Exercise 1. Consider an economy with two consumers (named Con-sumer 1 and Consumer 2) and two goods (called good 1 and good 2).Suppose that Consumer 1’s preferences are given by the Cobb-Douglasutility functionu1(x11, x21) = x11x21where x11 is the amount of good 1 that Consumer 1 consumes andx21 the amount of good 2 that Consumer 1 consumes. Suppose thatConsumer 1 is endowed with 9 units of good 1 and 2 units of good 2.Suppose that Consumer 2’s preferences are given by the Leontiefutility functionu2(x12, x22) = min{x12, x22}where x12 is the amount of good 1 that Consumer 2 consumes andx22 the amount of good 2 that Consumer 2 consumes. Suppose thatConsumer 2 is endowed with 1 units of good 1 and 8 units of good 2.Normalise the price vector so that p2 = 1.(1) The contract curve is defined as the set of Pareto optimal alloca-tions that make each consumer at least as well off as they are atthe initial allocation. Find the contract curve. [Hint: Think beforeyou start calculating! It may help to start with a draft answer topart (3).](2) Find the equilibrium price and the equilibrium allocation.(3) Represent this economy in an Edgeworth Box diagram. Clearlymark the initial endowment, the equilibrium allocation, and theequilibrium price vector. Draw the indifference curves through theinitial endowment for the two consumers, labelling clearly whichis which. Also draw the contract curve and clearly label it. Drawyour Edgeworth Box as neatly and accurately as you can.Date: First Semester, 2024.1
Question
Exercise 1. Consider an economy with two consumers (named Con-sumer 1 and Consumer 2) and two goods (called good 1 and good 2).Suppose that Consumer 1’s preferences are given by the Cobb-Douglasutility functionu1(x11, x21) = x11x21where x11 is the amount of good 1 that Consumer 1 consumes andx21 the amount of good 2 that Consumer 1 consumes. Suppose thatConsumer 1 is endowed with 9 units of good 1 and 2 units of good 2.Suppose that Consumer 2’s preferences are given by the Leontiefutility functionu2(x12, x22) = min{x12, x22}where x12 is the amount of good 1 that Consumer 2 consumes andx22 the amount of good 2 that Consumer 2 consumes. Suppose thatConsumer 2 is endowed with 1 units of good 1 and 8 units of good 2.Normalise the price vector so that p2 = 1.(1) The contract curve is defined as the set of Pareto optimal alloca-tions that make each consumer at least as well off as they are atthe initial allocation. Find the contract curve. Hint: Think beforeyou start calculating! It may help to start with a draft answer topart (3). Find the equilibrium price and the equilibrium allocation.(3) Represent this economy in an Edgeworth Box diagram. Clearlymark the initial endowment, the equilibrium allocation, and theequilibrium price vector. Draw the indifference curves through theinitial endowment for the two consumers, labelling clearly whichis which. Also draw the contract curve and clearly label it. Drawyour Edgeworth Box as neatly and accurately as you can.Date: First Semester, 2024.1
Solution
Sure, let's break down the problem step by step.
(1) The contract curve:
The contract curve is the set of allocations that are Pareto optimal, meaning that no one can be made better off without making someone else worse off.
For Consumer 1, the utility function is Cobb-Douglas, so the marginal rate of substitution (MRS) is x21/x11. For Consumer 2, the utility function is Leontief, so the MRS is either 0 or infinity, depending on whether good 1 or good 2 is the limiting good.
The contract curve is where these two MRS are equal. Since Consumer 2's MRS can only be 0 or infinity, the contract curve will be two straight lines: one where x11 = x12 (Consumer 2 is consuming all of good 1) and one where x21 = x22 (Consumer 2 is consuming all of good 2).
(2) The equilibrium price and allocation:
To find the equilibrium, we need to set up the budget constraints for the two consumers and solve for the quantities of goods 1 and 2.
For Consumer 1, the budget constraint is p1x11 + x21 = p19 + 2 (since p2 = 1). For Consumer 2, the budget constraint is p1*x12 + x22 = p1 + 8.
Solving these two equations will give us the equilibrium quantities of goods 1 and 2, and the equilibrium price p1.
(3) The Edgeworth Box diagram:
The Edgeworth Box diagram is a graphical representation of this economy. The initial endowment is represented by a point in the box, with the quantities of goods 1 and 2 on the axes. The indifference curves for the two consumers are drawn through this point.
The contract curve is drawn as two straight lines, as described above. The equilibrium allocation is represented by the point where these two lines intersect, and the equilibrium price vector is represented by the slope of the line tangent to the indifference curves at this point.
Remember to label everything clearly, including the initial endowment, the equilibrium allocation and price vector, the indifference curves, and the contract curve.
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