Which one of the statements below is false for Cobb-Douglas preferences over bundles of x1 and X2 with prices p1 and p2, respectively? a) The optimal bundle is always interior for a positive income, m. b) The Marshallian demand function for x; depends both on p; and pj, j i. c) These preferences are strictly convex and strictly monotone. d) The Engel curves for x, and x2 are positively sloped.
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- a. Determine the demand functions of x and y in the case of a Cobb-Douglas type utility function, in the following cases: α=0.40;β=0.60 Graph the demand functions of the two goods (price as a function of quantity) assuming the individual's income is $500 - Determine what is the quantity demanded of x and y, if the price of good x is USD 1, the price of good y is USD 4, and income is USD 500 - Now, explain what happens to the quantity demanded if the prices of the goods are doubles holding income constant.1/2 1/2 2. Cynthia has preferences represented by the utility function u(x) = x¹/² + x¹/² (a) Calculate Cynthia's marginal rate of substitution. (b) Calculate Cynthia's Marshallian Demand functions x(p, 1) and x(p,1). (c) Does Cynthia consider either of these goods to be Inferior Goods? (d) Show that Cynthia's demands generate an indirect utility function V(p,1)=√√IP₁P2 (e) Solve for Cynthia's expenditure function.A) Suppose a quasilinear utility function is given by u(x₁, x₂) = 2√√x₁ + 4x₂. 1. For this utility derive the demand functions xi (P₁, P2, 1) and x₂ (P₁, P2, 1). 2. For this utility derive the Engle function I = 1(x₂, P₁, P₂) for good 2. B) Do the same for the Cobb-Douglas utility function u(x₁, x₂) = x₁x². That is, 1. For this utility derive the demand functions x₁ (P₁, P2, 1) and x₂ (P₁, P2, 1). 2. For this utility derive the Engle functions I = 1(x₁1, P₁, P₂) and I = = 1(x₂, P1, P2).
- 5. Which one of the following statements is definitely correct for the marginal rate of substitution MRS(x1,x2) = dxX2/dx1 of some continuous preferences over bundles of x1 and x2 with prices p, and p2, respectively? a) If the preferences over x1 and x2 are strictly monotone, MRS(x1,x2) 0. c) For strictly monotone preferences, at a corner solution (x;,x;), MRS(x¡,x;) = P1/P2. d) For strictly monotone and strictly convex preferences, if at a point (x1,x2) on the budget line MRS(x1,x2) > P1/P2, then the optimum bundle (x, x;) lies where x X2.Slutsky.Consider a consumer with utility function U (x, y) = 2xy. Her income is I = 15, andprices are given by px = 2, and py = 3. Address the following questions:a) Find the demand functions.b) Find the optimal bundle.c) Find the price elasticity of good yAssume the price of good x increases up to px = 3.d) Find the income effect, the substitution effect and the total effect.e) Determine whether the good x is normal or inferiorf) Determine whether the good x is a Giffen goodg) Are goods x and y substitutes, complements, or independent goods? Note:- Do not provide handwritten solution. Maintain accuracy and quality in your answer. Take care of plagiarism. Answer completely. You will get up vote for sure.3 The following utility function is known as CES (constant elasticity of substitution) function: U (x, y) = (ax° + By')'/º, where a > 0, B > 0 %3D a) Is this function homothetic? b) How does the MRSY depend on the ratio x/y? Specifically, show that the MRSxy is strictly decreasing in the ratio x/y for all values d 1 and constant for 8 = 1. (Hint: take the derivative of the MRS with respect to the ratio x/y as z = and take the derivative with respect to z) x,y x/y c) Show that if x = y, the MRS of this function depends only on the ratio /B.
- Formulate and prove the Slutsky Equation for the effect on uncompensated demand for a good which results from a marginal own price change. Further, provide arguments for or against the following statement: "The optimal bundle chosen by a consumer with well-defined preferences (rational, continuous, strictly monotone and strictly convex) always contains luxury goods."5. Suppose that we can represent Pauline's preferences for cans of pop (the x-good) and pizza slices (y-good) with the utility function u(x, y) = min[4x,5y]. a) Find her Marshallian Demand Functions. b) Find her Hicksian Demand FunctionsA consumer’s preferences over two goods x and y are given bythe utility function U(x, y) = xαyβ with α, β > 0. The prices of the goods are px = 2 and py = 4.The consumer has an income of I > 0.(a) For what values of α and β are these utility functions strictly monotone?(b) For what values of α and β will the consumer demand (i.e., Walrasian demand) be more x than y?(c) For what values of α and β are these goods gross substitutes? For what values of α and β are these goods gross complements? Provide a justification for your answer.
- Suppose that the utility function for two commodities is: U(q1, q2) = q1α q2(1-α) Let the prices of the two commodities be p1 and p2 and let the consumer’s income be M. (1) Check the properties of marginal utilities. In particular, check whether it satisfies diminishing marginal utilities. (2) Assuming all income is spent on these two commodities, derive the demand curves for the two commodities. (3) What happens if U(q1, q2) = q1α q2β?Show that the two utility functions given below generate identical demand functions for goods X and Y UCXY) log(X)+log(Y C) UCxY) (XY05 The demand function for good X and the demand function for good Y for both utility functions equal O A. X= and Y Pv Px О В. Х- and Y 2Pv Ос. X Y= 21 PXPY O D. Х- and Y 2P. 2Py ОЕ. X= Y= РХPYAnswer typed .