An Example of Complete Vs. Incomplete Markets
Suppose there is an economy with two agents (Robinson and Jane) with identical log utility functions. There are two equally likely states of nature. If state 1 is realized, Robinson is endowed with 1 unit of wealth and Jane with 0. In state 2, Robinson gets 0 while Jane receives 1 unit of wealth. With Complete Markets there are two state contingent claims:
- pays 1 unit in state 1 and 0 otherwise.
- pays 1 unit in state 2 and 0 in state 1.
Before the realization of the uncertainty, the two agents can trade the state contingent securities. In equilibrium, the two Arrow-Debreu securities have the same price and the allocation is as follows:
- Robinson buys 0.5 of and sells 0.5 of .
- Jane buys 0.5 of and sells 0.5 of .
The main outcome in this economy is that both Robinson and Jane will end up with 0.5 units of wealth independently of the state of nature that is realized.
If the market is incomplete, meaning one or both of the securities are not available for trade, the two agents can't trade to hedge against a bad realization of nature and thus remain exposed to the possibility of the undesirable outcome of having zero wealth. In fact, with certainty, one of the agents will be 'rich' and the other 'poor'.
This example is an extreme case of market incompleteness. In practice, agents do have some type of savings or insurance instrument. The main point here is to illustrate the potential welfare losses that can arise if markets are incomplete.
Read more about this topic: Incomplete Markets
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