# Backward Bending Slutsky Equation

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• Published : September 29, 2012

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Backward Bending Slutsky Equation

In the Labor Supply Model, consumer has a choice between consumption and leisure. If they were to reduce their leisure and allocate more time working, they will be able to consume more. The amount of labor and consumption are determined by the interaction of consumer’s preferences and budget constraint. In this model, the utility function to be maximized is U(C,L), where individual cares about consumption (C) and leisure (L). The utility function is subjected to the budget constraint of

“Pc” denoted the price of consumption goods and “C” indicates the consumption. “W” is the wage rate or the opportunity cost of forgoing leisure and the portion of (-R) is the number of hourly works, or labor hours (L). “M” stands for the non-labor income or unearned income. The equation can be rearranged into: P*C + W*R = W* +m

“M”, the non-labor income, divided by price of consumption (Pc), will provide us with the amount of endowment consumption, “”. Therefore, = M/pc. The equation will be P*C + W*R = W* + P* This equation stated that the value of a consumer’s consumption plus his or her leisure has to equal to the value of endowment of consumption and the endowment of time. The right hand-side of the equation (W* +P* ) represents a person’s full income (S). In other word, it’s an amount that a person could earn if he devoted all his time to work. Through this equation of budget constraint, it’s clear that the utility maximization problem is just a standard consumer choice problem with “C” and “L” as the two commodities that can be bought in the market. The budget constraint when rewrite the equation we get: C = + (W*)/P – (W/P)*R

The slope of budget constraint is (-W/P). The endowment is the point where they spend all hours on relaxation and do not work at all; their endowment consumption is .

P*C + W*R = W* +m
C

Endowment
Leisure

R

labor

leisure
Optimal choice
consumption
M/P=
C

Endowment
Leisure...