# Infinitely Repeated Games

Christopher Makler

Stanford University Department of Economics

Econ 51: Lecture 13

"Present Value" for two periods

# Beyond Two Periods

If you save \(s\) now, you get \(x = s(1 + r)\) next period.

The amount you have to save in order to get \(x\) one period in the future is

Remember how we got this...

If you save \(s\) now, you get \(x = s(1 + r)\) next period.

The amount you have to save in order to get \(x\) one period in the future is

If you save for **two** periods, it grows at interest rate \(r\) again, so \(x_2 = (1+r)(1+r)s = (1+r)^2s\)

Therefore, the amount you have to save in order to get \(x_2\) two periods in the future is

If you save for **two** periods, it grows at interest rate \(r\) again, so \(x_2 = (1+r)(1+r)s = (1+r)^2s\)

Therefore, the amount you have to save in order to get \(x_2\) two periods in the future is

If you save for \(t\) periods, it grows at interest rate \(r\) each period, so \(x_t = (1+r)^ts\)

Therefore, the amount you have to save in order to get \(x_t\), \(t\) periods in the future, is

Therefore, the amount you have to save in order to get \(x_t\), \(t\) periods in the future, is

We call this the **present value **of a payoff of \(x_t\)

The present value of an **income stream** is the sum of the present values in each period:

# Evaluating Infinite Payoffs

The **present value** of a stream of payoffs

(\(\pi_0\) now, \(\pi_1\) in the next period, \(\pi_2\) two periods from now, etc)

may be given by the sum

The **present value** of a stream of payoffs of \(x\) in every period is

Value of getting payoff \(x\) forever, starting now:

Value of getting payoff \(z\) forever, starting next period:

Value of getting payoff \(y\) now and *then* payoff \(z\) forever after:

#### Copy of Econ 51 | Spring 23 | 1 | Welcome and Intertemporal Consumption

By Chris Makler

# Copy of Econ 51 | Spring 23 | 1 | Welcome and Intertemporal Consumption

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