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- 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.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.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= РХPY
- 3. Suppose you consume two goods, whose prices are given by p₁ and p2, and your income is m. Solve for your demand functions for the two goods, if (a) your utility function is given by U(x₁, x2) = ax₁ + bx₂ (b) your utility function is given by U(x₁, x2) = max{ax₁, bx₂}3 8. Let be the utility function U(x, y) =(xi + 2ya), , where x and y are the quantities consumed of two goods. (a) Show that the functional equation U(x,y) U(2, 2) defines y as a function of x in a neighborhood of point (2, 2). (b) If the quantity consumed of good x increases by 1 unit, by how much does the quantity have to increase or decrease? Consumed of the good and to keep utility constant? (c) If the quantity consumed of good x increases by 1%, by how much does the quantity have to increase or decrease? Consumed of the good and to keep utility constant?10. Construct an indirect utility function that corresponds to the direct function U = a In q1 + q2. Use Roy's identity to construct demand functions for the two goods. Are these the same as the demand functions derived from the direct utility function?
- 6. Suppose that a fast-food junkie derives utility from three goods - softdrinks (x), 0.5 hamburgers (y), and ice cream sundaes (z) – according to the Cobb-Douglas utility function U(x, y, z)=x5y5 (1+z)05. Suppose also that the prices for these goods are given by p=0.25, p, = 1, and p. = 2 and that this consumer's income is equal to 2. a) How much will this consumer buy of each good to maximize utility? b) Show that the utility function is maximized (Use |H₂| to identify whether it is a maximum or a minimum). c) What is the maximum utility of the fast-food junkie?8. Let be the utility function U(x, y) =(xi + 2ya), , where x and y are the quantities consumed of two goods. (a) Show that the functional equation U(x, y) U(2,2) defines y as a function of x in a %3D neighborhood of point (2, 2). (b) If the quantity consumed of good x increases by 1 unit, by how much does the quantity have to increase or decrease? Consumed of the good and to keep utility constant? (c) If the quantity consumed of good x increases by 1%, by how much does the quantity have to increase or decrease? Consumed of the good and to keep utility constant?2. Consider a consumer who purchases two goods, x and y. The consumer’s utility function is U(x, y) = xy. Assume initially that the consumer’s income is $160, the price of x is Px= $8, and the price of y is Py= $1.a) Find the utility-maximizing bundle of x and y-So the utility-maximizing bundle of good "x" and good "y" is equal to 10 units of good "x" and 80 units of good "y". b) Find the total utility at the utility-maximizing bundle. total utility is equal to 800.c) Now assume the price of x decreases to $4. Re-compute the values from part a) at the new price. So the utility-maximizing bundle of good "x" and good "y" is equal to 20 units of good "x" and 80 units of good "y". d) Determine the decomposition basket that identifies the substitution and income effects as the consumer moves from the optimal basket in part a) to the optimal basket in part c).e) Identify the substitution and income effects as the consumer moves from the initial consumption basket to the final consumption…
- 8. Suppose there are 2 goods in the economy and the demand function for Good 1 is given by m P₁ + ap₂ Is this a normal or inferior good? What if the demand function was instead given by x₁(m, P₁, P₂) = x₁ (m, p₁, P₂) = P₁ m+ ap₂3. Assume that the price of goods x increases. If x is a normal good, the substitution effect alone leads to a decrease in consumption of x.2. Consider a consumer who purchases two goods, x and y. The consumer’s utility function is U(x, y) = xy. Assume initially that the consumer’s income is $160, the price of x is Px= $8, and the price of y is Py= $1.a) Find the utility maximizing bundle of x and y.b) Find the total utility at the utility maximizing bundle.c) Now assume the price of x decreases to $4. Re-compute the values from part a) at the new price.
