# 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}

By Chris Makler

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