UW-Madison ECON 312 - Lecture 2 - Labor Supply

Unformatted text preview:

Lecture 2Labor SupplyNoah WilliamsUniversity of Wisconsin - MadisonEconomics 312Spring 2010Williams Economics 312The Representative HouseholdWe will now begin formal modeling by consideringindividual household behavior.As an abstraction, we will think of one household as astand in for the whole economy.Who is the representative household?Robinson Crusoe in a desert island.Justification: aggregation.Some problems with aggregation.Williams Economics 312What are we going to do?Think about the goods existing in the economy.Think about what does Robinson prefer.Think about his constraints.Think about what will Robinson do given his preferencesand his constraintsWilliams Economics 312Commodity Space2 goods, consumption c and leisure l.Total time of h hours, N =labor ⇒ N = h − l.Each good’s set:1c ∈ R+2l ∈ [0, h]Then (c, l) ∈ R+× [0, h]Williams Economics 312PreferencesPreferences: binary relation  defined over pairs (c, l):(ci, li)  (cj, lj)Basic assumptions on preferences:1Complete: for ∀ (ci, li) , (cj, lj) ∈ R+× [0, h] either(ci, li)  (cj, lj) or (cj, lj)  (ci, li).2Reflexive: for ∀ (ci, li) ∈ R+× [0, h] (ci, li)  (ci, li).3Transitive: for ∀ (ci, li) , (cj, lj) , (ck, lk) ∈ R+× [0, h], if(ci, li)  (cj, lj) and (cj, lj)  (ck, lk) ⇒ (ci, li)  (ck, lk).Violations of transitivity lead to “Dutch books,” andagents would become a “money pump.”Williams Economics 312Indifference CurvesLoci of pairs such that:(ci, li)  (cj, lj) , (cj, lj)  (ci, li) ⇒ (cj, lj) ∼ (ci, li)Additional assumptions on preferences:1Monotonicity: If ci≥ cj, li≥ ljthen (ci, li)  (cj, lj).2Convexity: If (ci, li)  (ck, lk) and (cj, lj)  (ck, lk) then∀λ ∈ [0, 1]λ (ci, li) + (1 − λ) (cj, lj)  (ck, lk)Under these assumptions, the indifference curves are:1Negatively sloped.2Convex.Williams Economics 312Copyright © 2005 Pearson Addison-Wesley. All rights reserved.4-2Figure 4.1 Indifference CurvesWilliams Economics 312Utility FunctionsWorking directly with binary relations difficult.Can we transform them into a function?Definition: a real-valued function u : R2→ R is called autility function representing the binary relation  definedover pairs (c, l) if for ∀ (ci, li) , (cj, lj) ∈ R+× [0, h],(ci, li)  (cj, lj) ⇔ u (ci, li) ≥ u (cj, lj).Theorem: if the binary relation  is complete, reflexive,transitive, strictly monotone and continuous, there exist acontinuous real-valued function u that represents  .Williams Economics 312Utility Functions IIu simply represents the indifference curves. Indifferencecurves are level sets {(c, l) : u(c, l) = ¯u}.u is only defined up to a positive monotone transformation.If f is an increasing function f (u(c , l)) also represents theindifference curves.We’ll always assume u is continuous and differentiable.The properties of preferences imply properties of u:1Monotonicity ⇒ uc≥ 0, ul≥ 0.2Convexity ⇒ ucc≤ 0, ull≤ 0Notation: uc(c, l) =∂u∂c(c, l).Williams Economics 312Utility Function and Indifference Curves012300.51−6−5−4−3−2−1012ConsumptionUtility FunctionLeisureUtility0.5 1 1.5 2 2.5 30.10.20.30.40.50.60.70.80.9ConsumptionLeisureIndifference CurvesWilliams Economics 312Marginal Rate of SubstitutionThe slope of an indifference curve is given by minus theratio of marginal utilities:u(c(l), l) = ¯uuc(c, l)c0(l) + ul(c, l) = 0c0(l) = −ulucMinus this slope is called the marginal rate of substitution.MRS =uluc.Monotonicity ⇒ MRS ≥ 0.Convexity ⇒ MRS decreasing in l.Williams Economics 312Copyright © 2005 Pearson Addison-Wesley. All rights reserved.4-3Figure 4.2 Properties of Indifference CurvesWilliams Economics 312Budget ConstraintLeisure l ⇒labor supply N = h − l.Wage w. Unearned income π.Thenc = Nw + π = (h − l)w + πWilliams Economics 312Copyright © 2005 Pearson Addison-Wesley. All rights reserved.4-4Figure 4.3 Representative Consumer's Budget Constraint (T > π) Williams Economics 312Copyright © 2005 Pearson Addison-Wesley. All rights reserved.4-5Figure 4.4 Representative Consumer's Budget Constraint (T < π)Williams Economics 312Household’s ProblemProblem for Robinson is then:maxc,Nu (c, h − N )s.t. c = Nw + πCan either impose constraint or form Lagrangian. easyhere to impose constraint:maxNu (Nw + π, h − N )First order condition:ucw − ul= 0uluc= wInterpretation: marginal rate of substitution equal torelative price of leisure.Williams Economics 312Copyright © 2005 Pearson Addison-Wesley. All rights reserved.4-6Figure 4.5 Consumer OptimizationWilliams Economics 312Copyright © 2005 Pearson Addison-Wesley. All rights reserved.4-7Figure 4.6 The Representative Consumer Chooses Not to WorkWilliams Economics 312A Parametric Exampleu (c, l) = log c + γ log lMRS =uluc=γ1l1c= γclFOC+Budget constraint:γc∗h − N∗= wc∗= N∗w + πThen:N∗=wh − γπ(1 + γ)wWilliams Economics 312Income and Substitution EffectWe will follow the Hicksian decomposition.Income Effect: changes in w induce changes in total incomeeven if l∗stays constant. Reduces work incentive: use moreincome to “buy” leisure.Pure income effect: Increase in π.Substitution Effect: changes in w make leisure change itsrelative price with total utility constant. Increases workincentive.(Almost) pure substitution effect: One-time change inwage, say in peak sales period.Williams Economics 312Copyright © 2005 Pearson Addison-Wesley. All rights reserved.4-8Figure 4.7 An Increase π – T for the ConsumerWilliams Economics 312Copyright © 2005 Pearson Addison-Wesley. All rights reserved.4-9Figure 4.8 Increase in the Real Wage Rate—Income and Substitution EffectsWilliams Economics 312Income and Substitution Effects in the ExampleN∗=wh − γπ(1 + γ)wIncome effect:∂N ∗∂π= −γ(1 + γ)w< 0Suppose π = 0, thenN∗=h1 + γLabor supply does not respond to the wage at all! Soincome and substitution effects completely offset.With π > 0 income effect only partly offsets substitutioneffect.Williams Economics 312Labor SupplyLabor supply curve N (w) plots response of labor suppliedby households to a change in wage, holding fixed unearnedincome (and preferences).For individual workers, slope of labor supply unclear.Depends on income and substitution effects. For highenough wage, may be backward bending. That isN0(w) > 0 for low w but N0(w) < 0 for w high enoughIn the aggregate, labor supply supply curve embodies bothintensive and extensive margins,


View Full Document

UW-Madison ECON 312 - Lecture 2 - Labor Supply

Download Lecture 2 - Labor Supply
Our administrator received your request to download this document. We will send you the file to your email shortly.
Loading Unlocking...
Login

Join to view Lecture 2 - Labor Supply and access 3M+ class-specific study document.

or
We will never post anything without your permission.
Don't have an account?
Sign Up

Join to view Lecture 2 - Labor Supply 2 2 and access 3M+ class-specific study document.

or

By creating an account you agree to our Privacy Policy and Terms Of Use

Already a member?