Consider a society of identical workers that each earns a wage W when they work.
ID: 1113288 • Letter: C
Question
Consider a society of identical workers that each earns a wage W when they work. Each worker faces a probability of sustaining an injury . If they sustain an injury, they have no earnings (W=0). In any case, however, they always have some outside income of 5. Workers have utility of the form: U = 1/2 log(C), where C=consumption=total income in the period (they do no saving). Assume there is no moral hazard.
a) What is the expression for the expected utility of each worker?
Now, suppose that the government introduces a worker’s compensation program. Under this system, individuals pay some fraction of their wage when they are employed (i.e., their wage
is taxed at a certain rate), and get a benefit when they are injured. The system must break even at a point in time; that is, the benefits paid to injured workers must be equal to the taxes
collected from employed workers.
b) What is the optimal workers compensation system? That is, what is the system that, subject to the constraint of breaking even, maximizes worker utility? Present both the tax rate and the benefit level for the system.
c) Are there welfare gains or losses from introducing the Workers’ Compensation system (you don’t actually have to measure the gains/losses – just sign them)? Why?
d) Would your answer to part c) change if utility was of the form: U=(1/2)C?
Now suppose that when workers get injured their spouses go to work. Each worker injured gets an amount kW from their spouse, where k is some constant and k < (1-) .
e) What is the expected utility now if there is no Workers’ Compensation program?
f) Now, reintroduce Workers’ Compensation, which once again must break even. What is the optimal Workers’ Compensation system now (both tax rate and benefit level)? How does this compare to your answer to part (b)? Why?
g) Are there welfare gains or losses now from introducing the system (once again, no precise measurement is necessary)? Intuitively (and not mathematically), are these gains or losses greater than in part c)? Why?
Explanation / Answer
aWe are given,
Wage of workers= W;
Probability of facing injury = ; thus, Probability of no injury = (1-).
Outside income= 5
Utility, U = 1/2 log(C),
Thus, expression for the expected utility of each worker:
E(U) = (1-) log(w+5) + log(5)
Where, (1-) =Probability of no injury,
(w+5) = Total income when no injury
= Probability of injury,
and, 5 is the income in case of injury
Let be the fraction of wage that the individuals pay when they are employed, or the tax rate.
In case of no injury (with Probability 1-) the individuals get a wage W and they pay a part of their wage in the form of tax.
Thus, w(1-)
Similarly, in case of no injury (with probability ) the individuals get no wage but only benefits say x. Thus,
x .
So, for break even the difference in these two cases should be equal to zero. Thus,
w(1-) -x = o
or, x = w(1-)
or, x = [ w(1-)]/
When, there is no injury (P=1-), then income is equal to wage plus 5 outside income. And, wage will be equal to w (1- ) because is the tax rate. So for example if tax rate, = 10% and Wage = $100, then the tax paid = $10.So income = $90. Or, 100(1-.10)=$90
When, there is injury (P= ), then income is equal to the $5 outside income and the benefit received x.
And, benefit received is found to be
x = [ w(1-)]/
Hence, to maximize worker utility,
E(U) = max[(1-)log(w(1- )+5) + log{5+(1- / ) w}]
As per the first order conditions:
If f(x) is a given function, then we need to choose x* such that the derivative of f is equal to zero:
f(x)=0
Thus, [(1-)/w(1- )+5](-w) + [/{5+(1- / ) w}]*(1- / ) w= 0
Or, [(-1)w/w(1- )+5]+ [(1-)w/{5+(1- / ) w}]= 0
Or, (-1)/{5+(1- / ) w}= (-1)/w(1- )+5]
Or, 5+[(1- / ) w]= w(1- )+5
Or, 5+[( w- w / )] = w- w +5
Or, ( w- w / ) = w- w
Or, w- w = w- w
Or, w= w
Or, =
Substituting the value of tax rate , in the benefit level equation of x, we get
x = [ w(1-)]/
Or, x = [ w(1-)]/
x = w(1-)
Thus, the tax rate is equal to and the benefit received is w(1-). The log utility shows welfare that the expected utility is greater after introducing compensation system as compared to before the introduction of benefits. Hence, there will be welfare gains.
Now suppose that when workers get injured their spouses go to work. Each worker injured gets an amount kW from their spouse, where k is some constant and
k < (1)
Thus, the expected utility is:
E(U)= (1-) log(w+5) + log(kw+5)
Where, (1-) =Probability of no injury,
(w+5) = Total income when no injury
= Probability of injury,
and, 5 is the income in case of injury and, kw is the income for injured worker from spouse.