Chapter 15. Consumption, income and wealth

ECON320Prof Mike Kennedy

We will start with the Keynesian consumption function

• The Keynesian consumption function assumed a positive MPC and that the APC would decline with the level of income

• Problems with this formulation:– Not clear that optimizing agents would look just at disposable income– Micro data do indicate that the rich tend to save more but macro data

show that the APC is pretty constant

€

Ct

Ytd = b +

a

Ytd

Consumption at the micro and macro level

For Canada (as well as other countries) the APC fluctuates around a constant over time which contradicts the Keynesian model

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Average propensity to consume

Average plus/minus 2 standard deviations

Average APC = 0.89

Need to examine the micro underpinnings of consumer behaviour

• Assume the inter-temporal utility function looks like:

• The term ϕ captures the consumers impatience• For a consumer living two periods, the amount of wealth in

the future is

• All is consumed in the future and the budget constraint in period 2 is the amount of wealth (V2) and after-tax income

The inter-temporal budget constraint

• Combing the previous two equations gives the inter-temporal budget constraint

• We can define human wealth (H1) as

• Now the inter-temporal budget constraint becomes

• It says that the PV of lifetime consumption must equal initial financial wealth plus human wealth

Allocating consumption over time

• The consumer’s problem is to choose a path of consumption so as to maximize lifetime utility subject to the budget constraint

• The consumer takes as given both V1 + H1 as well as r • We start by eliminating future consumption from the utility

function so as to have everything in terms of C1

• The consumer must choose C1 so as to maximize U thus

Allocating consumption over time con’t

• The condition on the previous slide yields

– The first condition says that, in equilibrium, the consumer will be indifferent between consuming an extra unit today or saving for consumption tomorrow

– The second says that the Marginal Rate of Substitution, written as MRS(C2:C1), must equal the slope of the budget constraint line

– In the second formulation, the slope of the budget line (1 + r) is the relative price of present consumption – the amount of future consumption that is given up to increase present consumption

– This equilibrium is illustrated by the diagram in the following slide

Allocating consumption over time con’t

Determining current consumption• We now introduce a specific form of the utility function

• We can illustrate this last condition by taking the derivative of the utility function

• Then and we know that

• In order to interpret σ, we need to introduce the concept of the Inter- temporal Elasticity of Substitution or IES €

dU(Ct )as σ →1

=dCt

Ct

€

dCt

Ct

⇔ d ln(Ct )€

U ' (Ct ) =dU (Ct )

dCt

=σ

σ −1

⎛

⎝ ⎜

⎞

⎠ ⎟σ −1

σ

⎛

⎝ ⎜

⎞

⎠ ⎟Ct

−1/σ = Ct−1/σ =

1

Ct

⎛

⎝ ⎜

⎞

⎠ ⎟

1/σ

€

u(Ct ) = lnCt for σ =1

The inter-temporal elasticity of substitution or IES

• The IES is defined as the percentage change in (C2/C1) implied by a one per cent change in MRS(C2:C1), the slope of the utility curve

• In the following slide we see the relationship between the ratio C2/C1 and the slope of the indifference curve as given by the MRS

• As the ratio C2/C rises, consumers become more reluctant to substitute future for present consumption (the slope becomes steeper)

• The coefficient σ measures the willingness of the consumer to engage in inter-temporal substitution of consumption

Interpreting the IES

The inter-temporal elasticity of substitution or IES con’t

• From slide 10 we know that which implies

and in terms of natural logs

• These definition will be helpful in defining the IES

€

U ' (Ct ) =1

Ct

⎛

⎝ ⎜

⎞

⎠ ⎟

1/σ

The inter-temporal elasticity of substitution or IES con’t

• We need to find expressions for the numerator and the denominator of the IES equation shown in slide 11

• The numerator is

• The denominator is

• Which implies

€

d ln(C2 /C1) =d(C2 /C1)

(C2 /C1)

€

d ln MRS(C2 : C1) =1

σ

⎛

⎝ ⎜

⎞

⎠ ⎟d(C2 /C1)

(C2 /C1)

€

IES =d[(C2 /C1)/(C2 /C1)]

1

σ

⎛

⎝ ⎜

⎞

⎠ ⎟d[(C2 /C1)/(C2 /C1)]

= σ

The shape the indifference curve and the IES

Deriving an equation for current consumption

• We start with the equilibrium condition that the MRS = 1 + r

• We can solve this for C2 in terms of C1

• Next we insert the above into the inter-temporal budget constraint to get

€

MRS(C2 :C1) = (1+φ)(C2 /C1) =1+ r

€

C1 +[(1+ r) /(1+φ)]σ

(1+ r)C1 = V1 + H1

Deriving an equation for current consumption con’t

• Which is

or

• This is the consumption function based on optimizing behaviour

• Given a constant IES, current consumption is proportional to the sum of financial and human wealth

• The proportionality factor (θ) will change with both r and ϕ.

€

C1 +[(1+ r) /(1+φ)]σ

(1+ r)C1 = V1 + H1

€

C1 =θ (V1 + H1), 0 <θ =1

1 + (1 + r )σ −1(1 + φ)−σ <1

The consumption function and the empirical evidence

• We are interested in the relationship between present consumption and disposable income

• Start by re-organising the term on the right hand side of the consumption function such that it is written in terms of Yd

• Writing this as

• The variable R is the ratio of future-to-current disposable income and v1 is the wealth-to-income ratio

€

C1 =θ Y1d +

Y2d

1 + r+ V1

⎛

⎝ ⎜

⎞

⎠ ⎟, Yt

d ≡ YtL −Tt , t =1, 2

The consumption function and the empirical evidence con’t

• Why does the cross-sectional evidence point to a low MPC out of current income?– Based on the above, all will depend on R, the ratio of future-to-current

disposable income– If the rise in current Yd is transitory, then R will fall; consumers save

the transitory part

• Why do aggregate data show that, over the long run, the MPC is roughly constant– We start by writing R in terms of growth to yield

– Growth and the wealth-to-income ratio (v1 ) tend to be fairly constant over time and the real rate of interest shows no particular tendency to trend up or down, implying that the above is constant

Consumption and wealth

• The average propensity does fluctuate around its longer-run trend, likely for three reasons– the expected growth rate of income (g)– the wealth to income ratio (v1) and– the real rate of interest (r)

• The next two slides show the simple empirical relationship between the APC and v1

Wealth and consumption

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Net wealth to disposable income

Average propensity to consume (right scale)

Wealth and consumption from another perspective

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1

f(x) = 0.0383612460673893 x + 0.700632727156711R² = 0.753883274995015

Net worth to disposable income

Aver

age

prop

ensit

y tp

o co

nsum

e

Consumption and interest rates

• Changes in interest rates affect the APC through the propensity to consume out of wealth (θ)

• The derivative of θ wrt to the interest rate (r) is:

• How C1 will respond will depend on the value of σ • The propensity to consume increases for σ < 1 and decreases for σ > 1• The parameter σ measures the strength of the competing income

(positive response) and substitution (negative response) affects• When σ = 1 the two effects balance each other and the coefficient θ

equal 1/[1 + (1 + ϕ)-σ] < 1 and is a constant• Empirical evidence suggests that σ < 1 implying a positive effect

€

dθ

dr=

−(σ −1)(1+ r)σ −2

[(1+ (1+ r)σ −1(1+φ)σ ]2 = (1−σ )(1+ r)σ −2θ 2

Consumption and interest rates con’t

• The other two channels are through– The market value of financial assets (v1)

– The value of human capital (H1)• A rise in interest rates will lower both financial and human

wealth which will lower the APC• Not surprisingly, it has been difficult to identify a relationship

between interest rates and consumption (see next slide)

Real interest rates and the APC:Not a well-defined in a simple relationship

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f(x) = − 0.00246132441808557 x + 0.896577042696481R² = 0.00973104715274331

Real interest rates

Aver

age

prop

ensit

y to

cons

ume

Consumption and taxation:A temporary tax cut

• We need to re-write the consumption function to highlight taxes (Ti, i = 1, 2)

• The effect of a temporary tax cut in period 1 is:

• A tax cut in period 1 (T1 falls) will raise consumption by θ units• The coefficient θ is less than one reflecting consumption

smoothing – consumers save some part of the tax cut

€

C1 =θ Y1L −T1 +

Y2L −T2

1 + r+ V1

⎛

⎝ ⎜

⎞

⎠ ⎟

Consumption and taxation:A permanent tax cut

• A permanent tax cut would have a larger effect on C1

• If consumers want to spend equal amounts in each period (implying that r =ϕ and perfect smoothing) then θ becomes

implying

• With perfect consumption smoothing, there is no need to save something extra for the future since Yd will rise by the same amount in each period€

θ =1

1+1+ r

1+φ

⎛

⎝ ⎜

⎞

⎠ ⎟

σ1

1+ r

⎛ ⎝ ⎜

⎞ ⎠ ⎟

=1

1+1

1+ r

=1+ r

2 + r

€

∂C1

∂T1

+∂C1

∂T2

= −θ 1+1

1+ r

⎛

⎝ ⎜

⎞

⎠ ⎟= −θ

2 + r

1+ r

⎛

⎝ ⎜

⎞

⎠ ⎟

What about the government:Introducing the government’s budget constraint

• In a two period model the government’s debt level (D) in the next period is

where G is government spending• In period 2 the government must have a surplus sufficient to

payoff its debts and cover spending T2 = D2 + G2

Based on these two equations the government’s inter-temporal budget constraint is

• Fiscal policy is sustainable when the debt level plus the PV of spending equals the PV of revenues (taxes in each period)

Debt versus taxes and Ricardian Equivalence • Following a tax cut, the government must either reduce G2 or raise T2 in

order for the budget constraint to hold• Assuming that they raise taxes, then the revenues must be sufficient to

payoff the incurred debt

which implies

• A tax cut not accompanied by a planned cut in present or future spending will not affect current spending

• The other implication is that taxes and debt are equivalent – hence the title “Ricardian Equivalence”

Private and public saving in Canada:There is a tendency for them to move in opposite

directions

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1Recessions

Private saving (% GDP)

Government saving (% GDP)

Why Ricardian Equivalence may not hold

• Consumers may not be that rational or have the means and sophistication to make complex calculations

• Finite horizons and inter-generational distribution effects– Future taxes are paid by others – Parents and bequests

• Intra-generational redistribution• Distortionary taxes

– The implicit assumption is that taxes are lump sum• Credit constraints

– It also assumes that all consumers can borrow or lend at the going rate of interest – some households cannot and will spend the money

– Evidence suggests that consumption is more sensitive to current income than theory would suggest

Employment and the average propensity to consume

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Average propensity to consume

Employment rate (right scale)

Employment and the average propensity to consume: Another perspective

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f(x) = 1.03737916094496 x − 0.0706213144520798R² = 0.209776765793347

The consumption function function

• A general form of the consumption function

€

C1 = C(Y1d

(+)

, g(+)

, r(− )

, V1(+)

)

Using the Lagrange multiplier and the Euler equation• Here we show a useful way to solve the two period consumption problem• We start by forming the Lagrangian (L) which is

• The equation say that we are going to maximize utility (U) subject to the inter-temporal budget constraint – the term in brackets

• We now substitute into the above the general form of the utility function

• Maximizing L wrt to the C1 and C2 yields two equations

• We can form the Euler equation by eliminating λ, which yields

• This identical to the condition that we arrived at in slide 8€

U ' (C1) = λ and U ' (C2 ) /(1+φ) = λ /(1+ r)

€

U ' (C1) /U ' (C2) = (1+ r) /(1+φ)

€

L =U (C1) +U (C2 )

1+φ+λ V1 +Y1

d +Y2

d

1+ r−C1 −

C2

1+ r

⎛

⎝ ⎜

⎞

⎠ ⎟

€

L =U (C1, C2 ) +λ V1 +Y1d +

Y2d

1+ r−C1 −

C2

1+ r

⎛

⎝ ⎜

⎞

⎠ ⎟

Using the Lagrange multiplier and the Euler equation

• We can use the specific form of the utility function and the budget constraint to arrive at the consumption function shown in slide 17

• Recall that for any C we have

• Using the relationship on the previous slide we get

• This is the same relationship we found in slide 16• Substituting the relationship between C2 and C1 into the inter-

temporal budget constraint, and remembering that yields the consumption function in slide 17

€

U (C) =σ (C (σ −1)/σ )

σ −1 and U ' (C) =

1

C

⎛

⎝ ⎜

⎞

⎠ ⎟

1/σ

€

U ' (C1) /U ' (C2 ) =C2

C1

⎛

⎝ ⎜

⎞

⎠ ⎟

1/σ

=1+ r

1+φ or C2 =

1+ r

1+φ

⎛

⎝ ⎜

⎞

⎠ ⎟

σ

C1

€

C1 =θ (V1 + H1), 0 <θ =1

1 + (1 + r )σ −1(1 + φ)−σ <1€

H1 =Y1d +Y2

d /(1+ r)