Consider the utility function u(x, y) \x+2y+1/ (a) Find the equation for the marginal utility for x and one for marginal utility of y. (b) Find an equation of the form y = f (x) that describes the indifference curve for utility equal to -1/2. (c) Argue that the utility values all lie inſthe interval [-1,0]
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- 10. Consider the utility function U(x, y) = xy + x. Which of the following bundles is noton the same indifference curves as the others. Note: The first entry is the amount of x, thesecond is the amount of y.(a) (5,1)(b) (2,4)(c) (4,2)(d) (2.5,3)ASAP 1. Consider a consumer with utility u(x1,x2) = .5lnx1 + .5lnx2 (b) Now, consider an equivalent representation of the above utility functionx51 x52 Does this utility function have the "increasing difference property"? Howabout the "strict increasing difference" property?1. For each of the three utility functions below, find the substitution effect, the income effect, and the total effect that result when prices change from p = (2,1) to p' = (2,4). Assume the consumer has income I = 20. (a) Before doing any calculation, make an educated guess about the relative magnitude of the three substitution effects and the three income effects to be found below. (b) u(x1, x2) = x1 + x2. (c) u(x1, x2) = x1x2. (d) u(x1,x2) = min {x1,x2). (e) Rank the substitution effects and the income effects found above by their magnitude. To what extent do they conform to your guess?
- 4. Consider the utility function U(r,g) r + Iny. (a) Find the marginal rate of substitution. MRS of this function. Interpret the result (b) Find the equation of the indifference curve for this function (c) Compare the marginal utility of r and y. How do you interpret these functions? How might a consumer choose between e and y as she tries to increase utility by, for example, consuming more when their income increases?Q1: (I)Asma eats only apples and almonds. Her level of income is $36, Price of apple (Pa) is $2 per kg and price of almond (Pl) is $6 per kg. The preferences of Asma are represented by the following utility function Utility U= 10*Qa + 40*Ql where Qa and Ql are Asam’s consumption of Apples and almonds. (a) which kind of preferences Asma has? (b) Find Utility per dollar spent on apple and almond (c) Find Marginal rate of substitution and interpret it (d) Find utility maximizing values of apples and almond and show graphically1. A consumer has a utility function defined over two goods X and Y. Let the quantity of Good X be x ≥ 0 and the quantity of Good Y be y ≥ 0. The utility function is given below: u(x, y) = xy + 2y. Assume that the consumer has income m and that prices are px and py. (a) Explain whether the preferences underlying this utility function satisfy completeness and transitivity. = (b) Explain whether the preferences underlying this utility function satisfy monotonicity and convexity (c) Find the consumer's Marshallian demands for Good X and Good Y at prices px > 0 and py > 0. (d) Show that goods X and Y are normal goods and explain whether either good is a substitute for the other. (e) Assume that px = 10, Py = 5 and m = 100. Suppose that px increases to p how much of the change in demand for Good X is via the substitution effect and how much is via the income effect? Note: You may assume an interior solution (i.e. x > 0 and y > 0). = 15,
- 14. Consider the following utility function: U(x, y) = xy+x with MUx = y+1 and MUy = x.As the consumer buys more x, the marginal utility of x:(a) Diminishes(b) Remains constant(c) Increases(2) Consider the utility function u(x, y) = Vx+ay. Your budget constraint is P1x+p2y = М. (a) Find the marginal utilities for these goods. (b) Show that these preferences are quasilinicar. (c) Solve the consumer's optimization problem.21. A consumer has a utility function defined over two goods X and Y. Let the quantity of Good X be x ≥ 0 and the quantity of Good Y be y ≥ 0. The utility function is given below: u(x, y) = xy + 2y. Assume that the consumer has income m and that prices are på and py. (a) Explain whether the preferences underlying this utility function satisfy completeness and transitivity. (b) Explain whether the preferences underlying this utility function satisfy monotonicity and convexity. (c) Find the consumer's Marshallian demands for Good X and Good Y at prices px > 0 and Py > 0. (d) Show that goods X and Y are normal goods and explain whether either good is a substitute for the other. (e) Assume that px 10, Py = 5 and m = 100. Suppose that px increases to px = 15, how much of the change in demand for Good X is via the substitution effect and how much is via the income effect? Note: You may assume an interior solution (i.e. x > 0 and y> 0). =
- 5. What kind of preferences are represented by a utility function of the form u(x1, x2) = x1 + V*2? Is the utility function v(x1, x2) = xỉ + 2x1Vx2+x2 a monotonic transformation of u(x1, x2)? 6. Consider the utility function u(x1, x2) = Vx1x2. What kind of pref- erences does it represent? Is the function v(1, x2) transformation of u(x1, x2)? Is the function w(x1, x2) = xỉx, a monotonic transformation of u(x1, x2)? = x²x2 a monotonic4. Let u(z) be a utility function that represents the preferences of a household. We say that the function v(r) is a monotonic transformation of u if f(-) is a strictly increasing function and v(z) = f(u(z)). (a) Show that if v(z) is a monotonic transformation of u, it represents the same preforences. (b) Can you explain why taking a monotonic transformation of a utility function does not change the marginal rate of substitution? (c) What kind of preferences are reprosented by a utility function of the form u(a1, 12) = VEi +#2? What about the function v(z1, r2) = 13z + 13r2? (d) Consider the function u(r1, 2) = VE1#2. What kind of preferences does it represent? Is the function (11, r2) jz a monotonic transformation of 1 u(r1, 12)? Is the function w(r1, 12) u(r1, 12)? = zfr a monotonic transformation ofConsider a consumer whose preferences can be represented by the utility function u(x, y) = x + y (a) Originally, px = 1, py = 2 and m = 1. What bundle does the consumer choose, and what is his utility from this bundle? (b) The price of good x then rises to 3. What bundle does the consumer choose after the price change, and what is his utility from this bundle? (c) Calculate the compensating variation. (Hint: at the new price ratio, what good will he spend his income on?) (d) Calculate the equivalent variation.