When you are asked to provide a numerical value of a variable, you do not have to give a round-up decimal number.

When you are asked to derive a result, please go step by step. Each step counts. Stating the result with no reasoning gets no credits.

When you are asked to answer a question and explain the answer, answering with no explanation gets no credits.

relevant textbook: The economics of Money, Banking, and financial markets, Frederic S. Mishkin, Columbia University

II. Liquidity-provision banking theory (30%) There are three dates, 0, 1, and 2. There is a [0, 1] continuum of ex-ante identical agents. Each agent is endowed with one unit of goods at the start of date 0. There are two technologies, referred to as storage and investment. Storage turns a units of date- good to r units of date-+1 good, f = 0, 1. Investment turns a units date-0 good into Rr units of date-2 good, R > 1; but if an agent liquidates r units of investment at date 1, he only gets /x units date-1 good, I 1. 1. To derive the the optimal coalition allocation, i.e., the first best al- location for the whole economy, we use the constraints mig = 1 - / and many = RI. For these constraints to make sense, we need (a) it is not optimal to liquidate investment at date 1 as a source for c, and (b) it is not optimal to carry storage to date 2 as a source for ca. Verify (a) and (b). 2. Let * = #2 = 0.5, R =3, a = 2, and / =0.3. What is the per-capita (date-0) investment in the first best allocation ? 3. Let * = my = 0.5, R = 3, a = 2, and / = 0.3. What is the optimal (date-0) investment if a single agent does not interact with any other agents? 4. Let #1 = #2 = 0.5, R = 3, a = 2, and / = 0.3. Denote by /* the per capita investment, by 1 - /* per capita storage 1 - /', by c; type-l consumption, and by of type-2 consumption in the first best allocation. (4.1) Describe a demand-deposit contract by which a bank may implement the first best consumption (of, of) without knowing each agent's type; by implementation, we mean that each type-i agent gets of at date i). (4.2) Verify that the bank can indeed implement the first best consump- tion by the contract in part 4.1; that is, verify that the following is an equilib rium: after all agents deposit their goods at date 0, type 1 agents withdraw at date 1 and type 2 withdraw at date 2. (4.3) Show that the bank is illiquid in that the value of its date-1 liability is greater than the value of its asset that is liquid at date 1. (4.4) Show that the bank is subject to panic runs. 2