During a pandemic, each industry has a representative competitive firm whose output sells for a unit price of$1 and has the linear production functionAL(1I) whereLis there total number of employees andIis the share of employees who are sick. Employees can get sick at work or during leisure so thatI=Iw+Iland each employee earns a wagewindependent of their health status. The firm can exert social distancing effortaat costc(a,L) =21a2Land the share of people infected at workisIw=(1a).

(a) The model assumes that employees earn a full wage independent of their health status. In one or two sentences, describe a realistic government regulation that would make this assumption reasonable.

(b) Write the firm's profit in terms ofA,L,Il,,aandw.

(c) Find the firm's optimal choice ofaas a function of parameters.

Now considerLworkers who choose amounts of consumptioncand non-home leisurel. If a worker is sick they receive disutilityso that a representative agent has preferences over consumption, non-home leisure, and the share of people infected and chooses

maxc+l(Iw+Il)s.t.c=wT

c,l where the utility function isu(c, l) =c+l(Iw+Il),wis the wage (paid

by the firm),Tis taxes paid, and the non-workplace infection rate is given by the

functionIl=(l)l, a linear function whose slope increases in the average amount

of non-home leisurel. (d) Find an individual's optimal choice oflas a function of parameters assuming

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they do not account for their effect on infection rates (that is, assumel, andT

are exogenous to the agent's problem).

Hint: Substitute constraintsIl=l(l)andc=wTto get a one-variable maximization problem.

(e) Write out (but do not solve) the problem of a benevolent social planner who choosesc,l,andato maximize the agent's utility subject to the constraint that total consumption cannot exceed the difference between firm profitand healthcare costs(Iw+Il)L.

(f) In the competitive equilibrium all agents make the same choices so thatl=l and competitive leisure and abatement choices are not efficient. Explain the source of the inefficiency and whether each oflandaare too big or too small in the competitive equilibrium.

Hint: You can do this without solving for the optimal outcome. Take a close look at the SPP derived in (e) and compare with the firm's profit and household's utility maximized in parts (b) and (d), what's missing from (b) and (d)?

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