Question

In an economy with two types of consumer, each type has the respective utility function and endowments:

u1(x1,x2)=x1x2 and u2(x1,x2)=x1x2
​ and
​ω1=(8,2)ω2=(2,8)
​
(a) Draw an Edgeworth box for this economy when there is one consumer of each type. (b) Characterize as precisely as possible the set of allocations that are in the core of this two-consumer economy. (c) Show that the allocation giving x1​ =(4,4) and x2 =(6,6) is in the core. (d) Now replicate this economy once so that there are two consumers of each type, for a total of four consumers in the economy. Show that a double copy of the previous allocation, giving x1q =(4,4) and x2q=(6,6) for replicas q=1,2 is not in the core of the replicated economy.

Answer 1

An Edgeworth box is a graphical representation of the distribution of goods between two consumers. The core of a two-consumer economy consists of allocations that cannot be improved upon by voluntary agreements. The allocation (4,4) and (6,6) is in the core, but in the replicated economy with four consumers, the double copy of the previous allocation is not in the core.

Step-by-step explanation:

(a) Edgeworth Box

An Edgeworth box is a graphical tool used to represent the distribution of goods between two consumers in an economy. In this specific case, since there is one consumer of each type, we have two utility functions and endowments: u1(x1,x2)=x1x2, u2(x1,x2)=x1x2, and ω1=(8,2), ω2=(2,8). To draw the Edgeworth box, we plot the endowment points ω1 and ω2 on the box's axes. The box represents all possible allocations of goods between the two consumers.

(b) Core of the Economy

The core of a two-consumer economy consists of allocations that cannot be improved upon by voluntary agreements. In this case, the precise characterization of the core is not provided. However, the core generally includes allocations where neither consumer can be made better off without making the other consumer worse off.

(c) Allocation (4,4) and (6,6) in the core

To show that the allocation x1=(4,4) and x2=(6,6) is in the core, we need to demonstrate that no voluntary agreement can make either consumer better off without making the other consumer worse off. Since both consumers have the same utility function, any reallocations must result in a decrease in both consumers' utility. Therefore, the allocation (4,4) and (6,6) is in the core.

(d) Replicated Economy and Core

In the replicated economy with four consumers, the allocation (4,4) and (6,6) for both types of consumers is not in the core. This is because voluntary agreements can be made within each type to reallocate goods and improve the utility of one consumer without decreasing the utility of the other consumer. Therefore, the double copy of the previous allocation is not in the core of the replicated economy.

Answer 2

In this two-consumer economy, (a) the Edgeworth box is a 45-degree line connecting the intercepts of the two consumers' endowments, (b) the core consists of allocations within the Edgeworth box, (c) the allocation (4,4) and (6,6) is in the core, and (d) in a replicated economy with four consumers, the double copy of the previous allocation is not in the core.

Step-by-step explanation:

(a) To draw an Edgeworth box for this economy with one consumer of each type, we can use the endowments of each consumer as the intercepts for their respective axes. Consumer 1's endowment is (8,2) and consumer 2's endowment is (2,8). Plotting these intercepts on the axis, we can draw the Edgeworth box. The resulting box will be a 45-degree line connecting the intercepts of both consumers.

(b) The core of a two-consumer economy consists of allocations where no consumer can be made better off without making another consumer worse off. In this case, the set of core allocations will be the entire Edgeworth box.

(c) The allocation x1=(4,4) and x2=(6,6) is in the core because it lies within the Edgeworth box and satisfies the condition mentioned above, where no consumer can be made better off without making another consumer worse off.

(d) In a replicated economy with two consumers of each type, the double copy of the previous allocation x1q=(4,4) and x2q=(6,6) for replicas q=1,2 is not in the core. The core of the replicated economy will be a subset of the Edgeworth box, and this particular allocation does not satisfy the condition of no consumer being able to be made better off without making another consumer worse off.