A consumer has utility u(x₁, x2) = x₁x² 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 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 z(p, w) = (3p₁¹ 3p²) w 2w (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.
A consumer has utility u(x₁, x2) = x₁x² 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 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 z(p, w) = (3p₁¹ 3p²) w 2w (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.
Chapter1: Making Economics Decisions
Section: Chapter Questions
Problem 1QTC
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Question
why convex:
The budget set consists of all convex combinations of the points
How to know these: can you show each the points can be derive
(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)
how to solve it?
Transcribed Image Text:A consumer has utility u(x₁, x2) = x₁x² 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 x₁ = 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) = (3p₁' 3p2)
w2w
(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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