Infinitely Repeated Games

Christopher Makler

Stanford University Department of Economics

Econ 51: Lecture 13

c_1 + \frac{c_2}{1+r} = m_1 + \frac{m_2}{1 + r}

"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

s(1 + r) = x
s = {x \over 1 + r}

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

s(1 + r) = x
s = {x \over 1 + r}

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

s(1 + r)^2 = x_2
s = {x_2 \over (1 + r)^2}

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

s(1 + r)^2 = x_2
s = {x_2 \over (1 + r)^2}

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

s(1 + r)^t = x_t
s = {x_t \over (1 + r)^t}

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

s(1 + r)^t = x_t
s = {x_t \over (1 + r)^t}

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

PV(x_t) = {x_t \over (1 + r)^t}

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

PV(x_0,x_1,x_2) = x_0 + {x_1 \over 1 + r} + {x_2 \over (1 + r)^2}

Evaluating Infinite Payoffs

V(\pi_0,\pi_1,\pi_2, \cdots) = \pi_0 + {\pi_1 \over 1 + r} + {\pi_2 \over (1 + r)^2} + \cdots
V(x,x,x, \cdots) = x + {x \over 1 + r} + {x \over (1 + r)^2} + \cdots

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

x + {x \over r}

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:

{z \over r}
y + {z \over r}