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- The demand function for a particular good is x = a + bp. What are the associated direct and indirect utility functions?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.How can I analyze and find the Marshallian Demands of the following Utility function? u(x)=min{x1,x2}x3 I understand that goods x1 and x2 are perfect complements with each other, but what is the relation with x3?
- The demand function for a particular good is x = a + bp+cm. What are the associated direct and indirect utility functions?Consider two consumers, A and B, who consume two goods, x and y. Assume that the utility function for A is U₁ = A₁ and the utility function for B is UB = XBYB. Consumer A is endowed with 10 units of x an 10 units of y. Consumer B is endowed with 10 units of x and 40 units of y. Assume the price of y is set to $1. What is the expression for Consumer A's demand for x? XA 10p, +10 Pr 10p, +40 Pr 10pz +10 4pz 10p, +40 4pz1/2 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.
- 2. Ping receives a weekly allowance to purchase soda (x) and juice (y) at school. His preferences over these two goods can be represented by the utility function U(x,y) =0.5x+ 5lny where x represents the number of cans of soda and y represents the number of juice boxes. a) Given his preferences find his Marshallian demand functions for soda (x) and juice (y). b) Suppose that the price of a can of soda is $1.5 and that Ping has $30 to spend on soda and juice. Write Ping's demand curve for juice (y). Illustrate his demand curve. c) Suppose that the price of a box of juice is $1 (the price of soda and income remain $1.5 and $30, resp.). Use your demand functions to find his best bundle. In an indifference curve diagram illustrate his best bundle at these prices. For the remainder of the question assume that Ping's allowance (income) rises to $36 per week and the prices of the two goods are unchanged at Px= $1.5 and Py= $1. d) Use your demand functions to find his new best bundle. e)…1.Let x and y denote the amount of goods X and Y. Find the demand functions of X (do not need to find that of Y) when your preferences are represented by the utility function U = x^2y. Is X normal good? Can you confirm law of demand for X? What is the relationship of X with Y? Answer all of them by using the demand curve you derived. 2.Let x and y denote the amount of goods X and Y. Find the demand functions of X (do not need to find that of Y) when your preferences are represented by the utility function U = x + xy + y. Is X normal good? Can you confirm law of demand for X? What is the relationship of X with Y? Answer all of them by using the demand curve you derived 3.There are two goods F and C. Let MU and P denote marginal utility and price of each good. SupposeMUF = 3, MUC = 4, PF = 2, PC = 2.Are you maximizing your satisfaction? If not, what would you do to increase your satisfaction? Explain.1. Suppose an individual consumes goods and y and has income I. The prices are pr and p, respectively. Find (i) the optimal (Marshalian) demands for x and y as functions of prices and income (ii) the value function (indirect utility function), V(Pa, Py, I) (iii) and deduce the expenditure function when the utility function is (a) U (x, y) = ax + y
- Harry consumes 2 goods, X and Y and he spends N$60 per month. The price of good X is N$4 and the price of good Y is N$10. Harry's utility function is U(X, Y) = XY. (i) What is Harry's marginal rate of substitution? (5) (ii) What is the expression for Harry's budget constraint? (3) (iii) What is the slope of Harry's budget constraint? (2) (iv)Find the values of good X and Y maximizes Harry's utility. Show all your work. (6) (v) Illustrate Harry's utility maximizing combination on a clearly labelled graph. (4) (vi) Suppose the price of good X increased to N$6. Illustrate Harry's income and substitution effects on a well labelled graph. Show your work. (10) (vii) Suppose Genie is consuming two goods, coffee and beer. She spends all her income on a combination of coffee and beer where MUcoffee/Peoffee is 5 and MUbeer/Pbeer is 3. Explain why this combination is not maximizing her satisfaction. What should she do to maximize her utility? (5)An individual consumes two goods, X and Y, and receives utility according to a Cobb-Douglas production function of the form: U(X,Y)= X1/3Y2/3. Give that MUX= 13(MUYMUX)2/3 and MUY= 23(MUXMUY)1/3 find the consumers optimal demand functions for X and Y in terms of PX, PY, and I. Explain why I could have solved the problem above using the utility function U(X,Y)= XY2 and would have obtained the same answer. What is it about utility that makes this possible? Economists define the ‘certainty equivalent’ of a risky stream of income as the amount of guaranteed money an individual would accept instead of taking a risk. The certainty equivalent varies between individuals based on their risk preference.Consider a market with two goods, x and z. The consumer’s utility function is U = x^0,2z^0,8 A. Derive the demand function for x and z. B. Let ? = 2, = 4 and = 50. Find the equilibrium quantities demanded of x and z.
