If Alice and Bob have reached ideal levels of specialization, that implies they have equal marginal prices.
Alice and Bob having the same prices does not, by itself, imply they are optimally specialized. If you add in an additional assumption of non-increasing marginal returns (e.g. if doubling the amount of land devoted to apples will give you at most twice as many apples), then it implies optimality. Otherwise, Alice and Bob could be in a local maximum that is not a global maximum.
We are assuming that the marginal exchange is the same whether going upwards or downwards. This is a fairly natural assumption for a continuous system where you can make infinitesimal steps in either direction. In a discontinuous system, prices would need to be represented by something more complicated than a real number and basically you’d end up saying that Alice’s and Bob’s spreads of prices need to overlap, rather than that they need to be identical.
Checking my understanding:
If Alice and Bob have reached ideal levels of specialization, that implies they have equal marginal prices.
Alice and Bob having the same prices does not, by itself, imply they are optimally specialized. If you add in an additional assumption of non-increasing marginal returns (e.g. if doubling the amount of land devoted to apples will give you at most twice as many apples), then it implies optimality. Otherwise, Alice and Bob could be in a local maximum that is not a global maximum.
We are assuming that the marginal exchange is the same whether going upwards or downwards. This is a fairly natural assumption for a continuous system where you can make infinitesimal steps in either direction. In a discontinuous system, prices would need to be represented by something more complicated than a real number and basically you’d end up saying that Alice’s and Bob’s spreads of prices need to overlap, rather than that they need to be identical.
All correct?
Correct.