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- 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)."If the utility function of an individual takes the form: U = U ( x 1, x2) = (x1 + 2) 2 (x2 + 3) 3 Where U is total utility, and x1 and x2 are the quantities of two commoditiies consumed: (a) Find the marginal-utility function of each of the two commodities (b) Find the value of the marginal utility of the first commodity when 3 units of each commodity are consumed.I need help with this question in my homework. Suppose a consumer’s utility function is given by U(X,Y) = MIN (5X, Y). Also, the consumer has $60 to spend, and the price of good X is P(x) = $5. Let good Y be a composite good (good Y is the “numeraire”) whose price is P(y) = $1. So, on the Y-axis, we are graphing the amount of money that the consumer has available to spend on all other goods for any given value of X. What is the Indirect Utility Function? What is the Expenditure Function?
- Consider the following utility functions: a) U= xy b) U=(xy)l3 c) U=min(x,y/2) d) U=2x + 3y e) U=x?y? + xy 1. Check if any of utility functions a) – e) are homogeneous. If so, state the degree of homogeneity of each.Problem 1. Two consumers, Alice and Bob, have utility functions given by u₁(x, y) = x¹/²y¹/4 and ug (x, y) = 4x² + y², respectively, where x and y are the number of apples and bananas respectively. Suppose that apples and bananas cost p₁ and p2 dollars each respectively and that both consumers have M dollars to spend on apples and bananas. (a) On the same axes, sketch Alice's budget set and some contours u₁(x, y) = c for positive values of c. By referring to your sketch explain why the method of Lagrange multipliers can be used to find the maximum value of u₁(x, y) subject to her budget constraint. Hence use the method of Lagrange multipliers to find Alice's optimal bundle. (b) On the same axes, sketch Bob's budget set and some contours up (x, y) = c For positive values of c. By referring to your sketch, find Bob's optimal bundle.Rafe is optimally choosing to consume 6 apples and 3 bananas. The prices of apples and bananas are p. = 7 and pb - 7. Which of the following utility functions over quantities of apples (a) and bananas (b) could represent Rafe's preferences? = u(a, b)-a4/5 61/5 Ou(a, b) = a¹/5 64/5 Ou(a, b)-a2/3 b1/3 Ou(a, b)-a¹/3 2/3
- Consider the following utility functions. G(x,y) = x² (1+y) H(x,y) =In(x) + In(2y) +20 L(x,y) = (x+1)1/2 + (y+1) 1/2 U(x,y) = ex+y W(x,y) = 30 x - 4x² + 4 y Z(x,y) = x¹/2 y Which function or functions are homothetic?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 + 0.5x₂Q2. Suppose a consumer seeks to maximize the utility function U (x, y) = (x + 2) (y + 1), where x and y represent the quantities of the two goods consumed. The prices of the two goods and the consumer's income are pa, py, and I. (c) Show the bordered Hessian matrix, H for this problem. What does the second order condition require for this problem? Show if it is satisfied.
- For the next three questions, assume that there are two consumers in an economy that have utility functions UA(2,3)=2¹/42/4 U" (x,y)=1¹/2¹/2 The two consumers begin with equal endowments of the two goods === e = 50 29. If the price of z and y were both set to 1, there would be (a) An excess demand for r so the equilibrium price ratio must be less than 1 (b) An excess supply of r so the equilibrium price ratio must be less than 1 (c) An excess demand for y so the equilibrium price ratio must be greater than 1 (d) An excess supply of y so the equilibrium price ratio must be greater than 1 (e) No excess supply or demand for either good, so the equilibrium price ratio is 1 30. What is the equilibrium price ratio? (a) 2/3 (b) 5/2 (c) 3/5 (d) 1 (e) None of these 31. Consumer A increases his endowment of both goods to 100 (e = e = 100). This will cause (a) No change in the equilibrium price ratio (b) The equilibrium price ratio to increase, causing consumer B to decrease their consumption of…Suppose that David and his friend Wilson derive utility from consuming two types of snacks: onion rings (9₁) and chips (9₂). The utility function for each individual is U (9₁, 92) = 9192. Their indifference curves for these two goods are assumed to have the usual (convex) shape. Suppose David has an initial endowment of 35 onion rings and 10 chips, and Wilson's initial endowment consists of 5 onion rings and 20 chips. (1) Draw an Edgeworth box and show the initial allocation of goods, to be labelled e. Indicate the initial quantities of each person's goods on the four axes.Text: Consider a consumer whose preferences are characterized by the utility function u(x1, X2) = In x1 + x2. This utility function is often used to represent the tastes of a consumer to whom good 1 is essential, while good 2 is not. Furthermore, the tastes represented by this utility function satisfy the five standard assumptions about tastes introduced in the lectures. Based on this information, answer the following questions Questions: Which of the following maps of indifference curves could be the one associated with the utility function above? X2 X2 b d