2 marks, −0.667MicroeconomicsGeneral equilibrium and exchange economiesEdgeworth box and the contract curveHardabout 3 min

IIT JAM 2022 Question 26: General equilibrium and exchange economies

Consider a pure exchange economy with two goods $x$ and $y$ . Ravi and Suraj have utility functions $U_R = \beta\log(xy)$ and $U_S = \left(\dfrac{x}{y}\right)^{\alpha}$ . Endowments satisfy $x_R + x_S = A$ and $y_R + y_S = B$ . Then their contract curve is

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Answer and worked solution

Answer: (B)

For Ravi, $U_R = \beta(\ln x + \ln y)$ gives marginal utilities $\beta/x_R$ and $\beta/y_R$ , so his marginal rate of substitution is $\dfrac{y_R}{x_R}$ .

For Suraj, $U_S = (x/y)^{\alpha}$ is increasing in $x$ but decreasing in $y$ , so $y$ acts as a bad. Its ratio $\dfrac{\partial U_S/\partial x}{\partial U_S/\partial y}$ works out to $-\dfrac{y_S}{x_S}$ . Tangency sets Ravi's ratio equal to Suraj's: $\dfrac{y_R}{x_R} = -\dfrac{y_S}{x_S} = -\dfrac{B - y_R}{A - x_R}$ .

Cross multiplying, $y_R(A - x_R) = -x_R(B - y_R)$ , which simplifies to $A y_R + B x_R - 2 x_R y_R = 0$ , option (B).

The trap: (A)

A y_R - B x_R = 0

Treating both agents as standard (both goods 'goods') gives the diagonal $A y_R - B x_R = 0$ in option (A); the sign flip from $y$ being a bad for Suraj is what produces the extra $-2 x_R y_R$ term.

Plain-text version

Consider a pure exchange economy with two goods x and y. Ravi and Suraj have utility functions U_R = βlog (xy) and U_S = (x/y)^α. Endowments satisfy x_R + x_S = A and y_R + y_S = B. Then their contract curve is (A) A y_R - B x_R = 0 (B) A y_R + B x_R - 2 y_R x_R = 0 (C) A y_R + B x_R - y_R x_R = 0 (D) A y_R - B x_R + 2 y_R x_R = 0

Also written as: Consider a pure exchange economy with two goods x and y. Ravi and Suraj have utility functions U_R = betalog (xy) and U_S = (x/y)^(alpha). Endowments satisfy x_R + x_S = A and y_R + y_S = B. Then their contract curve is

General equilibrium and exchange economies has come up 69 times in 7 exams since 2004.

What this question tests

This question is on Edgeworth box and the contract curve, part of General equilibrium and exchange economies in Microeconomics.

The geometry of trade between two consumers and the set of efficient allocations.

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