Limit in math?

When the money is spent on goods and services, those that receive the money also spent some of it. The people receiving some of the twice-spent money will spend some of that and so on. Economists call this chain reaction the multiplier effect. In a
hypothetical isolated community, the local government begins the process by spending R ringgits. Suppose that each recipient of spent money spends 100c% and 100s% of the money that he or she receives. The values c and s are called the marginal propensity to consume and the marginal propensity of save and of course,c + s = 1.

i. Let Sn be the total spending that has been generated after n transactions. Find an equation for Sn .
ii. Show that limit(n going to infinity) Sn=kR,where k=1/s

iii. The number k is called the multiplier. What is the multiplier if the marginal propensity to consume is 80%?

Answer:
Here's how to generate it mathematically.

Assume you start with a $100 and s = .4, c = .c

The first consumer will spend $60, save $40. The people who recieve the $60 will save $24, spend $36. The 3rd tier will spend $21.60, save $14.40

It is clear to see that the spending are $60 + $36 + 21.60..
This is the same as $100 * .6 + $100 *.6 * .6 + $100 * .6 * .6 *.6....

This means that spending is R * c^0 + R*c^1 + R * c^2 + R * c^3.....R*c^n.

That means you have summation 0=k to n with R * c^k.

The summation clearly converges to R * (1 + c/1-c) as n -> infinity..
That simplifies to R* (1/1-c); and since s + c = 1; then 1-c = s.

So the limit is R * 1/s.

Now you can plug away. If consumption is .8, s = .2; so the multiplier is 1/.2 = 5.

Five.

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Limit in math?

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