III: Labor Supply in Two Periods

Transcription

1 III: Labor Supply in Two Periods Dynamic Macroeconomic Analysis Universidad Autoónoma de Madrid Fall 2012 Dynamic Macroeconomic Analysis (UAM) III: Labor Supply in Two Periods Fall / 37

2 1 Outline 2 The representative agent in two periods Budget constraint Preferencias The Individual Problem 3 The Complete Solution Optimal Consumption Decisions (for given x) Optimal Labor Supply Elasticity of Intertemporal Substititution of Labor Complete solution Consumption levels Optimal labor supply 4 Concluding remarks Hechos Estilizados (EE.UU.) Dynamic Macroeconomic Analysis (UAM) III: Labor Supply in Two Periods Fall / 37

3 Summary In the first theme we analyzed labor-leisure decisions in a static framework Next, we analyzed consumption and savings decisions in a two-period setting with inelastic labor supply The logical next step is to endogenize the labor-leisure decisions in this two-period setting with endogenous saving. For the moment we will take the salaries, denoted by w t, as given. Agents will sacrifice leisure in periods with relatively high wages and they use the credit market to smooth consumption. The endogenous determination of salaries is explained in Theme 4. Dynamic Macroeconomic Analysis (UAM) III: Labor Supply in Two Periods Fall / 37

4 Budget constraints Denoting the salaries in period 1 and 2 by w 1 and w 2, respectively, we can write the one-period budget constraints as: c 1 + b 1 = w 1 l 1, c 2 = w 2 l 2 + (1 + R)b 1. The only difference w.r.t. the previous lecture is that income is now endogenous as y t = w t l t. The corresponding life-time budget constraint is given by: c 1 + c R = w 1l 1 + w 2l R Dynamic Macroeconomic Analysis (UAM) III: Labor Supply in Two Periods Fall / 37

5 Preferences To complete the description we need to specify the agent s preferences over consumption and leisure in the two periods. Like before we assume time-separable preferences. In addition we assume that preferences are additive in consumption and leisure. Formally, U(c 1, l 1, c 2, l 1 ) = u(c 1 ) + v(1 l 1 ) + β[u(c 2 ) + v(1 l 2 )], where c t denotes consumption and 1 l t leisure. Note that future consumption and leisure are discounted at the common rate β. Finally, utility is strictly concave in consumption and leisure. Dynamic Macroeconomic Analysis (UAM) III: Labor Supply in Two Periods Fall / 37

6 The optimization problem The problem of a representative agent is given by max u(c 1 ) + v(1 l 1 ) + β[u(c 2 ) + v(1 l 2 )] c 1,c 2,l 1,l 2 s.t. c R c 2 = w 1 l R w 2l 2 The Lagrangian associated to the above maximization problem is: L = max {u(c 1 ) + v(1 l 1 ) + β[u(c 2 ) + v(1 l 2 )] c 1,c 2,l 1,l 2 where λ is the multiplier. + λ[w 1 l R w 2l 2 c R c 2]}, (1) Dynamic Macroeconomic Analysis (UAM) III: Labor Supply in Two Periods Fall / 37

7 FOCs There are four FOCs: c 1 : u (c 1 ) λ = 0, c 2 : βu (c 2 ) λ R = 0, l 1 : v (1 l 1 ) + λw 1 = 0, l 2 : βv (1 l 2 ) + λ w R = 0. Dynamic Macroeconomic Analysis (UAM) III: Labor Supply in Two Periods Fall / 37

8 Intertemporal consumption decisions Combining the first (λ = u (c 1 )) and second FOC yields the standard Consumption Euler eqn.: u (c 1 ) = u (c 2 ) β(1 + R), }{{}}{{} MC MB Once again, therefore, the agent will choose a constant consumption stream when β(1 + R) = 1. Dynamic Macroeconomic Analysis (UAM) III: Labor Supply in Two Periods Fall / 37

9 Labor supply With the use of the first FOC we can rewrite the FOC associated with l 1 in the following manner: L l 1 = v (1 l 1 ) + λw 1 = 0 v (1 l 1 ) }{{} MC = u (c 1 ) w 1 }{{} MC The above condition equalizes the marginal cost in utility terms from the loss of leisure to the marginal gain from additional consumption. Similarly, since λ = u (c 1 ) and u (c 1 ) = β(1 + R)u (c 2 ): L l 2 = βv (1 l 2 ) + λ w R = 0 (β)v (1 l 2 ) }{{} MC = (β)u (c 2 )w 2 }{{} MB Dynamic Macroeconomic Analysis (UAM) III: Labor Supply in Two Periods Fall / 37

10 Intertemporal Labor Supply To obtain the equilibrium levels of labor supply we solve the FOCs for l 1 and l 2 in terms of λ: L l 1 = v (1 l 1 ) + λw 1 = 0 = λ = v (1 l 1 ) w 1 L l 2 = βv (1 l 2 ) + λ w R = 0 = λ = βv (1 l 2 ) (1 + R) w 2. Dynamic Macroeconomic Analysis (UAM) III: Labor Supply in Two Periods Fall / 37

11 Intertemporal labor supply Equalizing the right-hand sides of both conditions yields This condition can be rewritten as v (1 l 1 ) w 1 = βv (1 l 2 ) (1 + R) w 2.. v (1 l 1 ) v (1 l 2 ) = β(1 + R)w 1 w 2. The above condition implicitly defines the ratio of labor supply (l 1 /l 2 ) as an increasing function of the wage ratio (w 1 /w 2 ). Dynamic Macroeconomic Analysis (UAM) III: Labor Supply in Two Periods Fall / 37

12 Logarithmic utility max {ln c 1 + ln(1 l 1 ) + β (ln c 2 + ln(1 l 2 )) c 1,c 2,l 1,l 2 The associated FOC s are: + λ[wl R wl 2 c R c 2]}. c 1 : c 2 : β 1 c 2 λ R 1 c 1 λ = 0, (2) = 0, (3) l 1 : 1 1 l 1 + λw 1 = 0, (4) l 2 : β 1 1 l 2 + λ w R = 0. (5) Dynamic Macroeconomic Analysis (UAM) III: Labor Supply in Two Periods Fall / 37

13 The Optimal Consumption Decisions (given x) Combining the first two FOCs (corresponding to c 1 and c 2 ) we obtain the Consumption Euler eqn. in levels that we have seen before: c 1 = c 2 β(1 + R), Similarly, like before agents save a fixed proportion 1/(1 + β) of their lifetime resources x = w 1 l 1 + w 2l 2 1+R c 1 = x 1 + β c 2 = β x(1 + R) 1 + β Hence, if we were to take x as given, the consumption decisions would be the same as before. But x is NOT exogenous. Dynamic Macroeconomic Analysis (UAM) III: Labor Supply in Two Periods Fall / 37

14 Optimal labor supply Relative labor supply is again a function of the wage ratio w 1 /w 2, but this time we obtain a condition in levels. In a first step we solve the FOCs for l 1 and l 2 in terms of λ: L = 1 + λw 1 = 0 l 1 1 l 1 1 λ = w 1 (1 l 1 ) L = β 1 + λ w 2 l 2 1 l R = 0 β(1 + R) λ = w 2 (1 l 2 ). Dynamic Macroeconomic Analysis (UAM) III: Labor Supply in Two Periods Fall / 37

15 Optimal labor supply Equalizing the right hand sides of both conditions we obtain the following expression which can be rewritten as: 1 β(1 + R) = w 1 (1 l 1 ) w 2 (1 l 2 ), 1 l 1 1 l 2 = 1 w 2. (1 + R)β w 1 This arbitrage condition is key for the determination of labor supply in both periods. Dynamic Macroeconomic Analysis (UAM) III: Labor Supply in Two Periods Fall / 37

16 Optimal Labor Supply First of all, note that ( ) 1 l1 1 l 2 ( w2 w 1 ) > 0. Hence, an increase in the relative wage w 2 w 1 leads to a rise in 1 l 1 1 l 2 or equivalently a fall in l 1 l2. The agent decides to consume less leisure in the second period to benefit from the rise in the relative wage. Below we will see that the agent will use the credit market to consume part of the rise in w 2 l 2 in the first period. Formally, the real wage w t is a measure of the opportunity cost of leisure in period t. Following an increase in w 2 /w 1 leisure in the second (first) period is therefore relatively more expensive (cheaper). Dynamic Macroeconomic Analysis (UAM) III: Labor Supply in Two Periods Fall / 37

17 Optimal Labor Supply It is also important to note that ( ) 1 l1 1 l 2 (1 + R) < 0. After an increase in the real interest rate the agents demand relatively more leisure in the second period of life. First explanation: the higher interest rate reduces the cost of future consumption and raises the return to saving. This is an incentive to work more and save more in the first period of life. Second explanation: The rise in R is equivalent to a fall in the present w value of future wages, 2 1+R. This reduces the opportunity cost of leisure in the second period. Dynamic Macroeconomic Analysis (UAM) III: Labor Supply in Two Periods Fall / 37

18 Elasticity of Intertemporal Substitution of Labor This elasticity measures the percentage change in the ratio of leisure 1 l 1 1 l 2 per percent change in the wage ratio w 2 w 1. [( 1 l1 1 l 2 )] ( 1 l1 1 l 2 ) [( w2 w 1 )] ( w2 w 1 ) ( w2 [( )] ) = 1 l1 1 l 2 w 1 [( )] ( ), w2 1 l1 w 1 1 l 2 When this elasticity is high, agents are willing to substitute a relatively large amount of current leisure for future leisure in response to a rise in the wage ratio w 2 /w 1. Dynamic Macroeconomic Analysis (UAM) III: Labor Supply in Two Periods Fall / 37

19 Example: log utility Previously, we have seen that 1 l 1 1 l 2 = 1 w 2. (1 + R)β w 1 Hence, and so, [( 1 l1 1 l 2 )] [( w2 w 1 )] [( 1 l1 1 l 2 )] [( w2 w 1 )] = ( w2 w 1 ) ( 1 l1 1 l 2 ) = 1 (1 + R)β, 1 (1 + R)β ( ) w2 w 1 1 w 2 (1+R)β w 1 = 1. In ( other ) words, the elasticity is ( equal to 1 and so a x percent increase in w2 w 1 leads to a x % rise in 1 l1 1 l 2 ). Dynamic Macroeconomic Analysis (UAM) III: Labor Supply in Two Periods Fall / 37

20 Optimal consumption levels Combining the FOCs for l 1 and c 1 we obtain the following equality 1 1 l 1 + λw 1 = 0. 1 c 1 = λ, 1 1 l 1 = 1 c 1 w 1. Hence, c 1 = w 1 w 1 l 1 = w 1 l 1 = w 1 c 1. Dynamic Macroeconomic Analysis (UAM) III: Labor Supply in Two Periods Fall / 37

21 Optimal Consumption Levels Similarly, combining the FOC for l 2 β 1 1 l 2 + λ w R = 0. and the condition that λ = 1 c 1, we obtain: β 1 1 l 2 = 1 c 1 w R. Hence, and so: c 1 β(1 + R) = w 2 w 2 l 2, w 2 l 2 = w 2 c 1 β(1 + R) = w 2l R = w R c 1β. Dynamic Macroeconomic Analysis (UAM) III: Labor Supply in Two Periods Fall / 37

22 Optimal consumption levels Since w 1 l 1 = w 1 c 1 we get w 2 l R = w R c 1β. Using the fact that c 1 = x = w 1 l 1 + w 2l R = w 1 c 1 + w R c 1β. x 1+β, we can therefore write: (1 + β)c 1 = w 1 c 1 + w R c 1β, Dynamic Macroeconomic Analysis (UAM) III: Labor Supply in Two Periods Fall / 37

23 Solutions for c 1 and c 2 The final solutions: c 1 = [ 1 w 1 + w 2 2(1 + β) (1 + R) c2 1 = c 1 β(1 + R) = β(1 + R) 2(1 + β) ]. [ w 1 + w ] 2 (1 + R) Dynamic Macroeconomic Analysis (UAM) III: Labor Supply in Two Periods Fall / 37

24 Solution for l 1 To obtain the value for l1, we substitute the expression for c 1 condition w 1 l 1 = w 1 c 1, into the Hence, w 1 l 1 = w 1 c 1 1 = w 1 2(1 + β) w 1 l 1 = w 1 [1 l 1 = ] 1 2(1 + β) [ w 1 + w ] 2 (1 + R) 1 2(1 + β) 1 + 2β 2(1 + β) 1 1 w 2. 2(1 + β) 1 + R w 1 w 2 (1 + R) Comparative statics: l 1 / w 1 > 0 ; l 1 / w 2 < 0 ; l 1 / R > 0 Dynamic Macroeconomic Analysis (UAM) III: Labor Supply in Two Periods Fall / 37

25 Solution for l 2 Similarly, since wl 2 = w c 1 β(1 + R), w 2 l 2 = w 2 c 1 β(1 + R) [ β(1 + R) = w 2 w 1 + w ] 2 2(1 + β) (1 + R) β(1 + R) 1 β(1 + R) = w 2 [1 ] 2(1 + β) 1 + R 2(1 + β) w 1. So, l 2 = 2 + β β(1 + R) w 1. 2(1 + β) 2(1 + β) w 2 Comparative statics: l 2 / w 1 < 0 ; l 2 / w 2 > 0 ; l 2 / R < 0 Dynamic Macroeconomic Analysis (UAM) III: Labor Supply in Two Periods Fall / 37

26 A special case Suppose for the moment that β = 1 and R = 0. The constant consumption level satisfies: [ c1 1 = w 1 + w ] 2 = 1 2(1 + β) (1 + R) 4 [w 1 + w 2 ] = c2, while the solutions for l 1 and l 2 reduce to: l 1 = 1 + 2β 2(1 + β) 1 w 2 = w 1 l2 = 2 + β 2(1 + β) w 1 = w 2 2(1 + β) β(1 + R) w 1 2(1 + β) w 2 1 w R w 1 Dynamic Macroeconomic Analysis (UAM) III: Labor Supply in Two Periods Fall / 37

27 A special case Given the optimal choices for l t, total labor income satisfies x = w 1 l 1 + w 2 l 2 = w 1 ( w 2 w 1 ) ( 3 + w = w w 2 + w w 1 = 1 2 (w 1 + w 2 ) and the agent consumes half of x in each period. Finally, if w 2 = w 1, then l 1 = l 2 = 1 2. Hence, the agent works half time in both periods and consumes his or her entire labor income, so b 1 = 0. w 1 w 2 ) Dynamic Macroeconomic Analysis (UAM) III: Labor Supply in Two Periods Fall / 37

28 Concluding remarks In this section we have analyzed labor supply and consumption decisions in a two-period setting with exogenous prices. For given labor supply choices, the consumption decisions are the same as before, but labor supply is endogenous and we need to solve the jointly optimal consumption and labor supply decisions The optimizing agents will concentrate their labor supply in periods with a relatively high wage rate and they will use the credit market to smooth consumption over time. The elasticity of intertemporal substitution of labor measures the responsiveness of labor-leisure choices to changes in the relative wages w 2 /w 1. Suppose, realistically, that wages are high in booms and low in recessions. This would lead to pro-cyclical fluctuations in labor supply. Dynamic Macroeconomic Analysis (UAM) III: Labor Supply in Two Periods Fall / 37

29 W eekly hours worked Horas Trabajadas (USA 2000) Age Men Women Dynamic Macroeconomic Analysis (UAM) III: Labor Supply in Two Periods Fall / 37

30 Hourly W age Salario Medio por Hora (USA 2000) Age Men Women Dynamic Macroeconomic Analysis (UAM) III: Labor Supply in Two Periods Fall / 37

31 Hourly W age Horas Trabajadas y Salario Medio por Hora (USA 2000) Male Female Age Hourly wage Weekly hours worked Graphs by Sex Dynamic Macroeconomic Analysis (UAM) III: Labor Supply in Two Periods Fall / 37

32 Hourly W age Horas Trabajadas y Salario Medio por Hora (Mujeres-USA 2000) Never married Ever married Age Hourly wage Weekly hours worked Graphs by evermar Dynamic Macroeconomic Analysis (UAM) III: Labor Supply in Two Periods Fall / 37

33 (mean) hourwage Salario horario por Nivel de Educación (USA 2000) Age Less HS HS Some college College + Dynamic Macroeconomic Analysis (UAM) III: Labor Supply in Two Periods Fall / 37

34 (mean) uhrswork Horas Trabajadas por Nivel de Educación (USA 2000) Age Less HS HS Some college College + Dynamic Macroeconomic Analysis (UAM) III: Labor Supply in Two Periods Fall / 37

35 Mean houly wage Salario y Horas Trabajadas por Nivel de Educación (USA 2000) Less HS HS Some College College Age Hourly Wage Weekly Hours of work Graphs by educ Dynamic Macroeconomic Analysis (UAM) III: Labor Supply in Two Periods Fall / 37

36 Horas Trabajadas y Salario Medio (Histórico Anual) Dynamic Macroeconomic Analysis (UAM) III: Labor Supply in Two Periods Fall / 37

37 Tendencias en Horas Trabajadas Cambio Porcentual Dynamic Macroeconomic Analysis (UAM) III: Labor Supply in Two Periods Fall / 37

38 Participación Femenina y Trabajo Doméstico Dynamic Macroeconomic Analysis (UAM) III: Labor Supply in Two Periods Fall / 37