The gentlemenly C level of understanding

Consider a geometric series d + ad + a²d + º+ a^t-1d whose sum as t Æ € is [d/( (1 - a))] The multiplier is obtained by suming a geometric series

The B level of understanding

where do the terms come from: Household get their income at the end of the period and spend it in the next period. Each period the firms maintain the DI level of investments and that is the first term in each period. The other terms are b times the income received at the end of the previous period. Thus in period 4 the 2nd through the 4th terms come from b times (DI + b DI + b²DI) and so on.

Dynamic Problems

3. Given model above with a = 200 b = 0.75 and I₀ = 150:

a) Suppose the economy is in equilibrium initially. What is the equilibrium Y?

Solution

Solve using the solution to the static model:

k = 1/[1 - 0.75] = 4 Y = k[a + I] = 4[350] = 1400

b) Suppose I increases from 150 to 170 in period 1. Find Y₀, Y₁, Y₂, and Y₃.

Solution

Use a table. On the quiz and tests I will not give you the format for the table because there are several ways you could set up the table and I do not want arguments with the grader. This is a sample table with an extra column relating what you are doing to the geometric series.

t	a + It	bYt-1	Yt	Yt is
1	370	1050	1420	Y0 + DI
2	370	1065	1435	Y0 + DI + bDI
3	370	1076.25	1446.25	Y0 + DI + bDI + b2DI

c) What is the final equilibrium Y?

Solution

Use the same aproach as the initial equilibrium.

Y = k[a + I] = 4[370] = 1480

Note Y_t = Y₀ + DI + bDI + b²DI + º+ b^t-1DI thus Y_€ = Y₀ + kDI = 1400 + 4 ·20 = 1480

To solve dynamic problems use the following worksheet.