Question

Suppose individuals face a probability of \(u\) that they will be unemployed next year. If they are unemployed they will receive unemployment benefits of \(b,\) whereas if they are employed they receive \(w(1-t),\) where \(t\) is the tax used to finance unemployment benefits. Unemployment benefits are constrained by the government budget constraint \(u b=t w(1-u)\) a. Suppose the individual's utility function is given by \\[ U=\left(y_{i}\right)^{\delta} / \delta \\] where \(1-\delta\) is the degree of constant relative risk aversion. What would be the utility-maximizing choices for \(b\) and \(t ?\) b. How would the utility-maximizing choices for \(b\) and \(t\) respond to changes in the probability of unemployment, \(u\) ? c. How would \(b\) and \(t\) change in response to changes in the risk aversion parameter \(\delta\) ?

Short answer

Answer: As the probability of unemployment (u) increases, the optimal tax rate (t) decreases, while the optimal unemployment benefits (b) increases. As the risk aversion parameter (δ) increases, the optimal tax rate (t) decreases, and the optimal unemployment benefits (b) also decrease.

Step-by-step solution

Find the expected utility

Given the individual's utility function and the probability of being employed and unemployed, we can find the expected utility (Eu) as follows: \[ Eu = u\frac{(b)^\delta}{\delta} + (1-u)\frac{((1-t)w)^\delta}{\delta} \]

Set up the constraint

The government budget constraint is given as: \[u b = t w (1-u)\]

Substitute the budget constraint into the expected utility equation

Express b in terms of t from the budget constraint: \[ b = \frac{t w (1-u)}{u} \] Now substitute this back into the expected utility equation: \[ Eu = u\frac{(\frac{t w (1-u)}{u})^\delta}{\delta} + (1-u)\frac{((1-t)w)^\delta}{\delta} \]

Differentiate the expected utility

To find the utility-maximizing choices for b and t, we need to differentiate Eu with respect to t and set the resulting expression equal to 0: \[ \frac{\partial Eu}{\partial t} = u\frac{(\frac{w (1-u)}{u})^\delta \cdot w}{\delta} - (1-u)\frac{(w)^\delta}{\delta} = 0 \]

Solve for the optimal tax (t)

From the equation in Step 4, we can express t as: \[ t = \frac{u^{1-\delta}}{1-\delta} \]

Find the optimal unemployment benefits (b)

Substituting the optimal tax value back into the constraint equation, we get the optimal unemployment benefits (b): \[ b = \frac{\frac{u^{1-\delta}}{1-\delta} w (1-u)}{u} \]

Analyze the effect of changes in the probability of unemployment (u)

For part (b), we can see from our solution that as u increases, the optimal tax rate t would decrease, while the optimal unemployment benefits b would increase. This is because when the probability of unemployment is higher, it is preferred to have a larger payout from the unemployment benefits, while simultaneously reducing the tax burden on employed individuals.

Analyze the effect of changes in the risk aversion parameter (δ)

For part (c), as the risk aversion parameter δ increases, the optimal tax rate t would decrease since the denominator (1-δ) increases, while the optimal unemployment benefits b would also decrease. This means that more risk-averse individuals prefer more insurance against unemployment, so they are willing to have a lower tax rate and receive lower unemployment benefits. Conversely, less risk-averse individuals want less insurance and thus require higher tax rates and unemployment benefits.