1.Use the same model as in my notes on the Diamond Dybvig model, except for the following changes. Assume that, instead of the standard investment technology that allows for interruption in period 1, there are two investment technologies. The first technology – the short-term technology – converts one unit of investment in period 0 into one unit of consumption goods in period 1, and the other – the long-term technology – converts one unit of investment in period 0 into R units of consumption goods in period 2, where R > 1. For example, if an individual invests x units of their endowment in period 0 in the short-term technology in period 0 and 1 − x units in the long-term technology, then the payoffs are x units of consumption in period 1 and (1 − x)R units of consumption in period 2.
(a) Determine what happens in autarky, if individuals can only invest on their own. Make sure to allow for the fact that individuals may choose not to invest in the long-term technology.
(b) Next, assume that there is a securities market which opens in period 1, where the price of a claim to one unit of investment in the longterm technology is p units of consumption goods. In period 1, each individual will choose x, the fraction of their endowment to invest in the short-term technology, with 1 − x the fraction invested in the long-term technology. Then, in period 1, an early consumer has no decision to make, as they will consume the proceeds from short-term investment, x, and will sell their long-term investment for p(1−x), no matter what p is. The late consumer has a nontrivial choice to make,though. For the late consumer in period 1, let y1 denote the quantity of consumption goods exchanged for long-term investment, and let y2 denote the quantity of long-term investment traded for consumption goods. Solve the problem by first determining what the late consumer does in period 1, given their choice x from period 0, and the price p.
Then, determine the optimal choice x for any individual in period 0,given their optimal behavior in period 1. Impose market-clearing in period 1, and determine an equilibrium. How does this equilibrium compare to autarky, in part (a)? Explain your results.
(c) Finally, assume that there are banks, as in my notes, that offer deposit contracts, under which a depositor deposits their endowment with the bank in period 0, with the promise that the depositor can withdraw r1 in period 1 and r2 in period 2. Determine the equilibrium banking contract, compare this to what you obtained in parts
(a) and (b), and explain your results. Is a bank run possible, given the equilibrium banking contract? Explain why or why not.