Consider the following Indirect Utility Function U*PM derived from Linear Expenditure System (LES): ● M-P₁ B₁-P₂ B₂ P₁ P1-x Derive the Marshallian Demand functions for good X₁ and X2 U :
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- Suppose that there are two goods, gl and g2, that Jane consumes. Demand functions for these two goods are given as follows: gl * = gl(pg1), pg2 ,Inc) g2 * = g2(pg1), pg2 ,Inc) where pg1, pg2 ,Inc refer prices of the goods and income, respectively. Explain and demonstrate (on the graph) the substitution and income effect for both an increase and decrease in price good g1. Use the notations given in the question.Rick consumes 2 goods, Chicken McNuggets (M) with Szechuan sauce (S). His utility function is U(M, S) = M2/3S1/3 and his income is m. The price of Chicken McNuggets is p, and the price of Szechuan sauce is 1. d. Use the equation from part (c) and the budget constraint from part (a) to find Rick’s demand for each of the two goods. e. Suppose m=100 and p=1. How much of each good does Rick consume? Draw a graph showing Rick’s budget constraint and indifference curve passing through his chosen consumption bundle. f. Suppose m=100 and p=2. How much of each good does Rick consume? On the same graph from part (e), show Rick’s budget constraint and indifference curve passing through his new chosen consumption bundle.2. Suppose that a consumer's utility function is given by xy U (x, y) = x + y Denote the price of good x by pæ, the price of good y by Py, and consumer's income by w. Find this consumer's Marshallian demand functions and indirect utility function.
- A student has a lunch utility function U(S, P) 8S + 3P, where S is number of salads per month and P is number of pizzas per month. The price of a salad is $5, the price of a pizza is 8, and his monthly income for lunch is $80. a) Derive the equation representing the consumer's demand for Salad. Represent the demand for Salad graphically. b) The price of Salad goes up to $25 and the price of Pizza does not change. Compute and represent graphically the new equilibrium of the student. What can you tell about the substitution effect on Salad and pizzaPlease draw the income-consumption (or income-offer) curve for each of the following utility functions, indicating the slope of each: (a) U = (A)(x^a)(y^b) (b) U = min(ax, by) (c) U = ax + byA consumer's utility function is given by u (r_y) := r'/2y}/2 for any nonnegative x and y, representing the consumption quantities of goods x and y, respectively. Suppose that the price of good y is constantly ¢1, and that the consumer is given income m cedis (and nothing else). Denote p, for the price of good r. Based on the quantity demand (p, m) for good r. a. Given that p, > 1/(4m), calculate i (Pz, m) and (Pz, Py, m), then use the Slutsky equation to deduce what the substitution effect r° (pz, y) is equal to. b. Given that p, < 1/(4m), calculate i (P, m) and i (P2, Py, m), then use the Slutsky equation to deduce what the substitution effect r° (Pz; y) is equal to. c. What is your interpretation of the your result in a and b above?
- what problem could arise with consumers' net demand functions as any price becomes zero? what axiom of consumer theory is involved here? how might it be changed to exclude this possibility?8. Fabian consumes X and Y and the following utility function represents his utility: U = 2XY a. With a utility function of U = 2XY, Fabian's MU, = 2Y and his MU, = 2x ,where MU is marginal utility. Write an equation for Fabian's marginal rate of substitution (MRS). b. Suppose Fabian's income is $120 and Px = S6 and Py-$2, where Px is the price of X and Py is the price of Y. Write an equation for Fabian's budget constraint. c. Write the equation that equates the slope of Fabian's budget constraint to the slope of Fabian's indifference curve: d. If Fabian is spending all of his income, how much X and how much Y will he choose? Use the equations you wrote in part's b and c to answer this question. e. What will Fabian's utility be with this combination of X and Y? f. Now suppose the price of X increases to P.=10. Write the equation that equates the slope of Fabian's budget constraint to the slope of Fabian's indifference curve. How will this change the optimal combination of X and Y? What…2. Lola consumes good x and y, and her marginal rate of substitution is 3/2, regardless of how many units of x and y that she consumes. Assume Lola has an income of $200 and that P-$5. (a.) Find Lola's optimal consumption bundle when px = 6. (b.) Find Lola's demand for x as a function of p,, assuming the price can be any non- negative amount.
- 4.13 CES indirect utility and expenditure functions In this problem, we will use a more standard form of the CES utility function to derive indirect utility and expenditure functions. Suppose utility is given by U(x, y) = (x° +y®)'/8 [in this function the elasticity of substitution o = 1/(1 – 6)]. a. Show that the indirect utility function for the utility function just given is V = I(p, + p,)¬/", where r = 8/(ò – 1) = 1 – 0. b. Show that the function derived in part (a) is homogeneous of degree zero in prices and income. c. Show that this function is strictly increasing in income. d. Show that this function is strictly decreasing in any price. e. Show that the expenditure function for this case of CES utility is given by E = V(p', + p,)''". f. Show that the function derived in part (e) is homogeneous of degree one in the goods' prices. g. Show that this expenditure function is increasing in each of the prices. h. Show that the function is concave in each price.Consider the two bundles A and B. Note: each bundle is a pair of values, (c1, c2).Calculate the expenditure for the each of the two bundles. Put your answers in theblue highlighted cells for Q5.Suppose that we have a utility function involving two goods that is linear of the form U =ax+by.a. Derive demand functions for x and yb. Derive indirect utility functionsc. Derive the expenditure function for this utility function. Hint: The expenditure functionwill have kinks at various price ratios.