LaborSupply2011A

LaborSupply2011A - Econ145.LaborSupply2011A John Pencavel...

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Econ145.LaborSupply2011A John Pencavel LABOR SUPPLY OF ONE INDIVIDUAL Most people work for pay as employees and most sell their time. Consider the prototype single individual who allocates her fixed endowment of time, T , between time at work for pay, h , and time not at work, l , “leisure”: T = h + l . Her budget constraint is p.x = w.h + y . Labor income is w.h and non-labor income is y . Or the budget constraint may be written: p.x = w.(T-l) + y or p.x + w.l = w.T + y . Suppose p and y are given to the individual and also suppose she may choose any hours to work at a fixed wage. When will this happen? Graph the budget constraint showing the trade-off between consumption, x , and work hours, h : x = (w / p ).h + y / p . Why is the slope of the budget constraint a straight line? What does this graphed budget constraint look like if w , p , and y all increase by the same proportion?
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2 The individual’s preferences toward consumption and work time are contained in her utility function U = f ( x , h ) where M U / x > 0 and, at least in the neighborhood of actual work hours, U / h < 0 . Indifference curves between x and h are convex to the origin. If they were concave, small changes in wages would involve people moving from one extreme to the other, i.e., from h = 0 to h = T . An indifference curve slopes negatively because, if x rises, h must increase ( l decrease) to keep utility unchanged. Convexity implies that, along an indifference curve, it becomes increasingly difficult to substitute time for consumption. U 3 > U 2 > U 1 The slope of an indifference curve is -[( U/ h)/( U / x)] . The individual selects x and h given the budget constraint p.x = w.h + y and given the time constraint : T = h + l . In this problem, what are the endogenous variables and what are the predetermined variables? Optimizing behavior yields a commodity demand function x = x( p , w , y) and an hours of work (or labor supply) function h = h (p, w , y ) .
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3 dU dh fwp f dh f w p f w p f     1 1 2 2 2 11 2 12 22 0 20 .. . Because x = w.p -1 .h + p -1 .y , the problem can be simplified to one of choosing h to maximize U = f ( w.p -1 .h + p -1 .y , h ) . The foc requires d U / d h = 0 and the soc requires d 2 U / d h 2 < 0 . where f 1 = M U / x , f 2 = U / h , f 11 = 2 U / x 2 , f 12 = 2 U / x. h , and f 22 = 2 U / h 2 . The soc will be satisfied if indifference curves are convex to the origin as we have assumed. The foc may be written w / p = -f 2 / f 1 > 0 and this tangency condition may be graphed as follows: The slope of the budget constraint - w/p equals the slope of the highest attainable indifference curve f 2 / f 1 at the hours-commodities combination h * -x * .
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4 Comparative Statics Let y increase with w and p constant. This is an “income effect” : an increase in opportunities with prices and wages constant. Typically, work hours fall. Let w increase with p and y constant.
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LaborSupply2011A - Econ145.LaborSupply2011A John Pencavel...

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