estion 5 A utility function is given by u(x₁, x2)=-4x+80x18x+100x2-x1x2. Suppose that prices are p₁ = 5, and p2 = 1. Income is I. 1. Specify the Lagrangean using all three constraints. 2. Use the Lagrangean determine the range of values of income / for which the different constraints may bind. Note: these preferences are not monotone. As a consequence, the budget constraint itself may not bind.
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- 1. Suppose that U (f,c) = f + 8c² 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. Derive an expression for her marginal utility (Uf) from a small increase in f holding c fixed. Also find the marginal utility for custard (U.). b. What is Amelia's marginal rate of substitution (MRS)? Give a brief (2 sentences maximum) intuitive description of what MRS represents. If Amelia has 4 units of custard, holding her utility constant, how many 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 goods on the indifference curve. d. Does the fact that Amelia's indifference curve intersects with the custard axis violate any of the 5 properties of indifference curves? Briefly support your answer. a. Give another utility function that…8. Show that if strict preferences > over three outcomes x, y and z, do not satisfy transitivity, so that x > y > z > x, then these preferences cannot be represented by a utility function.2) Which of the following utility functions represent the same preferences? Explain. a) U (x₁, x₂) = X₁ X₂ b) W (x₁, x₂) = 5lnx₁ +5lnx₂ c) V (x₁, x₂) = x₁¹/3x₂ ¹/3 - 0.8 d) Z(x₁, x₂) = 0.5x₁ + 0.5x₂
- 1 Consider a consumer who consumes only two goods, 21 and z2. His utility function is u (21, 22) = In z1 + 2. where In represents the natural logarithm. 1.a Derive the consumer's MRS of good 1 for good 2 using calculus to calculate his marginal utility from z and his marginal utility from z2. 1.b If the price of z1 is $2 per unit and the price of z2 is $4 per unit, and the consumer's income is $100, What is the equation of this consumer's budget line? 1.c What are the optimal consumption choices, z and z5, for this consumer? Show your work.4. The CES utility function is given by x°+y° U (x, y) (i). Show that the first-order conditions for a constrained utility maximum with this function require individuals to choose goods in the proportion = Pa 8–1 Py (ii). Show that the result in part (i) implies that individuals will allocate their funds equally between x and y for the case 8 = 0. (iii). How does the ratio depend on the PyY value of 8? Explain your results intuitively. (iv). Derive the indirect utility and expenditure functions for this case and check your results by describing the homogeneity properties of the functions you calculated.Problem 2 Suppose that John's preferences over meat (M) and vegetables (V) are represented by the following utility function U(M,V) = a ln(M) + (1 − a) ln(V) where 0 < a < 1. ¹No claim of realism is made for the numbers in this example. 1 (a) Write down the Lagrangian for John's optimization problem. (Recall that John max- imizes utility given an income, I, and prices på and på for the goods.) (b) Solve for John's optimal consumption bundle (M*, V*) (as a function of income and prices) using the Lagrangian method. = (c) Suppose a = . Suppose also that John has income I = 1 and pv 2. What is the value of John's optimal happens if John's income doubles to I = 400? 200 and faces prices p consumption bundle? What =
- 1. Andrew is a deeply committed lover of croissants. Assume his preferences are Cobb-Douglas over croissants (denoted by D on the x-axis) and a numeraire good (note: we use the notion of a numeraire good to represent spending on all other consumption goods - in this example, that means everything other than croissants - its price is normalized such that P = $1). Assuming Andrew's utility function is given by U(C, N) = CN and his income is $64 a year, his Marshallian demand for croissants will be Dc (Pc, PN,Y)= The expenditure minimization problem yields his compensated (Hicksian) demand for croissants, his compensated (Hicksian) demand for the numeraire good, and his expenditure function: 2PC Hc = U C. 1/2 HN = U (PN) ² 1/2 = 0 (Pc) ¹/² E (PC, PN,U)= Pc* Hc + PN * HN = 2Ū(Pc * PN)¹/2 No need to derive these for the assignment, but you can solve for them on your own if you want extra practice! a. You've been hired by a government official considering a proposed piece of legislation that…For the case of two goods, give an example of a utility function U(x1, x2) that represents the preferences of a consumer who regards the two goods as perfect complements. Next, take the transformation f(U) = U³ of the your example utility function and explain if this newly gener ated function represents the original preferences. Further, provide clear arguments supporting or rejecting the claim that "f(U(x1, x2)) must be strongly increasing in (x1, x2)."Question 1. For the case of two goods, give an example of a utility function U(₁, ₂) that represents the preferences of a consumer who regards the two goods as perfect complements. Next, take the transformation f(U)= U³ of the your example utility function and explain if this newly gener- ated function represents the original preferences. Further, provide clear arguments supporting or rejecting the claim that "f(U (₁, ₂)) must be strongly increasing in (₁, ₂)."
- 2. Consider these three utility functions: u1 = U2 = In + In a2 Which of these functions can be said to represent the same preferences: Demonstrate whether uz and uz are monotonic transformations of the first function, u1.QUESTION ONE The only source of income available to Doreen, a level 200 student of UEW is her monthly stipend (M) from her parents. She neither saves nor borrows. Assuming she spends all her income on only two goods food (F) and Clothes (C), with Pr and Pc being the prices of the two goods respectively and having a utility function of the form: U = 6F°C i. Write an expression for Doreen's budget constraint. ii. Assuming good Fis on the vertical axis, produce a sketch of her budget constraint and determine the slope. iii. Find Doreen's monthly equilibrium demand functions for both goods. iv. With Pr and Pcgiven as ¢12 and ¢15 respectively and income of ¢300, calculate the equilibrium quantities of both goods. v. Compute the price elasticity of demand for both goods and interpret your results. vi. If income and prices of the two goods increase by 50%, calculate the equilibrium quantities of Food and Clothing.1 Alice has a monthly lunch budget of 360 dollars which she spends on either burgers (B) or pizza (Z). The price of each pizza is 16. dollars and the price of each burger is 8 dollars. Alice's utility is given by: u(B, Z) = B'/2 +4Z!/2 + 47!/2 a) Find Alice's budget constraint. Set up the Lagrangian and find her optimal consumption. What is her new consumption if the price of burgers becomes 16 dollars?