Question 2 2.1 A consumer has a utility function u(x₁, X₂) √1+2 and faces a budget constraint y = P₁x1 + P22. Assuming an interior solution to the consumer's utility maximisation problem (i.e. excluding the possibility of any corner solutions where x = 0 or x = 0), solve for xi (p, y) and x(p, y), the consumer's Marshallian demand functions. 2.2 Derive the consumer's indirect utility function. 2.3 Derive the consumer's expenditure function. 2.4 Suppose the assumption of an interior solution, as set out in question 2.1 above, is no longer maintained. Discuss how (under what conditions) a corner solution to this specific =

Question 2 2.1 A consumer has a utility function u(x₁, X₂) √1+2 and faces a budget constraint y = P₁x1 + P22. Assuming an interior solution to the consumer's utility maximisation problem (i.e. excluding the possibility of any corner solutions where x = 0 or x = 0), solve for xi (p, y) and x(p, y), the consumer's Marshallian demand functions. 2.2 Derive the consumer's indirect utility function. 2.3 Derive the consumer's expenditure function. 2.4 Suppose the assumption of an interior solution, as set out in question 2.1 above, is no longer maintained. Discuss how (under what conditions) a corner solution to this specific =

ENGR.ECONOMIC ANALYSIS

14th Edition

ISBN:9780190931919

Author:NEWNAN

Publisher:NEWNAN

Chapter1: Making Economics Decisions

Section: Chapter Questions

Problem 1QTC

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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.
Question 2 David spends his budget on chocolate and chip. His utility function is given by ?(?1,?2)= 2?1?2, where ?1 is the number of chocolates he consumers per week, and ?2 is the number of chips he buys per week. A chocolate costs 10 SEK, and a chip costs 20 SEK. David’s weekly budge for consuming on these two goods is 120 SEK. (1) What is David’s budge line? Draw the budget line on a graph with chocolate amounts on the horizontal axis and chip amounts on the vertical axis. Write explicitly at which points budget line crosses the axis. (2) What is David’s marginal utilities for the two goods, respectively? What is his marginal rate of substitution between the two goods? (3) What is David’s optimal choice? Calculate the numerical answer for the optimal bundle. Also draw an indifference curve for David on the same graph as question(1) and show the optimal bundle.
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 2 2.1 A consumer has a utility function u(x1, x2) = VT1 + x2 and faces a budget constraint y = P1x1 + P2x2. Assuming an interior solution to the consumer's utility maximisation problem (i.e. excluding the possibility of any corner solutions where x = 0 or x = 0), solve for x(p, y) and x(p, y), the consumer's Marshallian demand functions. 2.2 Derive the consumer's indirect utility function. 2.3 Derive the consumer's expenditure function. 2.4 Suppose the assumption of an interior solution, as set out in question 2.1 above, is no longer maintained. Discuss how (under what conditions) a corner solution to this specific utility maximisation problem may arise (i.e. where x = 0 or x = 0). Then re-write the solutions for xi(p, y) and x(p, y) (the consumer's Marshallian demand functions), as well as the new indirect utility function v(p, y), so as to allow for the occurrence of a corner solution.
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…
Consider the following utility functions. G(x,y) = x² + 3y² H(x,y) = (x+1)¹/2 + y + 1 L(x,y) = xey U(x,y) = In(x) + y² W(x,y) = 100 x - 4x² + 3 y Z(x,y) = (x)¹/² + y Which function or functions have strictly convex indifference curves (at every point where x>0 and y>0)?
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.
What kind of preferences are represented by a utility function of theform u(x1, x2) = √x1 + x2? What about the utility function v(x1, x2) =13x1 + 13x2?
Charlie's utility function is (1-a) U(xa, XB) = Ax“xr-) where A > 0 and a > 0
(a) A consumer with income I=120 facing prices pX = 4 and pY = 8 for two goods X and Y (for each good she prefers more to less, with diminishing MRS) chooses optimally to consume 12 units of X. If the prices change and now pX = 6 and pY = 4, what is the possible range for her new optimal X consumption? (b) Forget about (a). A consumer with I=$240 budget is shopping for apples (x) and oranges (y). Apples cost $1 each up until 60 units; thereafter each apple costs $2. Similarly, oranges cost $1 each up until 60 units, and thereafter $2 each. Draw the graph of feasible set of bundles for the consumer with relevant points and numbers (shade the feasible area), no explanation needed. (c)(HARD!) In (b), calculate the optimal bundle assuming the consumer’s utility function is u(x,y) = x5y.
Zeynep is at the supermarket buying her bi-weekly groceries. As she arrived at the store right before closing, not much is left on the shelves; so her purchases are limited to beef, B, and an assortment of organic vegetables, V. Beef costs £50/kg and vegetables Ł10/kg. Her utility function is expressed as U(B,V) = B0-6v0.3. She has ±150 to spend. Please answer the following questions using this information: A) Given the vegetables cost less than beef, explain why Zeynep would not only buy vegetables. B) Write Zeynep's constrained maximization problem (objective function subject to her budget constraint). C) Using the Lagrangian technique, find how much beef and vegetables would Zeynep buy? D) Show that for the preferences represented as above, demand for beef is a function of only its own price, pg, and income, m, but not the price of vegetables, py. Note: Rather than using the given prices and income, use PB, Py and m. Find and comment on the comparative statics, i.e., how do changes…
For the utility function U = Qx0.28Q (1-0.28) and the budget 137 = 11Qx+6Qy find the CHANGE in optimal consumption of X if the price of X increases by a factor of 1.6. Please enter your response as a positive number with 1 decimal and 5/4 rounding (e.g. 1.15 = 1.2, 1.14 = 1.1).