4. The CES utility function is given byx°+y°U (x, y)(i). Show that the first-order conditions for aconstrained utility maximum with thisfunction require individuals to choose goodsin the proportion =Pa8–1Py(ii). Show that the result in part (i) impliesthat individuals will allocate their fundsequally between x and y for the case 8 = 0.(iii). How does the ratiodepend on thePyYvalue of 8? Explain your results intuitively.(iv). Derive the indirect utility andexpenditure functions for this case andcheck your results by describing thehomogeneity properties of the functions youcalculated.

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ENGR.ECONOMIC ANALYSIS

14th Edition

ISBN:9780190931919

Author:NEWNAN

Publisher:NEWNAN

Chapter1: Making Economics Decisions

Section: Chapter Questions

Problem 1QTC

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Suppose that U(f,c) = f + 8c^(1/2)is a utility function that describes Amelia’s preferences over two goods: fish(f)and custard (c). For the following, think of fish as the good graphed on the horizontal axis.a. Derive an expression for her marginal utility (Uf)from a small increase in f holding c fixed. Also find themarginal utility for custard (Uc).b. What is Amelia’s marginal rate of substitution (MRS)? Give a brief (2 sentences maximum) intuitivedescription of what MRS represents. If Amelia has 4 units of custard, holding her utility constant, howmany units of custard would she be willing to give up in order to get one more unit of fish?c. Graph Amelia’s indifference curve for a utility level of 40. Be sure to specify at least 3 bundles of goodson the indifference curve.d. Does the fact that Amelia’s indifference curve intersects with the custard axis violate any of the 5properties of indifference curves? Briefly support your answer.e. Give another utility function that represents…
Exercise 4. In the lectures we have been working the the CRRA utility function, u : R+ which we defined, for any n≥0, as → R, Prove that u(x): -{ 21--1 1-n if ŋE [0,1) U (1,∞) In a if n = 1. 2--1 lim == Inz. →1 1-7 (7) Hint: Use L'Hôpital's Rule. Exercise 5. Consider the following two-period optimization problem we studied in class: max u (x1)+ẞu (x2), s.t. x1+px2=y, (1.22)ER² (8) where u is given by (7), x; represents consumption of good i = {1,2}, ẞ € [0, 1] is the discount factor, p = R++ is the relative price of good 2 in terms of good 1, and y = R++ is income expressed in terms of good 1. For the following two questions, consider an arbitrary n > 0. 1. Find the bundle (1, 2) that solves (8). 2. Notice that the solution (1, 2) depends on the parameters (n,3,p,y), which the con- sumer takes as given. Derive the expressions for ǝr/Op and ǝx/Əp.
Jose has utility for UC Davis sporting events equal to U = S²B where S is the number of Soccer games and B is the number of baseball games. Baseball games cost $5 per game and Soccer games cost $10 per game. How many soccer games will Jose attend in his optimal bundle for ever baseball game? In other words, in his optimal bundle, S = xB. Second, if Jose has $90 to spend on sports, how many baseball games will he attend? In other words, Jose will attend baseball games in his optimal bundle. Third, if Jose's budget increases to $180, now how many baseball games will he attend? Jose will attend baseball games. Fourth, for Jose, is baseball a normal or inferior good? Enter the word "normal" or "inferior" here
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You consume music (M) and concert tickets (C). Your utility function is U(M, C) = M1/4C3/4. The marginal utility for concert tickets, MUC is MUC =3/4C-1/2M1/4 and the marginal utility for music, MUM is MUm = 1/4C3/4M-3/4 (a) Calculate MRSMC using only the given marginal utilities. (b) Solve for the utility of bundle A where M = 16 and C = 16. Solve for your utility at bundle B where M = 128 and C = 8. Are the utilities the same? (c) Calculate MRSMC at bundle A and at bundle B. Are they the same? (d) Are your indifference curves convex? Draw the ICs. Make sure to label the quantities of the consumption bundles, the axis, and the MRS at those bundles.
John’s preferences for apples (A) and berries (B) are represented by U(A,B)= A+2B . Apples cost £2 and berries £1. Given that John’s monthly income is £30 answer the following questions: What type of goods are apples and berries for John? What is the proportion to which John is willing to exchange apples for berries? Illustrate and solve graphically John’s utility maximization problem. If his income increases every month by £10, how will John’s consumption choice be affected? Illustrate graphically the income expansion path and the Engel curve for each good. How will an increase in the price of berries to £6 affect John’s optimal consumption choice? (John’s income is £30) Graph John’s demand curve for each good. Assume that John wins a voucher of £20, redeemable only in apples. How would this affect John’s utility? (Assume that prices and income are as described initially) Assume that John is presented with two options: an apple voucher of £20 or just £6 to spend on any good he wants.…

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