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A consumer has utility u(x₁, x2) = x₁ for two goods. The government subsidizes half of the cost of the purchase of good 1 up to a maximum of 2 units. Thus if the consumer chooses the bundle (1, 2), she pays p₁/2 per unit up to a₁ = 2, and p₁ per unit thereafter. In particular, if she consumes more than two units, she receives a total subsidy of p₁. Assume throughout this question that w > p₁. Recall that the Marshallian demand for this utility function (in the absence of any subsidy) is x(p, w) = w 2w 3p1 3p2 (a) Carefully sketch the budget set. Be sure to identify relevant points of intersection with

A consumer has utility u(x₁, x2) = x₁ for two goods. The government subsidizes half of the cost of the purchase of good 1 up to a maximum of 2 units. Thus if the consumer chooses the bundle (1, 2), she pays p₁/2 per unit up to a₁ = 2, and p₁ per unit thereafter. In particular, if she consumes more than two units, she receives a total subsidy of p₁. Assume throughout this question that w > p₁. Recall that the Marshallian demand for this utility function (in the absence of any subsidy) is x(p, w) = w 2w 3p1 3p2 (a) Carefully sketch the budget set. Be sure to identify relevant points of intersection with

ENGR.ECONOMIC ANALYSIS
ENGR.ECONOMIC ANALYSIS
14th Edition
ISBN:9780190931919
Author:NEWNAN
Publisher:NEWNAN
Chapter1: Making Economics Decisions
Section: Chapter Questions
Problem 1QTC
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A consumer has utility u(x₁, x2) = x₁ for two goods. The government subsidizes half of the cost of the purchase of good 1 up to a maximum of 2 units. Thus if the consumer chooses the bundle (21,22), she pays p₁/2 per unit up to 2₁ = 2, and p₁ per unit thereafter. In particular, if she consumes more than two units, she receives a total subsidy of p₁. Assume throughout this question that w > p₁. Recall that the Marshallian demand for this utility function (in the absence of any subsidy) is x(p, w) w 2w 3p13p2 (a) Carefully sketch the budget set. Be sure to identify relevant points of intersection with the axes. Solution: The budget set consists of all convex combinations of the points (0,0), (0, w/p2), (2, (w - P₁)/p2), and (1+w/p₁,0). The upper-right boundary of this region consists of two line segments meeting at a kink, with the segment to the left of the kink being flatter than the one to the right. (b) Find the utility-maximizing bundle as a function of prices (p₁, p2) and wealth w. Solution: The "budget line" consists of two parts: below x₁ = 2, it is the same line as for wealth w and prices (p1/2, p2); above x₁ = 2, it is the same line as for wealth w+p₁ and prices (p1, p2). The corresponding Marshallian demands are (2w/3p1, 2w/3p2) and ((w + P₁)/3p1,2(w +P₁)/3p2). Thus if 2w/3p₁ < 2, the first is the demand; if (w + p₁)/3p1 > 2, the second is; otherwise, the demand is (2, (w - P₁)/p2).
Transcribed Image Text:A consumer has utility u(x₁, x2) = x₁ for two goods. The government subsidizes half of the cost of the purchase of good 1 up to a maximum of 2 units. Thus if the consumer chooses the bundle (21,22), she pays p₁/2 per unit up to 2₁ = 2, and p₁ per unit thereafter. In particular, if she consumes more than two units, she receives a total subsidy of p₁. Assume throughout this question that w > p₁. Recall that the Marshallian demand for this utility function (in the absence of any subsidy) is x(p, w) w 2w 3p13p2 (a) Carefully sketch the budget set. Be sure to identify relevant points of intersection with the axes. Solution: The budget set consists of all convex combinations of the points (0,0), (0, w/p2), (2, (w - P₁)/p2), and (1+w/p₁,0). The upper-right boundary of this region consists of two line segments meeting at a kink, with the segment to the left of the kink being flatter than the one to the right. (b) Find the utility-maximizing bundle as a function of prices (p₁, p2) and wealth w. Solution: The "budget line" consists of two parts: below x₁ = 2, it is the same line as for wealth w and prices (p1/2, p2); above x₁ = 2, it is the same line as for wealth w+p₁ and prices (p1, p2). The corresponding Marshallian demands are (2w/3p1, 2w/3p2) and ((w + P₁)/3p1,2(w +P₁)/3p2). Thus if 2w/3p₁ < 2, the first is the demand; if (w + p₁)/3p1 > 2, the second is; otherwise, the demand is (2, (w - P₁)/p2).
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