Reference no: EM133077787

During a pandemic, each industry has a representative competitive firm whose output sells for a unit price of $1 and has the linear production function AL(1 - I) where L is there total number of employees and I is the share of employees who are sick. Employees can get sick at work or during leisure so that I = Iw + Il and each employee earns a wage w independent of their health status. The firm can exert social distancing effort a at cost c(a,L) = 1/2 a2L and the share of people infected at work is Iw =γ(1-a).

(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 of A, L, Il ,γ, a and w.

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

Now consider L workers who choose amounts of consumption c and non-home leisure l. If a worker is sick they receive disutility β so that a representative agent has preferences over consumption, non-home leisure, and the share of people infected and chooses

max √c+ √l-β(Iw+Il)s.t.c=w-T

where the utility function is u(c, l) = √c + √l - β(Iw + Il), w is the wage (paid by the firm), T is taxes paid, and the non-workplace infection rate is given by the function Il = δ(l ¯) × l, a linear function whose slope increases in the average amount of non-home leisure l ¯.

(d) Find an individual's optimal choice of l as a function of parameters assuming they do not account for their effect on infection rates (that is, assume l ¯ and T are exogenous to the agent's problem).

Hint: Substitute constraints Il = lδ(l ¯) and c = w - T to get a one-variable maximization problem.

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

(f) In the competitive equilibrium all agents make the same choices so that l = l ¯ and competitive leisure and abatement choices are not efficient. Explain the source of the inefficiency and whether each of l and a are 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)?