Endogenous Income The consump0on-­‐leisure model Modifying consumer’s problem
Assume that the consumer is endowed with an ini0al bundle of goods, ( x, y )
and she has no other income sources. If the consumer can trade at the market prices px and py, then her budget constraint is xp x + yp y ≤ x p x + yp y
Budget constraint
y xpx
y+
py
y
x
x+
yp y
px
x Changes in prices
An increase in px y y
x
x Changes in prices
An increase in py y y
x
x Changes in prices
•  The impact of price changes in the consumer’s budget set is now a bit more subtle: an increase of the price of a good may make the consumer “richer” if she has a large endowment of the good. •  Since the ini0al endowment is always a feasible bundle (consumer can always consume her own endowment), changes in prices result in a rota0on of the budget line around her ini0al endowment. Consumer Demand
Assume the u ( x , y ) is differen0able, and that the system xp x + yp y = x px + yp y
px
py
yields an interior solu0on to the consumer’s problem ~x ( p x , p y ) , ~y ( p x , p y ) . Then MRS ( x, y ) =
~
x ( p x , p y ) = x * ( p x , p y , x p x + yp y )
~
y ( p x , p y ) = y * ( p x , p y , x p x + yp y )
where the func0ons on the RHS are the ordinary demands. Consumer Demand: example
u ( x , y ) = x y ; ( x , y ) = (2,1)
Calculate ordinary demands: Hence, x * ( px , p y , I ) =
2I
3 px
y * ( px , p y , I ) =
I
3 py
2(2 p x + p y ) 4 2 p y
~
x ( px , p y ) =
= +
3 px
3 3 px
~y ( p , p ) = 2 p x + p y = 1 + 2 p x
x
y
3 py
3 3 py
Income and substitution effects
•  Study the effect of an increase in px for u0lity func0on u ( x , y ) and endowment ( x , y )
y C B A u2 u1 y
x
x Income and substitution effects
•  Subs0tu0on effect is nega0ve SE = xB − xA < 0
•  Income effect is positve IE = xC − xB > 0
•  In this case, income effect is larger than subs0tu0on effect, so the total effect is posi0ve: an increase in p x leads the consumer to increase its consump0on of both goods TE = SE + IE = xC − x A > 0
Income and substitution effects
The sign of the income effect of a price increase (that is, whether a price increase makes the consumer poorer or richer) depends on whether the consumer is a net seller or a net buyer of the good whose price has increased: -­‐If the consumer is a net seller of the good, then the IE is nega0ve. -­‐If the consume is a net buyer of the good, then the IE is posi0ve. Formally: ∂x ∂x *
∂x *
=
|u=cte −(x − x)
∂px ∂px
∂I
Income and substitution effects Exogenous income Endogenous income y y B B C C A A u1 u1 u0 x u2 x The consumption-leisure model
Two goods: leisure (x-­‐axis) and consump0on (y-­‐axis) -­‐ Leisure, denoted by h and measured in hours. The wage per hour (or price of leisure) is denoted by w. -­‐ Consump0on, denoted by c and measured in euros. The price of c is therefore pc = 1). Ini0al endowment is (M,H), where: -­‐ M : ini0al exogenous wealth (or non-­‐labor income). -­‐ H : number of hours available for leisure and work. The consumption-leisure model
Budget set (recall p c = 1 ) c + hw ≤ wH + M
c + hw : market value on the consump0on-­‐leisure bundle. wH + M : market value of ini0al endowment. c
wH + M
slope = − w
M
H
h
The consumption-leisure model.
Labor supply The consumer problem is: Maxc , h
st
u ( c, h )
c + hw ≤ wH + M
0≤h≤ H
c≥0
The consumption-leisure model.
Labor supply. Example Maxc , h c + 2 ln h
st
c + wh = 16w + 4
0 ≤ h ≤ 16
c≥0
Interior solution requires
MRS (h, c) = w ⇔
And
2
2
= w ⇒ h( w) =
h
w
2
≥ 0 ⇔ ∀w > 0
w
2
1
h( w) =
≤ 16 ⇔ w ≥
w
8
h( w) =
The consumption-leisure model:
An example
Therefore, the consumer’s demands of leisure and consumptions are:
16
if w < 1 / 8
h(w) =
2
if w ≥ 1 / 8
w
4
if w < 1 / 8
c (w) =
€
2 +16w
if w ≥1/8
The consumption-leisure model:
An example
And her labor supply is
l ( w) = H − h( w) =
1/8
0
if w < 1 / 8
2
16 −
w
if w ≥ 1 / 8
The consumption-leisure model:
An example
•  Assume w' < w
•  For interior solu0ons, the consumer is a net supplier of leisure •  Total income effect: if leisure is a normal good, ∂ h / ∂ I > 0 , TIE is posi0ve, leading the consumer to demand less leisure (or supply more labor) −
∂h
(l − H ) > 0
∂I
>0 <0 •  Subs0tu0on effect : is always non-­‐posi0ve. Since leisure is cheaper, this effect leads the consumer to demand more leisure (or supply less labor) •  Total effect is ambiguous (it depends on the shape of the u0lity func0on) The consumption-leisure model:
Changes in wages
c
wH + M
w' H + M
A
B
C M
H
h
The consumption-leisure model.
Labor supply. Effect of changes in wages c
wH + M
w' H + M
A
B
C M
H
h
The consumption-leisure model.
Labor supply. Effect of changes in wages For w ∈ ( 0 , 10
) , SE dominates (leisure is more expensive and consumer offers more labor) For w ∈
( 10
, 20
) , TIE dominates (consumer is richer and does not need to work as much as before) Application: a tax on labor income
•  Impose t ∈ [0,1]
•  The new budget constraint is c + (1 − t ) wh ≤ (1 − t ) wH + M
•  The tax is equivalent to a reduc0on of wage: its impact on leisure consump0on (or labor supply) is ambiguous •  Its impact on welfare is unambiguous. Of course, tax policies have other objec0ves we are not considering here. Application: a tax on labor income c
wH + M
(1 − t ) wH + M
c*
M
h*
H
h
Application: a tax on labor income •  Alterna0ve: a non-­‐labor income tax T
•  The new budget constraint is c + wh ≤ wH + ( M − T )
•  If both goods are normal, then the introduc0on of T reduces their demands (increases labor supply, in par0cular) Application: a tax on labor income c
wH + M
wH + ( M − T )
M
M −T
h
H
Application: a tax on labor income Exercise: what if T=twh*? (Hint: is (c*,h*) op0mal for T?) c
wH + M
wH + ( M − T )
(1 − t ) wH + M
A c*
M
B M −T
h*
h