Multi-period Consumption I

Assumptions

• • two periods: current and future

  • income (current and future) and initial wealth are given

  • she is able to borrow or lend as much as she wants at the same real interest rate

The Budget Line

Let

Y = 28,000

Yf = 22,000

initial wealth = a = 12,000

The consumer chooses af (wealth at the beginning of the future period), c, and cf. One possible consumption point is c = a + y and cf = yf with s = y - c = 28,000 - 40,000 = -12,000.

af = s(1 + r) + a(1 + r)

af = (y + a - c)(1 + r)

so the household's budget constraint is cf = (y + a - c)(1 + r) + yf

Let r = 10%

when c = 0, cf = (28,000 + 12,000 - 0)(1.1) + 22,000) = 66,000

when cf = 0, c = (y + a) + yf/(1 + r) - cf/(1 + r) = (28,000 + 12,000) + (22,000)/(1.1) = 60,000

Present Value

Present value is the value of payments to be made in the future in term's of today's dollars. For example, with r=10%, $12,000 invested today is worth $13,200 in one year. So, the present value of $13,200 to be received in one year is $12,000.

PV = FV/(1+r)

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

PVLC = c + cf/(1+r)

The budget constraint means that PVLR = PVLC or c + cf/(1+r) = y + yf/(1+r) + a. The horizontal intercept of the budget line is c=PVLR, cf=0

Consumer Preferences

Indifference curves show combinations of c and cf that give the same utility.

• • slope downward: less consumption in one period requires more consumption in the other period to keep utility constant

  • higher indifference curves represent higher levels of utility

  • bowed in toward the origin: prefer consuming equal amounts in each period rather than a lot in one period and little in the other

Equilibrium

effects on c, cf, and s of

• • rise in y

  • rise in yf

  • rise in a

Permanent Income Theory

temporary increase in income: y rises but yf is unchanged

permanent increase in income: both y and yf rise

A permanent increase in income causes a bigger increase in PVLR than a temporary income increase. So, current consumption will rise more with a permanent income increase, and saving from a permanent increase in income is less than from a temporary increase in income.

Permanent changes in income lead to much larger changes in consumption than temporary income changes. Thus, permanent income changes are mostly consumed while temporary income changes are mostly saved.

Since recessions are short-lived, they shouldn't affect consumption very much according to the Permanent Income Hypothesis. So, aggregate consumption should be smoother than GDP and saving must decline during recessions. These assertions are supported by the data.

There seems to be excess sensitivity of consumption to changes in current income. This could be due to short-sighted behavior or to borrowing constraints (cannot borrow as much as they want; faced by 20%-50% of the population). Those who are liquidity constrained are unable to smooth out consumption when current income changes.