## Setting up Equations for Sums

This is actually a macroeconomics problem.

Section explaining the math:

Example: Application in Macroeconomics (Multiplier Effect): In a two-sector, simple Keynesian macro-model of the economy with a consumption function of C = bY, investment function of I = Io, and the income identity of Y = C + I, every one dollar increase in I will increase Y by the following amounts over time periods,

Time Period: 1, 2, 3, ....................., n
Increase Y: 1, b, (b^2),..............., b^(n-1)

Find the total increase in output after n periods (simple multiplier.)

Solution: The progression is a geometric progression with an initial value of 1 and a common ratio of b. Using the sum formula, S = (1 - b^n)/(1 - b). However, in this case b, which in macroeconomics is called marginal propensity to consume (MPC), is a number less than 1 (b < 1). Therefore, as n approaches infinity, b^n = 0 and S = 1/(1 - b).

Problems:

6- In a simple macro model of an economy C = 100 + .8Y, I = 200, and Y = C + I.

a- Assume that investment (I) increases by $100. What is the effect on Y after 2 years, 5 years, and 10 years? What is the final effect on Y? b- Assume that autonomous spending by consumers increases by$200. What is the effect on Y after 1 year, 4 years and 10 years? What is the final effect on Y?

I really don't know how to set these equations up for either problem. =/

Any help is appreciated!