3 Determining the output gap during the pandemic We study the problem of a policy maker who is trying to determine the output gap during the pandemic. Some (but not all) of the questions in this problem may be harder than the rest of the exam. We will assume that actual output is produced using the production function Yt=AtKLt1, where At is unobserved TFP, K is a known level of capital (which we will keep fixed for simplicity), and Lt is the number of workers actually working. On the other hand, potential output is defined as Yt=AtK(Lt)1, where Lt is the number of workers who would be willing to work at the prevailing wage, given the pandemic situation, described next. Workers supply labor but are afraid to work when the pandemic is severe. In particular, their labor supply is determined by the following optimization problem maxLtLtwtpt2aLt2subjecttoLt0andLt1 Here, pt is the current state of the pandemic, which is increasing in its current severity, while a>0 is a parameter. You can think about pt as probability of getting infected when working. The first term in the objective function, Ltwt, represents the wage income, while the second term, pt2aLt2, represents the disutility associated with risks of getting infected while working. The constraint Lt0 states that labor supply cannot be negative, while Lt1 states that total available labor supply (the number of all potentially available workers in the economy) is normalized to one. The solution to problem (3.3) represents the number of workers who are willing to work in the prevailing state of the pandemic, at the prevailing wage wt. This is the quantity of labor Lt used in the definition of potential output in expression (3.2). Question 3.1 First assume that there is no pandemic, pt=0. What is the labor supply Lt, determined as the solution to problem (3.3)? Hint: Notice the presence of the two constraints in (3.3). Question 3.2 Now assume that pt>0. Use the first-order condition in the problem (3.3) to determine the labor supply Lt. For simplicity assume that the parameter a is such that the solution determined by the first-order condition always satisfies the two constraints (so that you can abstract from the constraints when taking the first-order condition). How does the labor supply decision change when the pandemic becomes more severe? Question 3.3 Now consider a profit-maximizing firm that chooses labor to solve maxLtAtKLt1wtLt (the absence of rent costs is inconsequential here). By taking the first-order condition with respect to Lt, derive the labor demand curve of the firm. Denote LtD the labor demand as a function of the wage wt. Question 3.4 In equilibrium, the labor market has to clear. By imposing the appropriate labor market clearing condition, derive the equilibrium wage, denoting it wt. Treat separately the cases when pt=0 and pt>0. Question 3.5 Derive the equilibrium quantity of labor employed at the market-clearing wage, again separately for the cases pt=0 and pt>0. Use the notation Lt for this quantity Question 3.6 We now want to entertain the possibility that wages are rigid. In particular, assume that the prevailing wage wt is higher than the market-clearing wage wt. What impact does this higher wage wt have on the actual quantity of labor supplied relative to the quantity of labor at the market clearing wage wt ? Determine the output gap. Distinguish the cases pt=0 and pt>0. Question 3.7 For the following scenarios, decide whether the information that the policy maker has is sufficient to determine the output gap. If the answer is yes, describe how the output gap can be determined. If the answer is no, state the additional information that would be needed to determine the output gap. In all cases, assume that parameters of the model (all quantities without a time sub- script) are always known. The policy maker knows the production function (3.1) and the expression for the calculation of the potential output (3.2). The policy maker also understands that the quantity of employed labor can vary due to the impact of the pandemic on labor supply, as well as due to the impact of rigid wages when the prevailing wage may be above the market-clearing wage. The scenarios are as follows: 1. The policy maker knows that the economy is not in a pandemic (pt=0), observes the actual quantity of labor employed Lt, and understands the decision problem of the worker (3.3). However, the policy maker does not know how firms determine labor demand. 2. The policy maker understands the economy is in a pandemic (pt>0). The policy maker also understands the decision problem of the worker and of the firm, and observes actual quantity of employed labor Lt and prevailing wage wt. However, the policy maker does not exactly know pt, and does not know whether the prevailing wage wt is above the market clearing wage wt or not. 3 Determining the output gap during the pandemic We study the problem of a policy maker who is trying to determine the output gap during the pandemic. Some (but not all) of the questions in this problem may be harder than the rest of the exam. We will assume that actual output is produced using the production function Yt=AtKLt1, where At is unobserved TFP, K is a known level of capital (which we will keep fixed for simplicity), and Lt is the number of workers actually working. On the other hand, potential output is defined as Yt=AtK(Lt)1, where Lt is the number of workers who would be willing to work at the prevailing wage, given the pandemic situation, described next. Workers supply labor but are afraid to work when the pandemic is severe. In particular, their labor supply is determined by the following optimization problem maxLtLtwtpt2aLt2subjecttoLt0andLt1 Here, pt is the current state of the pandemic, which is increasing in its current severity, while a>0 is a parameter. You can think about pt as probability of getting infected when working. The first term in the objective function, Ltwt, represents the wage income, while the second term, pt2aLt2, represents the disutility associated with risks of getting infected while working. The constraint Lt0 states that labor supply cannot be negative, while Lt1 states that total available labor supply (the number of all potentially available workers in the economy) is normalized to one. The solution to problem (3.3) represents the number of workers who are willing to work in the prevailing state of the pandemic, at the prevailing wage wt. This is the quantity of labor Lt used in the definition of potential output in expression (3.2). Question 3.1 First assume that there is no pandemic, pt=0. What is the labor supply Lt, determined as the solution to problem (3.3)? Hint: Notice the presence of the two constraints in (3.3). Question 3.2 Now assume that pt>0. Use the first-order condition in the problem (3.3) to determine the labor supply Lt. For simplicity assume that the parameter a is such that the solution determined by the first-order condition always satisfies the two constraints (so that you can abstract from the constraints when taking the first-order condition). How does the labor supply decision change when the pandemic becomes more severe? Question 3.3 Now consider a profit-maximizing firm that chooses labor to solve maxLtAtKLt1wtLt (the absence of rent costs is inconsequential here). By taking the first-order condition with respect to Lt, derive the labor demand curve of the firm. Denote LtD the labor demand as a function of the wage wt. Question 3.4 In equilibrium, the labor market has to clear. By imposing the appropriate labor market clearing condition, derive the equilibrium wage, denoting it wt. Treat separately the cases when pt=0 and pt>0. Question 3.5 Derive the equilibrium quantity of labor employed at the market-clearing wage, again separately for the cases pt=0 and pt>0. Use the notation Lt for this quantity Question 3.6 We now want to entertain the possibility that wages are rigid. In particular, assume that the prevailing wage wt is higher than the market-clearing wage wt. What impact does this higher wage wt have on the actual quantity of labor supplied relative to the quantity of labor at the market clearing wage wt ? Determine the output gap. Distinguish the cases pt=0 and pt>0. Question 3.7 For the following scenarios, decide whether the information that the policy maker has is sufficient to determine the output gap. If the answer is yes, describe how the output gap can be determined. If the answer is no, state the additional information that would be needed to determine the output gap. In all cases, assume that parameters of the model (all quantities without a time sub- script) are always known. The policy maker knows the production function (3.1) and the expression for the calculation of the potential output (3.2). The policy maker also understands that the quantity of employed labor can vary due to the impact of the pandemic on labor supply, as well as due to the impact of rigid wages when the prevailing wage may be above the market-clearing wage. The scenarios are as follows: 1. The policy maker knows that the economy is not in a pandemic (pt=0), observes the actual quantity of labor employed Lt, and understands the decision problem of the worker (3.3). However, the policy maker does not know how firms determine labor demand. 2. The policy maker understands the economy is in a pandemic (pt>0). The policy maker also understands the decision problem of the worker and of the firm, and observes actual quantity of employed labor Lt and prevailing wage wt. However, the policy maker does not exactly know pt, and does not know whether the prevailing wage wt is above the market clearing wage wt or not