Abel & Bernake: Macro Ch3 Varian: Micro Ch10 Appendix 4.A A Formal Model of Consumption and Saving Micro-foundation of Macro Abel & Bernake: Macro Ch3 Varian: Micro Ch10

Optimization over time Current income Y, future income Yf : Endowment point: (a+Y, Yf) initial wealth a, wealth at beginning of future period af ; Choice variables: C = current consumption; Cf = future consumption Slope of lifetime BC = -(1+r)

Figure 4.A.1 The budget line

Present-Value Budget Constraint (PVBC) Present value of lifetime wealth: PVLW = a+ Y + Yf/(1+r) (4.A.2) Present value of lifetime consumption: PVLC = C + Cf/(1+r) The budget constraint means PVLC = PVLW C + Cf/(1+r) =a+ Y + Yf/(1+r) (4.A.3) Slope of PVBC≡ (△Cf/△C)= -(1+r) Price of current consumption=(1+r)：△Cf = -(1+r) △C

Consumer Preferences: Indifference Curves A person is equally happy at any point on an indifference curve 3 important properties of indifference curves Slope downward from left to right: Less consumption in one period requires more consumption in the other period to keep utility unchanged Indifference curves that are farther up and to the right represent higher levels of utility, because more consumption is preferred to less Indifference curves are bowed toward the origin, because people have a consumption-smoothing motive, they prefer consuming equal amounts in each period rather than consuming a lot one period and little the other period

Figure 4.A.2 Indifference curves

The Optimal Level of Consumption Optimal consumption point: the budget line is tangent to an indifference curve (Fig. 4.A.3) Tangency condition: ?

Fig 4.A.3 The optimal consumption combination

Saving (S), a lender or a borrower S≡Y-C If C=Y, S=0 If C<Y, S>0 If C>Y, S<0 af= 0, C=a+Y: no-borrowing, no-lending af>0, C< a+Y: lender, with interest income af< 0, C< a+Y: borrower, with interest payment

Income Effect (IE) or Wealth Effect When income or wealth (PVLW) increases, PVBC shifts outward, the opportunity set increases, the demand for normal goods (C and Cf ) increases. a↑, Y↑, Yf ↑: PVLW↑  C↑ and Cf ↑

Comparative Statics: △a, △Y, △Yf An increase in wealth: a↑ Increases PVLW, so LRBC shifts out parallel to old BC. with consumption smoothing, both current and future consumption increase IE: a↑ PVLW↑, C↑, Cf↑ (Y is unchanged, S:↓) An increase in current income: Y ↑ (Fig. 4.A.4) Y↑ IE: C↑, Cf ↑ (S↑) An increase in future income: Yf ↑ Yf↑ IE: C↑, Cf ↑ (S↓)

Fig 4.A.4 An increase in income or wealth

Permanent vs. temporary increase in income Different types of changes in income Temporary increase in income: Y rises and Yf is unchanged  ? Permanent increase in income: Both Y and Yf rise

The permanent income theory This distinction made by Milton Friedman in the 1950s and is known as the permanent income theory Permanent changes in income lead to much larger changes in consumption Thus permanent income changes are mostly consumed, while temporary income changes are mostly saved

Life-Cycle Model developed by Franco Modigliani and associates in the 1950s Patterns of income, consumption, and saving over an individual’s lifetime (Fig. 4.A.5) Real income steadily rises over time until near retirement; at retirement, income drops sharply Lifetime pattern of consumption is much smoother than the income pattern.

Life-cycle consumption, income, and saving hump-shaped of Y, S Figure 4.A.5

Ricardian equivalence: two-period model Suppose the government reduces taxes by 100 in the current period, r = 10%, and taxes will be increased by 110 in the future period Then the PVLW is unchanged, and no change in C. Or ?

Fiscal policy: △T<0 (a lump-sum tax↓ )  △Sd Assume closed economy: NFP=0, Assume TR=INT=0 for simplicity S= Y + NFP– C – G Spvt= Y + NFP – T + TR + INT – C Sgovt =T – TR – INT – G ???

Fiscal policy: △G>0  △Sd

Excess sensitivity of consumption Generally, life cycle or permanent income theory have been supported by looking at real-world data But data shows some excess sensitivity of consumption to changes in current income This could be due to short-sighted behavior Or it could be due to borrowing constraints

Borrowing constraints If a person wants to borrow and can’t, the borrowing constraint is binding A consumer with a binding borrowing constraint spends all income and wealth on consumption. So an increase in income or wealth will be entirely spent on consumption as well This causes consumption to be excessively sensitive to current income changes Perhaps 20% to 50% of the U.S. population faces binding borrowing constraints.

Comparative statics: change in r r↑(Fig. 4.A.6) one point on the old BC is also on the new BC: the no-borrowing, no-lending point Slope of new budget line is steeper

Fig 4.A.6 The effect of an increase in the real interest rate on the budget line

Intertemporal substitution effect (ISE) r↑makes future consumption cheaper relative to current consumption use cheaper Cf to substitute more costly C, C↓(S↑), Cf ↑

Fig 4.A.7 The substitution effect of an increase in the real interest rate

Income Effect (IE) For a person consume at no-borrowing, no-lending point, r↑ no IE If the person originally a lender, r↑ positive IE  C↑, Cf ↑ If the person originally a borrower, r↑ negative IE  C↓, Cf ↓

Δr: Total effect =ISE +IE IE and ISE together If a person consumes at no-borrowing, no-lending point (Fig. 4.A.7), ? For a lender, (Fig. 4.A.8) For a borrower,

Fig 4.A.8 An increase in the real interest rate with both an income effect and a substitution effect

Δr  aggregate saving The effect on aggregate saving of r↑ is ambiguous theoretically. Empirical research suggests that saving ↑ (Saving function is positively-sloped) But the effect is small

Temporary vs. Permanent increase in wage Optimization over time (Ch3) Max U(C, L, Cf, Lf) ISE between current C and future Cf ISE between current L and future Lf If temporary w↑: strong ISE + weak IE ISE > IE => L↓, h ↑ If permanent w↑ : weak ISE + strong IE ISE < IE => L ↑, h ↓ Empirical evidence support the implication.