Suppose a consumer has the following indirect utility function: v (p, w) = wp, p₂², where w > 0 denotes wealth and p₁ and p2 denote the prices of good one and two, respectively. (a) Formally derive the ordinary demand functions for the two goods. (b) Find the compensated demand functions for the two goods. (c) Verify that the Slutsky equation holds in this case.
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- Given the utility function U = X0.5+Y0.5 and budget constraint I=Px X+ Py Y,(a) Is this utility function is homothetic?(b) Derive the consumer’s demand functions for goods X and Y.(c) Obtain the numerical values of the own-price elasticity, cross-price elasticity, and the income elasticity of demand for X and Y.(d) Derive the indirect utility function and the expenditure function for this consumer.e) Obtain the compensated demand functions for goods X and Y. (f) Prove that the Slutsky equation holds in this case.The consumer’s utility function is u(x1, x2) = 2√x1 + √(x2/4) (a) Determine the equilibrium condition for MRS21 and prices p1, p2. (b) Express the general form of D1(p1, p2, m), D2(p1, p2, m). (c) Calculate the desired amounts x1, x2 for p1 = 8, p2 = 2, m = 80. (d) Express the general form of income elasticity for commodity x1. Please show working step by step for b,c,d with final values of solution like in aM -)a+P (- a + B (II) Consider indirect utility function U* = (- P `P, (a) Find Marshallian demand functions for X1. (b) Find Minimum Expenditure function. (c) Find Hicksian demand functions for X2
- A consumer has the following utility function U (X1, X2) = (x1 + 3) (x2+4) Prices of the two goods X₁ and x₂ respectively are p₁ and p2 and the consumer has income m. We assume that all prices and income are strictly positive. Furthermore, throughout this question we assume that m is high enough so that both goods are consumed in strictly positive amount in equilibrium. (a) Solve the consumer's optimization problem and express the demand for the two goods in terms of prices and income. (1) Compute the missing values of elasticities in the following table assuming M = 10, p₁ = 2, and p2 = 1. Show your work. Income elasticity Absolute value of own price elasticity Cross price elasticity X₁ T≤ X₂ (b) While x2 is produced locally, x₁ is transported from a different region. Assume P₂ = 1. Building a new railroad (that connects the regions) will reduce transportation cost which in turn will reduce the price of good 1 from p₁ = 2 to p₁ = 1. Railroad will be funded by taxes which will reduce…(a) Briefly describe the differences between the ordinary demand function and the compensated demand function. Provide an expression that relates the two. (b) A consumer enjoys two goods, x1 and 2, according to the following utility: u(x) = x1x2 + x2. The prices of the goods arep= consumer's ordinary demand functions and the compensated demand functions for x, and x2. (P1, P2), and the consumer's income is w. Find the (c) Suppose that the consumer's income is w = 125, and that the price vector de- creases from p = that would compensate the consumer for the price change and make him/her just as well of as s/he was before the price decrease. (25, 10) to p' = (20, 10). Find the minimum amount of moneyPlease answer all (a) to (e), whether they are True or False:(a) If a consumer spends her entire income, then she has a strictly monotone utility function.(b) The condition that ‘the marginal rates of substitution equal the ratio of prices’ is necessary but not sufficient for a given bundle to be a Walrasian demand.(c) If U, V: R2 → R are such that U is a strictly increasing transformation of V then U and V must represent the same preferences.(d) If the substitution effect is negative (in response to a price increase) then we know the Walrasian demand for the good in question (in response to the same price increase) will also be negative.(e) A consumer’s utility is continuous and strictly monotone and when prices are given by p and income is I her Walrasian demand yields a utility of 7. Then, any bundle that yields a utility of at least 8 must cost more than I
- Please answer all (a) - (e), whether they are True or False: (a) If a consumer spends her entire income, then she has a strictly monotone utility function. (b)The condition that ‘the marginal rates of substitution equal the ratio of prices’ is necessary but not sufficient for a given bundle to be a Walrasian demand. (c) If U, V: R2 → R are such that U is a strictly increasing transformation of V then U and V must represent the same preferences. (d) If the substitution effect is negative (in response to a price increase) then we know the Walrasian demand for the good in question (in response to the same price increase) will also be negative. (e) A consumer’s utility is continuous and strictly monotone and when prices are given by p and income is I her Walrasian demand yields a utility of 7. Then, any bundle that yields a utility of at least 8 must cost more than I.Consider a consumer with indirect utility function v(p, w) = w - B1p1 - B2p2 √P1P2 where 3₁ and 32 are nonnegative constants. (a) Find the consumer's expenditure function. (b) Find the Hicksian and Marshallian demands for Good 1. (c) Find expressions for the substitution and income effects on Good 1 associated with a marginal increase in the price of Good 2. (d) Suppose that the price of good 1 changes from pi to kp₁ where 0 < k < 1. Find the compensating variation of this price change.Suppose an individual has the following utility function: U(X,Y)= min{X, 2Y}, and they choose marshallian demands gx,gy. Suppose you have an increase in px. Which of the following is true about the change in demand for good X, gx? (a) No substitution effect, because they only consume one of the goods. (b) No substitution effect, because they always consume the goods in fixed proportion. (c) No income effect, because they only consume one of the goods. (d) No income effect, because they always consume the goods in fixed proportion. Which of the following production functions exhibits constant returns to scale? (a) f(k,l) = 0.310.7 (b) f(k, l) = kl (c) f(k, 1) = ln(k)+ln(1) (d) f(k, l) In(k) + In(1) =
- Suppose U = 2X + Y, I = 20, Px = 2, and Py = 2. (a) Find Marshallian demand for X and Y . (b) What is Marshallian demand for X and Y if the price of X increases to 5? How much of the change in demand for X is the income effect and how much is the substitution effect? (c) How much is compensating variation for the price change described in part (b)? (d) How much is equivalent variation for the price change described in part (b)? ( Please solve all the subparts ASAP I will give you thumbs up . )Using the homogeneity of indirect utility, which of the following cannot be an indirect utility function? (a) V: = (b) V: = (c) V: = [2 PzPy I Pz+Py p²+p² (d) V = 1 PzIf 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.