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ECON3102 Tutorial 06 - Week 7
Question 1
Suppose household preferences are given by:
u(c, l) = c + θlog(l)
where c is consumption, l is leisure and θ is a parameter. Households have a total of one unit of time and can supply labor in a competitive labor market at a wage w.
(a) Find an expression for the fraction of their time that households spend in market work.
(b) If this was the right model and one looked at households in different countries, how would hours of work correlate with wage levels? How does this compare to the empirical evidence?
Question 2
Suppose preferences for consumption and leisure are:
u(c, l) = log(c) + αlog(l)
and households solve:
max u(c, l)
c,l
s.t.
c = w(1 − τ )(1 − l) + T
(a) Find first-order conditions for the consumption-leisure decision.
(b) Use the budget constraint to solve for leisure l. You should get an explicit expression for l as a function of w, τ , T and α.
(c) Suppose T = 0. How does l respond to the tax rate τ? What does this mean?
Now suppose that in both Europe and the US we have:
α = 1.54
w = 1
But in the US we have:
τ = 0.34
T = 0.102
While in Europe we have:
τ = 0.53
T = 0.124
(d) Compute the amount of leisure chosen in the US and Europe. If we interpret 1 as your entire adult lifetime, what fraction of their adult lives do people in Europe and the US work? Comment on the respective role of taxes and transfers in this analysis using your answers to parts (b) and (c).
(e) The values for τ and T above are not arbitrary. If you did the calculations cor-rectly, you should find that both governments have balanced budgets (up to rounding error), i.e. they redistribute all the tax revenue back as transfers. Check that this is the case.
(f) Assuming the production function is Y = L = 1 − l, how much lower is GDP per capita in Europe compared to the US?
(g) Compute the relative welfare of Europe by solving for λ in the following equation:
u(cEurope, lEurope) = u(λcUS, lUS)
Interpret the number λ that you find
(h) How do the answers to questions (f) and (g) compare? Why?
(i) Suppose a European policymaker sees Prescott’s calculation and concludes that Eu-rope could increase its welfare by a factor of 1 λ by reducing its tax rate and level of transfers to US levels. Do you think they are right? Why? Don’t answer this question mechanically: think about what this calculation does and what it leaves out.
In any calculation of this sort, an important parameter is the elasticity of labor sup-ply. One definition of elasticity that is often looked at by labor economists is known as the ”Frisch” elasticity. It is based on the answer to the following question: ”suppose we increased wages but adjusted the household’s income so that consumption remained con-stant: how would labor supply change?” Let’s calculate the Frisch elasticity in Prescott’s model.
(j) Use your answer to part (a) to find an expression for labor supply 1 − l in terms of w(1 − τ ), c and α. Notice that now we are holding consumption constant, so the idea is that we don’t replace c from the budget constraint like we did in part (b).
(k) Use your answer to part (j) to find an expression for ∂w(1−τ)/∂(1−l) 1− l/w(1−τ)
, i.e. the elas-ticity of labor supply with respect to after-tax wages, holding consumption constant.
(l) Plug in the values of α, τ , w, c and l that you found for the US case into the expression for elasticity. What number do you get? Empirical estimates of this elasticity are usually in the range of 0.4 to 1. How does that compare to the elasticity implied by Prescott’s model? Why does that matter for our conclusions about tax policy?