Consider an economy in which every person’s utility function takes the form Here, c is consumption and h is hours of work. The parameter θ varies across individuals. Within the economy, θ is uniformly distributed on the unit interval. Each unit of consumption goods costs $1, and the wage rate is $2. a) Assume that there are no taxes or subsidies within the economy. How much work does a person of any type θ do? What is his consumption and what is his utility? If a lump-sum tax were imposed upon him, or a lump-sum subsidy given to him, how would that tax or subsidy affect his hours of work, consumption and utility? b) People with low va
Consider an economy in which every person’s utility function takes the form
Here, c is consumption and h is hours of work. The parameter θ varies across individuals. Within the economy, θ is uniformly distributed on the unit interval. Each unit of consumption goods costs $1, and the wage rate is $2.
a) Assume that there are no taxes or subsidies within the economy. How much work does a person of any type θ do? What is his consumption and what is his utility? If a lump-sum tax were imposed upon him, or a lump-sum subsidy given to him, how would that tax or subsidy affect his hours of work, consumption and utility?
b) People with low values of θ find work difficult. (A low θ might be interpreted as a physical disability which hampers a person’s ability to earn an income.) Suppose that the government decides to assist the people for whom θ is less than 1/4. Unfortunately, it cannot directly observe each person’s θ, so it instead adopts a policy of giving a subsidy of σ dollars to each person whose earned income is less than 1. This subsidy is paid for by imposing a tax of τ on each person whose earned income is greater than 1. This policy will induce some people who originally had earned income greater than 1 to reduce their hours of work and their earned incomes in order to qualify for the subsidy. If a person of type θ is indifferent between paying the tax and receiving the subsidy, every person whose θ is smaller than 
will elect to receive the subsidy and every person whose θ is above will elect to pay the tax
i) Find the equation that describes the value of for which the cost of the subsidy is exactly equal to the revenue from the tax.
ii) Consider the behaviour of a person whose θ is greater than 1/4. What is his utility if he pays a tax τ ? What is his utility if he reduces his hours of work so that he will receive a subsidy σ? Under what circumstances is he indifferent between receiving the subsidy and paying the tax? What equation describes for arbitrarily chosen values of τ and σ?
iii) Assume that the government chooses σ and τ so that is 3/8, and so that the cost of the subsidy is exactly covered by the revenue from the tax. Find σ and τ .
c) Now consider an alternative policy, which is also intended to assist the people for whom θ is less than 1/4. Under this policy, the people with earned incomes smaller than 3/4 receive the subsidy and the people with earned incomes larger than 3/4 pay the tax. Find the values of σ and τ such that is 3/8 and the cost of the subsidy is exactly equal to the tax revenue. Compare the alternative policy with the original policy. In particular, show that every person whose θ is between 0 and 3/16 is better off, as is every person whose θ is between 3/16 and 1/4.
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