Efficient Consumption In An Exchange Economy

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Efficient Consumption in an Exchange Economy

we can see directly which allocations of goods that are Pareto efficient.

  • Consider, for instance, point a. Can it correspond to an efficient allocation? Compare it to point b. In b, both Robinson and Friday are better off (as both are on a higher indifference curve). Consequently, b is a Pareto improvement as compared to a, and then a cannot be an efficient allocation. Note that all points within the grey area are Pareto improvements as compared to a.
  • Is then point b efficient? Compare b to c. In c, Robinson is better off while Friday is indifferent between b and c. Consequently, c is a Pareto improvement as compared to b, and then b cannot be efficient either.
  • In point c, One of Robinson’s indifference curves just barely touches one of Friday’s indifference curves. That is the criterion for a Pareto efficient allocation of goods. Compared to c, every other allocation makes either Robinson or Friday (or both) worse off. c is consequently a Pareto efficient allocation.

Remember that the slope of an indifference curve is the same thing as the marginal rate of substitution; MRS. The criterion for an efficient allocation of goods can then be written

where the subscripts refer to Robinson and Friday, respectively. In other words, for the allocation to be efficient, both agents are required to have the same marginal valuation of the goods. Point c is, however, not the only Pareto efficient allocation in the diagram. We could repeat the procedure above for every possible indifference curve, and find all points of tangency. If we would do that, and then connect all points to a curve, we would get the so-called contract curve.

If we assume that the initial allocation of coconuts and fish is as in point a and that we have free trade, then we would expect Robinson and Friday to start trading until they end up somewhere on the contract curve. Moreover, since only points in the grey area are Pareto improvements compared to a, we would expect them to end up on the part of the contract curve that lies within that area. Exactly where we they end up is, however, a question about negotiations between Robinson and Friday.

The Two Theorems of Welfare Economics

There are two important theorems regarding efficiency and competitive markets: the two welfare theorems:

  • 1st theorem of welfare economics: If all trade occurs in perfectly competitive markets, the allocation that arises in equilibrium is efficient.
  • 2nd theorem of welfare economics: Each point along the contract curve is a competitive equilibrium for some initial allocation of goods.

The first theorem is a variation of “the invisible hand”. It is enough to have perfect competition to get an efficient allocation. The second theorem states that there is no loss to efficiency from a reallocation. A competitive market will always find an efficient allocation.

Efficient Production

Regarding production, we can perform an analysis that is very similar to the one we did for consumption. Instead of two indifference maps, we put two isoquant maps together. We imagine that Robinson and Friday have two firms, one that produces coconuts and one that produces fish. In the production, they use labor and capital, and their access to these input factors is fixed. They have a certain number of fishing tools, tools to pick coconuts with, and a maximum number of working hours. This allows us to construct an Edgeworth box for production (see Figure 18.3). Let us start by assuming that Robinson and Friday have chosen point a. They consequently invest quite a large number of working hours, but not so much
capital, on fishing and, vice versa, a small number of working hours but quite much capital on picking coconuts. Point a is not efficient. If they instead choose point b, they get, employing the same total number of working hours and the same total amount of capital, more coconuts and just as much fish as they did in a. If they would choose point c instead of a, they get more fish and just as many coconuts as they did before. Consequently, both point b and c constitute efficiency gains compared to point a.

What is “wrong” with point a? The isoquant for fishing (the full line) has a slope that is smaller than the isoquant for coconuts (the broken line). we defined the marginal rate of technical substitution, MRTS, as the slope of an isoquant. Remember what MRTS means: If we use one unit less of labor, how much more capital must we use in order to produce the same quantity
of goods? The fact that the curves have different slopes implies that we can reduce the work in the fishing firm by a small amount and increase the capital in the same firm by a small amount to keep the quantity the same. This will free up labor that we can put into the coconut firm instead while we have to reduce the capital employed in that firm by a small amount. However, since the isoquant for coconuts is steeper than the one for fishing, the change means that they can now increase the production of coconuts. In Figure 18.3, we see that the criterion for having an efficient production is that the isoquant for fishing just barely touches the isoquant for coconuts. In such a point, the two curves have the same slope and the criterion can be expressed as

If we, similarly to before, find all such efficient combinations of work and capital for the two goods and joint them into a curve, we get the production contract curve.

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