A_StaticLS - The Basic Static Labor Supply Model Consider a...

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The Basic Static Labor Supply Model Consider a single individual with a utility function U ( y , ℓ) where y is income and ℓ is leisure. Both y and ℓ are “goods”, i.e. the consumer prefers more of each: U 1 > 0; U 2 > 0. Suppose this person has non-labor income of G , and can work as many hours, h , as she wishes at a wage of w per hour. Total time available for the only two possible activities, work ( h ) and leisure (ℓ) is T. If she allocates her time between work and leisure to maximize her utility, what can we say about her decisions, and about how these decisions will respond to changes in the exogenous parameters, w and G ? Using the constraints h + ℓ = T and y = wh + G , this problem can be written as a single-variable maximization problem without constraints: Choose h to maximize: U ( wh + G , T h ). (1) First-order conditions for a maximum are: ( 29 0 ) , ( , 2 1 = - + - - + h T G wh U h T G wh wU (2) or simply: ) ( ) ( 1 2 income MU leisure MU U U w = = (3) Diagrammatically, (3) represents the familiar tangency condition between the slope of the budget constraint ( w ) and the slope of an indifference curve (U 2 /U 1 ), shown for individual #1 below:
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y G G “Tangency” Solution “Corner” Solution y l l # hours worked # hours at home chooses not to work at all Individual #1 Individual #2 2
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Of course, it is also possible (e.g. when G is high and w low) for the solution to be at a “corner” where it is optimal not to work at all, as shown for individual #2. We could show this formally using Kuhn-Tucker conditions in the maximization but the intuition is perfectly clear from the diagram. To understand the comparative-static predictions of the model, assume an interior solution (as for individual #1), and totally differentiate equation (2), yielding: ( 29 ( 29 0 ) ( ) ( ) ( 21 11 22 21 12 11 21 11 1 = - + - - - + - + dG U wU dh U w U U w U w dw h U h wU U , or simply: 0 = + + CdG Bdh Adw (4) To help interpret this, note that, by the second-order condition for a maximum ( U as defined in equation 1 must be concave in h ), B < 0. (btw the SOC also require U 11 < 0 and U 22 <0). Note that w and G are exogenous and h endogenous, so we can define ceteris-paribus thought experiments where we introduce small changes in w and G one at a time, and consider their consequences for the utility-maximizing level of h : Effects of nonlabor income on labor supply: pos U wU B C dw holding dG dh 21 11 ) 0 ( - = - = = , (5) which can be greater, less than, or equal to zero. It will be negative for sure if U 21 >0, i.e. if the utility function is such that additional income always raises the marginal utility of leisure. It will also be negative as long as U 21 is not “too” negative, i.e. as long as additional income does not reduce the marginal utility of leisure “too much”. From here on we will define leisure as a normal good if the partial derivative defined by (5) is negative. (This is the standard definition: a good is normal if its optimal consumption increases when the budget constraint shifts outward without changing slope). Of course, if (5) is positive, we will say that leisure is inferior .
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