Topic 4: Consumption, Saving and Investment

National level of desired (optimal) consumption (Cd)

Aggregate quantity of goods and services that households optimally choose to consume, given income and other factors determining household economic opportunities

Aggregate level of desired (optimal) consumption

Sum of desired consumption of all households

Desired national saving

Level of aggregative saving when consumption is at its desired level

Desired national saving equation

Sd \= Y - Cd - G

Individual lifespans

Current period and future period

Budget constraint of current period

af \= y + a - c

Budget constraint of future period

cf \= yf + (1+r)af

Saving

s \= y - c

Saver

c < y

Dissaving (|s|)

c \> y

Borrower

c \> y + a

Lender

c < y + a

No borrowing, no lending case

c \= y + a

Consolidated budget constraint

c + (cf/(1+r)) \= y + a + (yf/(1+r))

Present Value of Lifetime Resources (PVLR)

y + a + (yf/(1+r))

Present Value of Lifetime Consumption (PVLC)

c + (cf/(1+r))

Slope of intertemporal budget constraint

-(1+r)

Household optimisation problem

max U \= u(c) + 𝛽u(cf)

𝛽 \> 0 in optimisation problem

Weight an individual gives to current and future consumption

𝛽 \= 1 in optimisation problem

Utils today equal future utils

𝛽 < 1 in optimisation problem

Utils today are valued more than utils in the future

Optimality condition (Euler equation)

u'(c) \= 𝛽(1+r)u'(cf)

Euler equation using u(c) \= ln c

cf/c \= 𝛽(1+r)

Consumption smoothing motive

the desire to have a relatively even pattern of consumption over time

Current period consumption (Euler)

c \= (1/(1+𝛽)) x PVLR

Future period consumption (Euler)

cf \= (𝛽/(1+𝛽)) x (1+r)PVLR

Consumption smoothing motive

The desire to have a relatively even pattern of consumption over time

Increase in individual current income

Pure income effect (increase in c and cf and dissaving)

Marginal propensity to consume (mpc)

Fraction of additional current or future income that the individual allocates to current or future consumption [0,1]

Increase in yf or a

Increased PVLR (both c and cf and dissaving increase by a pure income effect)

Increase in r (SE)

Decrease in c and an increase in s and cf

Increase in r for a lender (IE)

Rise in wealth so an increase in c and cf and dissaving

Increase in r for a borrower (IE)

Fall in wealth so a fall in c and cf and a rise in saving

Total effect of an increase in r on saving

Lender is indeterminate and a borrower sees an increase in savings

Left of point D (no lending, no borrowing point)

Saver/lender

Right of point D (no lending, no borrowing point)

Dissaver/borrower

Expected real after-tax interest rate (re,a-t)

(1-t)i - 𝜋e

FIscal policy assumption

No impact on aggregate supply (Y)

Affects of fiscal policy on Sd

Changing Cd and G both change Sd

Disposable income

Income available for consumption (YD \= Y - T)

G financed by raising current taxes (T)

Current after-tax incomes fall so consumption falls by mpc x YD

G financed by increased borrowing

Future taxes (Tf) rises, expected future after-tax incomes fall (Yf,D) and desired consumption now falls (Cd)

Lump-sum tax

Each taxpayer pays the same amount of tax

Lump-sum tax on consumption (Cd)

YD rises, current consumption rises by less, future taxes rise, Yf,D falls and Cd falls now

Ricardian equivalence proposition

Tax cuts don't affect consumption and do not affect national saving. If government purchases do not change, a cut in current taxes can affect the timing of tax collections but not the ultimate tax burden borne by consumers. With PVLR unaffected, desired consumption remains unaffected also

Sceptics

Consumers may not associate the increased government borrowing with higher taxes in the future so a tax cut may lead to higher consumption today

Permanent Income Theory of Consumption

Permanent one unit increase in income causes a larger increase in PVLR (y and yf) than does a temporary one unit increase in income (y)

Life-Cycle Model

Many-period model looking at patterns of income, consumption and saving over an individual's lifetime

Segments of the Life-Cycle Model (real income)

Real income steadily rises over time until near retirement; at retirement, it drops sharply

Segments of the Life-Cycle Model (Smoothness)

Lifetime pattern of consumption much smoother than the income pattern

Segments of the Life-Cycle Model (savings)

Low or negative early in working life; max saving occurs when income is highest (50-60) and dissaving occurs in retirement

Investment

Purchase/construction of capital goods

Firm’s desired capital stock

Amount of capital that allows the firm to earn the highest possible profit

Benefit of an additional unit of capital

Expected future value marginal product of capital (p.MPKf)

User cost of capital (uc)

Expected cost of using a unit of capital for a specified time period

User cost of capital formula

𝑢𝑐 = 𝑟p𝐾 + 𝑑p𝐾 − ∆p𝐾 = (𝑟 + 𝑑 − ∆𝑝𝐾/𝑝𝐾 ) ∙ pK

Capital Gain (Capital Loss)

∆p𝐾 > 0 (∆p𝐾 < 0)

Optimal capital level (maximum expected profit)

MPKf = uc

Factors that change the desired capital stock

Changes in r, d, pK, technological changes affecting MPKf

Return to capital (taxes)

(1-t)MPKf

Desired capital stock (taxes)

(1-t)MPKf = uc

Tax-adjusted uc

uc/(1-t)

Increase in t (MPKf)

Increases tax-adjusted uc and reduces desired K

Depreciation allowances

Reduce profit to be taxed

Investment taax credits

Reduce taxes when firms make new investments

Effective tax rate

Tax rate on firm revenue that would have the same effect on the desired capital stock as do the actual provisions of the tax code

Equation of motion of capital

Kt+1 - Kt = It - dKt

Net investment during period t

Kt-1 - Kt

Gross investment during period t

It = K\* - Kt + dKt (desired net increase in K over period t and the investment needed to replace worn-out capital)

Capital stock at the beginning of period t

Kt

Factors that lead to a change in K\* results in

An equal change in the rate of investment (I)

Yad (Closed economy)

Yad = Cd + Id + G (Id = Ip)

Goods market equilibrium condition

Y = Yad

Y > Yad

Iu > 0 (unplanned accumulation of unsold goods and an unwanted accumulation of inventory stock in a recession)

Y < Yad

Iu < 0 (Unplanned decrease in the value of firms’ inventory stocks, expansionary periods)

Y = Yad

Iu = 0; (Sd = Id)

r = r\*

Sd = Id and Y = Yad

r < r\*

Sd < Id and Y < Yad (r increases until r = r\*)

r > r\*

Sd > Id and Y > Yad (r decreases until r = r\*)

Shifts in savings curve (right)

Rise in current output and taxes (unless Ricardian equivalence holds), fall in expected future output and wealth and government purchases

Crowding-out effect

Increased government purchases cause private investment to decline

Shifts of the investment curve (right)

Any factor that decreases uc, a fall in t, rise in expected future MPK (MPKf)