Two-Period Model

Credit market clears when... Therefore, credit market equilibrium condition is:

net quantity that consumers want to lend in current period is equal to quantity government wishes to borrow Sp = B (Aggregate quantity of private savings is equal to quantity of debt issues by government in current period - NATIONAL SAVING IS 0

Lower-case letters: Upper-case letters: Prime variables:

refer to variables refer to aggregate variables denote future period variables

Interest rate changes GRAPH 8

- 1/(1+r) is relative price of future consumption goods, therefore a change to this will: Change the intertemporal relative price (income and substitution effects)

Increase in Future Income:

- Clearly implies increased future consumption, but consumption smoothing means she will also have higher current consumption - Does so by borrowing against future income and repaying loan when she starts working

Some of aggregate consumption is not consumption in the economic sense

- Consumption of durable goods is economically similar to investment expenditures, as will use consumption services over its lifetime (very volatile) - Consumption of nondurables is much smoother than real GDP -Explains difference in variability between aggregate consumption and GDP

One bond issued at r in current period is a promise to pay: Relative price of future consumption in terms of current consumption:

-1+r units of consumption good in future period - 1/(1+r)

Bond assumptions:

-Bonds are INDISTINGUISHABLE with no risk associated with holding a bond (in practice, credit instruments associated with risk levels e.g. govt. vs bank) -Bonds are TRADED DIRECTLY IN THE CREDIT MARKET (ignores channeling through financial intermediaries)

Assumptions of two-period model

-Consumer lives for two periods -All consumers pay same taxes, despite different incomes -Consumers do not make a work-leisure decision in either period, but simply receive EXOGENOUS INCOME -Allow us to focus attention on CONSUMPTION-SAVINGS DECISION

Excess variability explanations

-Imperfections in credit market (higher borrowing rate than lending rate) -Whilst trying to smooth consumption, MARKET PRICES CHANGE and this will affect future consumer behaviour

3 preference properties:

-More always preferred to less -Consumer likes diversity in his/ her bundle -Current consumption and future consumption are normal goods

Competitive equilibrium market:

-N consumers and government interact is the credit market, where consumers and government can borrow and lend -Credit market trading effectively trades future consumption goods for current consumption goods -The relative price at which future goods trade for current consumption goods is 1/(1+r)

Competitive equilibrium three conditions for two-period economy:

1) Each consumer chooses first and second period c and s, given r 2) Government present value budget constraint holds: G + (G'/(1+r)) = T + (T'/(1+r)) 3) The credit market clears

Ricardian Equivalence theorem:

A change in the timing of taxes by the government is neutral. In saying this, we mean that in equilibrium a change in current taxes, exactly offset in present-value terms by an equal and opposite change in future taxes, has no effect on r or on c. Means GOVERNMENT DEFICITS IN A SENSE DO NOT MATTER

Consumer's consumption-savings decision:

A dynamic decision, which involves a tradeoff between current and future consumption By saving, consumer gives up consumption in exchange for present assets, so as to consumer MORE IN FUTURE By dissaving (borrowing), consumer sacrifices future consumption for more current consumption, as loan will be repaid

If all consumers act to smooth consumption relative to income?

Aggregate consumption should be smooth relative to aggregate income: REAL AGGREGATE CONSUMPTION IS LESS VARIABLE THAN REAL GDP

Ricardian Equivalence Theorem

Conditions under which govt. debt is irrelevant and does not affect macro variables or economic welfare of individual

Real interest rate assumption

Consumer can lend at the same r as that which a consumer can borrow (typically borrow at higher rates than they lend)

Bond as financial asset that is traded:

Consumer lends - buys bonds Consumer borrows - sells bonds

Consider timing of tax changes that govt. budget constraint continues to hold at r

Current taxes change by (change in t) Future taxes change by (-change in t x (1+r)) r,y,y',N,G,G' remain unaffected in consumer lifetime wealth equation above Therefore, consumer makes same decisions and chooses the same quantity of current and future consumption

GRAPH 6 Increase in future income:

Future consumption increase (AF) is less that future income increase (AD), as desire for smooth consumption This time change in savings is less than 0, so SAVINGS DECREASES

Government current period budget constraint: Government future period budget constraint (LHS vs RHS)

G = T + B (bonds issued are Govt. way of borrowing) (B>0) G' + (1+r)B = T' LHS - total government outlays in future RHS - taxes finance government outlays

Interest changes (case of a borrower)

GRAPH 10

Ricardian Equivalence Graph

GRAPH 12

Interest rate changes (case of a lender)

GRAPH 9

Relevance of temporary vs permanent effects

Government taxes can be expected as temporary or permanent by consumers, so can have varied effects

GRAPH 7 Temporary vs Permanent increases in income

HL reflects the temporary increase in income, while point J is shown by new budget constraint as intersecting I2 This reflects an increase in current consumption less than the increase in current income LM reflects the permanent increase in income, while point K is shown by an upward shift in budget constraint as intersecting I3 The new point c3 shows the further increase in current consumption from a permanent increase in income, as there need not be an increase in saving

Permanent income hypothesis:

Milton Friedman argued that a primary determinant of a consumer's current consumption is their permanent income This ties in with concept of lifetime wealth Permanent income changes have large effects on lifetime wealth and current consumption

Ricardian Equivalence application:

N consumers share an equal amount of the total tax burden in current and future periods: T = Nt, T' = Nt' G + (G'/(1+r)) = Nt + (Nt'/(1+r)) t + (t'/(1+r)) = (1/N)(G + (G'/(1+r)) Present value of taxes for a single consumer is the consumer's share of the present value of government spending C = Y-T ----- T = Y - C c+(c'/(1+r)) = y + (y'/(1+r)) - (1/N)(G + (G'/(1+r)) This is CONSUMER'S LIFETIME BUDGET CONSTRAINT WITH THE PRESENT VALUE OF TAXES Aggregate market clears so that Y = C + G

Consumption smoothing

Natural forces that cause consumers to want a smooth consumption path, rather than a choppy one

Why this model shows Ricardian Equivalence holds?

New timing of taxes, same r, each consumer is optimizing, govt. present value budget holds, and credit market clears (Y = C + G). r is therefore still the equilibrium real interest rate Change in timing of taxes has no effect on equilibrium consumption or the real interest rate

Lifetime budget constraint states that

PRESENT value of lifetime consumption equals PRESENT value of lifetime income minus PRESENT value of lifetime taxes

GRAPH 1 endowment point:

Point E, where consumption bundle consumer gets if both current and future disposable income are consumed This is ZERO SAVINGS in current period Anywhere inside or on AB is feasible

Key variable for two-period model

Real interest rate which determines future price of consumption in terms of present consumption

GRAPH 4

Shows where a consumer is a borrower y-t = y'-t' < c'* = -s

GRAPH 3

Shows where a consumer is a saver y-t = y'-t' > c'* = +s

GRAPH 2 SHOWS INDIFFERENCE CURVES for future and current consumption

Slope m = -MRS MRS is minus the slope of a tangent to the indifference curves Derive by taking derivative

Government present-value budget: Implications?

Solve for B: B = (T' - G')/(1+r) (B=T+G) G + (G'/(1+r)) = T + (T'/(1+r)) This implies that government must eventually pay off all of its debt by taxing its citizens

National income identity:

Sp + Sg = I + CA No capital accumulation, so Sg = -B because I=0, CA=0 in CLOSED ECONOMY ALSO NOTE: S = Sp + Sg (national savings therefore also 0)

Derive income expenditure identity

Sp = Y - C - T and B = G - T Since Sp = B: Y - C - T + G - T OR Y = C + G

Lifetime Budget constraint

Take two budget constraints above and write as single lifetime budget constraint: s = (c' - y' + t')/(1+r) c + ((c' - y' + t')/(1+r)) = y - t FINALLY: c+ (c'/(1+r)) = y + (y'/(1+r)) - t - (t'/(1+r))

Martingales

The best prediction of its value tomorrow is its value today E.g. the case of a stock price This is because a difference in today's value vs future value will change the stock price value today until it is equal to the expected price tomorrow. Therefore, any change to stock price is expected to be permanent, so amplifies effects of a change in stock prices STOCK PRICES MUCH MORE VOLATILE THAN CONSUMPTION

General empirical conclusion of relative volatility of aggregate consumption and aggregate income?

There is some EXCESS VARIABILITY of aggregate consumption relative to aggregate income, however consumption is still smoother than income

Future consumption and current consumption as perfect complements

They are consumed in fixed proportions: c' = ac GRAPH 11

GRAPH 5 two period effects of change in current income:

This behaviour arises because of the consumer's DESIRE FOR SMOOTH CONSUMPTION over time

Intertemporal choice

Tradeoff between current and future consumption

Present value

Value in terms of period 1 consumption goods

Lifetime wealth (we) =

c + (c'/(1+r) This is the quantity of resources that the consumer has available to spend on consumption over his lifetime in present-value terms

Consumer budget constraint example:

c + 0.909c' = 190 Found by inputting values for y, y' and t, t'

Consumer budget constraint:

c + s = y - t consumption plus savings in the current period must equal disposable income in the current period

GRAPH 1 Slope intercept form: Vertical intercept: Horizontal intercept: Lifetime budget constraint slope:

c' = -(1+r)c + we(1+r) VI is what could be consumed in future period, if consumer saves all of current period HI is what could be consumed if the consumer borrows the maximum amount possible Slope of lifetime budget constraint is -(1+r)

Future period disposable income/ interest/ consumption

y'-t' is disposable income (1+r)s is the interest received on savings c' = y' - t' + (1+r)s (s<0, consumer pays interest on loan) Consumer finishes period with no assets (as this is FINAL PERIOD)