Endogenous Income

Endogenous Income

The consumption-leisure model

Modifying consumer’s problem

• For the moment, assume there is no additional exogenous income

• Consumer’s income is the market value of her initial endowment,

• Given market prices p

and p

, the consumer’s budget constraint is

) y , x (

y x

y

x

yp x p y p

xp + £ +

Budget constraint

x y

x

y

y x

p p y + x

x y

p p x + y

Changes in prices

An increase in p

y

x ^x

y

Changes in prices

An increase in p

y

x

y

x

Changes in prices

• The initial endowment vector is on the consumer’s budget set whatever the prices – it is always feasible to not trade.

• The consumer’s budget set does not shrink when the price of a good increases: the rotation of the budget line around the initial endowment vector implies that the consumer can afford new bundles with more of the now relative cheaper good.

• If the consumer’s initial bundle contains a large

amount of the good now relatively more expensive,

then the consumer becomes richer.

Consumer Demand

• Assume derivable and the system

yields an interior solution to the consumer’s problem ,

• That is,

where the functions on the RHS are the ordinary demands.

) ,

~(

y x p p

x ^~y(p_x^{, p}_y⁾

y x

y x

y x

p y p

x MRS

p y p

x yp

xp

=

+

= +

) , (

) , ( yx u

) ,

, (

* )

,

~(

) ,

, (

* )

,

~(

y x

y x

y x

y x

y x

y x

p y p

x p

p y

p p

y

p y p

x p

p x

p p

x

+

=

+

=

Consumer Demand: example

;

Calculate ordinary demands:

Hence,

y x y

x

u( , ) =

( x , y ) = ( 2 , 1 )

y y

x

x y

x

p I I

p p y

p I I

p p x

) 3 , , (

*

3 ) 2 , , (

*

=

=

y x y

y x

y x

x y x

y x

y x

p p p

p p p

p y

p p p

p p p

p x

3 2 3 1 3

) 2 ,

~(

3 2 3 4 3

) 2

( ) 2 ,

~(

+ + =

=

+ + =

=

Income and substitution effects

• Study the effect of an increase in p_x for utility function and endowment

) , ( yx u

) , (x y

x

y

x y

u₁

u₂

A B

C

Income and substitution effects

• Substitution effect is negative

• Income effect is positve

• In this case, income effect is larger than substitution effect, so the total effect is positive: an increase in leads the

consumer to increase its consumption of both goods

px

< 0 -

= x_B x_A SE

> 0 -

= x_C x_B IE

> 0 -

= +

= SE IE x_C x_A TE

Income and substitution effects

Whether income is endogenous or exogenous, the substitution effect of an increase of the price of a good over its demand is negative.

However, when income is endogenous the income effect decomposes into two effects:

(1) Ordinary income effect, whose sign depends on whether the good is normal (N) or inferior (I).

(2) Endowment effect, whose sign depends on whether at the original prices the consumer was a net seller (S) or a net buyer (B) of the good.

The sign of the total income effect (TIE) is negative in the cases N-B and I-S, whereas it is ambiguous in the cases N-S and I-B.

Income and substitution effects

Total effect = substitution effect (SE) + (ordinary income effect + endowment effect)

= SE + TIE.

Formally,

)

* (

* |

*

| *

~ *

x I x

x p

x

I x x

I x x

p x p

x

cte u x

cte u x x

¶ - - ¶

¶

= ¶

¶ + ¶

¶ - ¶

¶

= ¶

¶

¶

=

=

Income and substitution effects

¿When is the TIE negative?

• If TIE < 0, then SE + TIE < 0.

• If TIE > 0, then the sign of SE + TIE is ambiguous.

Normal good ^!"_!#^∗ > 0 Inferior good ^!"_!#^∗ < 0 Net buyer

𝑥 − ̅𝑥 > 0 𝑇𝐼𝐸 < 0 𝑇𝐼𝐸 > 0

Net seller

𝑥 − ̅𝑥 < 0 𝑇𝐼𝐸 > 0 𝑇𝐼𝐸 < 0

Income and substitution effects

• When is the total income effect positive (the consumer is richer)?

If the good is normal and the consumer is a net seller of the good

• In any case, notice that positive income effect (“richer consumer”) does not mean more

consumption: we have to take into account the substitution effect

0

and

* 0

0 )

*(

<

-

¶ >

> ¶

¶ -

- ¶ x x

I if x

x I x

x

Income and substitution effects

Exogenous income Endogenous income

u₂ u₁

x y

B

A C

u₁ u₀

y

x

A B C

The consumption-leisure model.

Labor supply

• Two goods: leisure (x-axis) and consumption (y-axis)

- Leisure, denoted by h and measured in hours. The wage per hour (or price of leisure) is denoted by w.

- Consumption, denoted by c and measured in euros. The price of c is therefore p_c = 1).

• Initial endowment is (M,H), where:

- M : initial exogenous wealth (or non-labor income).

- H : number of hours available for leisure and work.

The consumption-leisure model.

Labor supply

• Budget set (recall )

: expenditure on leisure : monetary value of initial

endowment

=1 pc

M wH

hw

c + £ +

h c

H M

M wH +

w slope = - hw

M wH +

The consumption-leisure model.

Labor supply

• Solve the problem as usual, but watch out for additional constraints

h

Max

u ( h c , )

st ^{c + hw £ wH + M}

0 0

³

£

£ c

H

h

The consumption-leisure model.

Labor supply. Example

Interior solution requires

And

h

Maxc_{, c+ 2lnh}

st c +wh =16w+4

0

16 0

³

£

£ c

h

w w h h w

w c

h

MRS 2

) 2 (

) ,

( = Û = Þ =

8 16 1

) 2 (

0 2 0

) (

³ Û

£

=

>

"

Û

³

= w w w

h

w w w

h

The consumption-leisure model.

Labor supply. Example

Therefore,

= ) (w h

8 / 1

16 if w <

8 / 1

2 if w³ w

= ) (w c

8 / 1

4 if w<

The consumption-leisure model.

Labor supply. Example

And labor supply

= -

= ( )

)

(w H h w

l

8 / 1

0 if w <

8 / 1

16 - 2 if w ³ w

1/8

The consumption-leisure model.

Labor supply. Effect of changes in wages

• Assume

• For interior solutions, the consumer is a net supplier of leisure

• Total income effect: if leisure is a normal good, , TIE is positive, leading the consumer to demand less leisure (or supply more labor)

• Substitution effect : is always non-positive. 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 utility function) w

w'<

0 /¶ >

¶h I

0 ) ( - >

¶

- ¶ l H

I h

>0 <0

The consumption-leisure model.

Labor supply. Effect of changes in wages

c

h

M

H

M wH +

M H

w' + _A

B C

The consumption-leisure model.

Labor supply. Effect of changes in wages

c

M wH +

M H

w' +

M

H

h

B C

A

The consumption-leisure model.

Labor supply. Effect of changes in wages

For , SE

dominates (leisure is more expensive and consumer offers more labor)

For , TIE

dominates (consumer is richer and does not need to work as much as

before)

) 10 , 0 Î( w

) 20 , 10 Î( w

Application: a tax on labor income

• Impose

• The new budget constraint is

• The tax is equivalent to a reduction of wage: its impact on leisure consumption (or labor supply) is ambiguous

• Its impact on welfare is unambiguous. Of course, tax policies have other objectives we are not considering here.

] 1 , 0 Î[ t

M wH

t wh

t

c + ( 1 - ) £ ( 1 - ) +

Application: a tax on labor income

c

h

M

H

M wH +

M wH

t + - )

1 (

* h

* c

Application: a tax on labor income

• Alternative: a non-labor income tax T.

• The new budget constraint is

• If both goods are normal, then the introduction of T reduces their demands (increases labor supply, in particular)

) ( M T wH

wh

c + £ + -

Application: a tax on labor income

M wH +

) (M T wH + -

M

T

M -

h

c

H

Application: a tax on labor income

• Exercise: what if T = tw(H-h*)?

• Hint: is (c*,h*) optimal for T

?

*

h

h

* c

c

M wH +

M wH

t + - )

1 (

) (M T wH + -

M T M -

A

B