MACROECONOMICS OF COVID-19 (more advanced)

The objective of this exercise is to illustrate how one can use the model studied in class to think about current events. More than getting the correct answers, I want you to explain carefully how you set up the problem and how you understand it.

Consider the production model presented in your textbook. The economy is described by an aggregate production function, an exogenous supply of labor, and an exogenous supply of capital. During a pandemic, participation to market activities (work, consumption…) generates infections. Suppose that the number of infections is proportional to the number of employed workers and is equal to T = π L where π is a parameter capturing how contagious the virus is. The social cost of an infection is f . Think of this cost as the pain from being sick, the risk of death, the cost of hospitalization, and so on. Consider a benevolent social planner (government) who chooses the amount of capital (K) and labor (L) used in production (e.g., by imposing lock-downs) in order to maximize aggregate output net of the cost of infections, Y − T f . Aggregate production is given by the Cobb-Douglas production function with capital share equal to 1/3, as in the textbook. Recall that K and L cannot be larger than the amount of capital in the economy (K ¯) and the labor force (L ¯).

What is the optimal use of labor and capital? Solve in closed form the share of the labor force that is deemed essential (optimal) to participate as a function of all exogenous variables and parameters, including π and f.

How does the share of the essential workers vary if the virus is more contagious (π increases) or more deadly (fincreases)?