Economics 3330B, Problems for the Final Exam Steve Williamson University of Western Ontario April 2022 Instructions: 

These are six problems that deal with material covered in the third section of the course. Feel free to work in groups, or alone, on solutions to these problems. For your final exam, Sunday April 24, 2-4 pm, TC 304, I will choose two of these questions, which you will have to answer in the exam. Of course, I wont tell you in advance which ones Im choosing. You can ask me questions about the material I have taught, or about solutions to the problem sets you have done, but I will not answer specific questions about how to solve these six problems, in advance of the exam. If you have clarifying questions about any of these problems, Im happy to answer those. 1. There are three periods, T = 0, 1, 2, and a continuum of economic agents with unit mass. In period 0, each agent receives one unit of an investment good as an endowment. Agents do not know their type in period 0, and they learn whether they are impatient or patient in period 1. Type is private information. An agent is impatient with probability t, and patient with probability 1 t. If impatient, the agent gets utility u(c1), where c1 is consumption of the impatient person in period 1, and if patient the agent receives utility u(c21 + c2), where c21 is consumption of the patient agent in period 1, and c2 is consumption of the patient agent in period 2. There is an investment technology that either converts one unit of the period 0 investment good into one unit of the consumption good in period 1, if the investment is interrupted, or converts one unit of the period 0 investment good into R units of the consumption good in period 2, if the investment is not interrurpted. Assume that R > 1, t > 21 , and cu(c) > 1. There are banks that maximize period 2 profits, and offer u (c) deposit contracts. A deposit contract specifies that, in exchange for one unit of the investment good in period 0, the bank will give a depositor the opportunity to either withdraw r1 units of consumption from the bank in period 1, or r2 units of consumption from the bank in period 2. Banks are Nash competitors. In period 1, each depositor chooses whether to visit the bank or not. In period 1, all t impatient depositors will visit the bank. If s patient depositors visit the bank to request withdrawal in period 1, assume that each of the patient depositors gets a random place in line among the first s depositors requesting withdrawal, and the remaining t 1 impatient depositors get a random spot among the remaining t places at the end of the line. Assume that the bank commits to paying withdrawing customers r1, according to the banking contract, in period 1, until it runs out of resources. (a) Show that there is an equilibrium in which s = 0 (there is no run), and determine the consumption of patient and impatient depositors in equilibrium. (b) Can there be an equilibrium with s = 1 t (a run equilibrium)? If so, determine consumption of patient and impatient depositors in equilibrium. If the run equilibrium exists, how does it compare to the equilibrium in part (a)? (c) Determine whether or not a run equilibrium equilibrium exists in which 0 < s < 1 t. If such an equilibrium exists, this would be a mixed strategy equilibrium in which patient depositors are indifferent between requesting withdrawal from the bank in period 1 and waiting until period 2. If the equilibrium exists, determine the consumption of impatient and patient agents in equilibrium. (d) Explain your results in parts (a)-(c). 2. There are three periods, T = 0, 1, 2, and a continuum of economic agents with unit mass. In period 0, each agent receives one unit of an investment good as an endowment. Agents do not know their type in period 0, and they learn whether they are impatient or patient in period 1. Type is private information. An agent is impatient with probability t, and patient with probability 1 t. If impatient, the agent gets utility u(c1), where c1 is consumption of the impatient person in period 1, and if patient the agent receives utility u(c21 + c2), where c21 is consumption of the patient agent in period 1, and c2 is consumption of the patient agent in period 2. There is an investment technology that either converts one unit of the period 0 investment good into one unit of the consumption good in period 1, if the investment is interrupted, or converts one unit of the period 0 investment good into R units of the consumption good in period 2, if the investment is not interrupted. Assume that R > 1, and cu (c) > 1. There u (c) are banks that maximize period 2 profits, and offer deposit contracts. A deposit contract specifies that, in exchange for one unit of the investment good in period 0, the bank will give a depositor the opportunity to either withdraw r1 units of consumption from the bank in period 1, or r2 units of consumption from the bank in period 2. Banks are Nash competitors. In period 1, each depositor chooses whether to visit the bank or not. In period 1, any depositors who want to request withdrawal from the bank receive a place in line at the bank at random, and then each is served in sequence. The bank commits to a withdrawal policy whereby the first t depositors in line in period 1 receive r1, while any depositors with a place in line greater than t receive y units of consumption each, where y 0. 2 (a) Show that there is an equilibrium in which there is no bank run, and determine the consumption of patient and impatient depositors in equilibrium. (b) Under what conditions (if any) are there equilibria in which there is a bank run, that is where more than t depositors request withdrawal from the bank in period 1? Make sure to account for cases where all patient depositors run to the bank, and where only some patient depositors run to the bank. In any bank run equilibria, determine the consumption of the impatient and patient agents. (c) Explain your results in parts (a) and (b). 3. There are three periods, T = 0, 1, 2, and a continuum of economic agents with unit mass. In period 0, each agent receives one unit of an investment good as an endowment. Agents do not know their type in period 0, and they learn whether they are impatient or patient in period 1. Type is private information. An agent is impatient with probability t, and patient with probability 1t. If impatient, the agent gets utility u(c1), where c1 is consumption of the impatient person in period 1, and if patient the agent receives utility u(c21 +c2), where c21 is consumption of the patient agent in period 1, and c2 is consumption of the patient agent in period 2. There is an investment technology that either converts one unit of the period 0 investment good into one unit of the consumption good in period 1, if the investment is interrupted, or converts one unit of the period 0 investment good into R units of the consumption good in period 2, if the investment is not interrupted. Assume that R > 1, and cu(c) > 1. There are banks u (c) that maximize period 2 profits, and offer deposit contracts. A deposit contract specifies that, in exchange for one unit of the investment good in period 0, the bank will give a depositor the opportunity to either withdraw r1 units of consumption from the bank in period 1, or r2 units of consump- tion from the bank in period 2. In this case, though, the bank commits to a withdrawal policy that depends on how many depositors show up to withdraw in period 1. Assume that, when depositors arrive at the bank to request withdrawal, that they all arrive at the bank simultaneously. So, in period 1, the bank will observe s people arriving at the bank requesting withdrawal. Then, the bank determines r1 = f(s), and r2 = g(s). That is, the bank determines r1 and r2 contingent on the number of people s who show up to withdraw in period 1. (a) Given s, determine how the bank chooses r1 and r2 in period 1. (b) Determine s, r1, and r2 in equilibrium, and determine the equilibrium consumption of the impatient and the patient. Do there exist any bank run equilibria where s > t? (c) Explain your results in parts (a) and (b). 4. Suppose a model of money and secured credit. There is a representative 3 household, which maximizes t u ( c at ) + u ( c bt ) n t , t=0 where cat denotes consumption of the cash good, cbt is consumption of the credit good, and nt is labor supply. One unit of labor produces one unit of either good. A household cannot consume its own output, but sells this output to other households in exchange for money and credit. The households asset market constraint is bt +cat = (1+Rt1)bt1 +mt1 +t, 1+t where bt denotes the quantity of nominal government bonds acquired in period t, and mt is the quantity of money acquired in period t, both in units of consumption goods. As well, t denotes the inflation rate, Rt is the nominal interest rate on one-period nominal government bonds, and t is the transfer made in period t to each household by the government. The households collateral constraint is cbt (1 + Rt)bt + tnt, where 0 < t < 1. That is, credit goods can be purchased with credit secured by government debt, and a fraction of current income, t, is avail- able to purchase credit goods. The governments budget constraint is mt mt1 +bt = (1+Rt1)bt1 +t, 1+t 1+t where m1 = b1 = 0. The government follows the policy rule vt = mt + bt, where vt is exogenous. So, the central bank chooses Rt exogenously, and the fiscal authority chooses vt exogenously. (a) Derive equations that solve for consumption of cash goods, consump- tion of credit goods, labor supply, aggregate output, and inflation, if the collateral constraint binds. (b) Derive equations that solve for consumption of cash goods, consump- tion of credit goods, labor supply, aggregate output, and inflation, if the collateral constraint does not bind. (c) Determine the effects of an increase in t on consumption of cash goods, consumption of credit goods, labor supply, aggregate output, and inflation, and explain your results. 4 5. Suppose a model of money and secured credit. There is a representative household, which maximizes t u ( c at ) + u ( c bt ) v ( n t ) , t=0 where cat denotes consumption of the cash good, cbt is consumption of the credit good, and nt is labor supply. Assume that v(nt) is a strictly convex function with v(0) = 0 and v(h) = for some h > 0. One unit of labor produces one unit of either good. A household cannot consume its own output, but sells this output to other households in exchange for money and credit. The households asset market constraint is bt +cat = (1+Rt1)bt1 +mt1 +t, 1+t where bt denotes the quantity of nominal government bonds acquired in period t, and mt is the quantity of money acquired in period t, both in units of consumption goods. As well, t denotes the inflation rate, Rt is the nominal interest rate on one-period nominal government bonds, and t is the transfer made in period t to each household by the government. The households collateral constraint is cbt (1+Rt)bt, That is, credit goods can be purchased with credit secured by government debt. The governments budget constraint is mt mt1 +bt = (1+Rt1)bt1 +t, 1+t 1+t where m1 = b1 = 0. The government follows the policy rule vt = mt + bt, where vt is exogenous. So, the central bank chooses Rt exogenously, and the fiscal authority chooses vt exogenously. (a) Derive equations that solve for consumption of cash goods, consump- tion of credit goods, labor supply, aggregate output, and inflation, if the collateral constraint binds. (b) If the collateral constraint binds, determine the effects of changes in Rt and vt on cat , cbt , nt, and t, and explain your results. (c) Derive equations that solve for consumption of cash goods, consump- tion of credit goods, labor supply, aggregate output, and inflation, if the collateral constraint does not bind. (d) If the collateral constraint does not bind, determine the effects of changes in Rt and vt on cat , cbt, nt, and t, and explain your results. 5 6. Suppose a model of money and secured credit. There is a representative household, which maximizes t u ( c at ) + u ( c bt ) n t , t=0 where cat denotes consumption of the cash good, cbt is consumption of the credit good, and nt is labor supply. One unit of labor produces one unit of either good. A household cannot consume its own output, but sells this output to other households in exchange for money and credit. The households asset market constraint is bt +cat = (1+Rt1)bt1 +mt1 +t, 1+t where bt denotes the quantity of nominal government bonds acquired in period t, and mt is the quantity of money acquired in period t, both in units of consumption goods. As well, t denotes the inflation rate, Rt is the nominal interest rate on one-period nominal government bonds, and t is the transfer made in period t to each household by the government. The households collateral constraint is cbt (1+Rt)(1t)bt, That is, credit goods can be purchased with credit secured by government debt. But, in this case, it is possible for the household to abscond (run away with) fraction t of the collateral it has posted, in the event that it defaults on its debt. Here, 0 < t < 1. The governments budget constraint is mt mt1 +bt = (1+Rt1)bt1 +t, 1+t 1+t where m1 = b1 = 0. The government follows the policy rule vt = mt + bt, where vt is exogenous. So, the central bank chooses Rt exogenously, and the fiscal authority chooses vt exogenously. (a) Derive equations that solve for consumption of cash goods, consump- tion of credit goods, labor supply, aggregate output, and inflation, if the collateral constraint binds. (b) Derive equations that solve for consumption of cash goods, consump- tion of credit goods, labor supply, aggregate output, and inflation, if the collateral constraint does not bind. (c) Determine the effects of an increase in t on consumption of cash goods, consumption of credit goods, labor supply, aggregate output, and inflation, and explain your results. 6