Macroeconomics Problem Set
Labor supply: multi-period problem.
Consider a two-period (t and t + 1) problem facing the representative household:
for χ, η > 0 constant parameters. Household faces the intertemporal (or lifetime) budget constraint
where (1 + r) = R is a constant given gross real rental rate in the economy and wt is the real wage rate. Household chooses consumption, Ct; and hours of work, Lt; to maximize his lifetime utility. His only income is labor income, where he takes wt as given. β is the household's discount factor.
1. Set up the Lagrangian for this problem, use λ to denote the Lagrange multiplier on the budget constraint. Note that there is just one constraint, so we need just one Lagrange multiplier. Derive the optimality conditions characterizing the equilibrium of this model. Interpret these conditions.
2. What is the interpretation of λ in equilibrium?
3. Calculate the elasticity of labor supply with respect to the real wage: d ln Lt/d ln wt.
4. Derive an intertemporal labor supply condition that relates relative labor supply across periods t and t + 1, Lt/Lt+1; to the relative wage, wt/wt+1. This could be done by using a t and t + 1 versions of the first-order condition you derived for employment. Is the relative labor input increasing or decreasing in relative wage? Interpret this condition. Does this effect remind you of the substitution effect of wages we studied in class?
5. Substitute the first-order conditions you derived for consumption and employment as functions of λ into the lifetime budget constraint and derive an expression for λ in terms of wages. Is λ increasing or decreasing in wt? What about in wt+1? Interpret this condition. Does this effect remind you of the income effect of wages we studies in class?
6. Consider a permanent increase in wages by a factor θ > 1 due to the causes external to this model. This means that both wt and wt+1 increase by a factor of θ : wtnew = θwt and wt+1new = θwt+1. What will be the effect of such an increase on the relative labor supply (use the equation you derived in part 4)? How about the level of labor supply (Lt)? What happens to consumption, Ct?
7. Consider a temporary increase in real wage, wt. This means that only wt increases by a factor of θ : wtnew = θwt. Evaluate the effects of this increase on relative labor supply, on the level of labor supply (Lt), on consumption (Ct). Compare your results with those in part 6 of this question.
8. Compare your findings in parts 6 and 7 of this problem to the effects of wages on labor supply we derives in the static model with the same preferences.