,67) (2,106) I (3,134) (4,156) 2 Number of units of product 4 XA consumer with $120 income and the utility function u(x,y) = (x^2)y where x is food and y clothing can go for shopping in one of the two stores A or B, where the prices for food and clothing are respectively pAx = 1, PAy = 80 and PBx = 8, PBy = 1. (a) If she can visit only one of the stores, which one would she go and what would be her optimal bundle there? Could she afford this very bundle at the other store? (HINT: How happy would she be if she chooses to go to store A for shopping? To store B? Which one is higher and what bundle would she buy there?) (b) If she can freely visit both stores (so that she buys each good from the store where it is sold cheaper), what would be her optimal bundle? What would she buy from each store? (c) How much money would she be willing to give up (out of her $120 budget) for the ability to visit both stores as in (b) rather than being restricted to shopping in only one store (the one she chooses in (a)? (HINT: suppose she pays $A for this privilege.…