# Workshop #4 Problem Answers Using Aggregate Supply and Demand Models

## Question #1

Assume an aggregate supply curve of the following sort:

P = 1.41 + .0001Y
where P = the average price level and Y = GNP

### 1. Now, derive an aggregate demand function from the following data:

A consumption function in which C = 100 + .9Yd -20P
An investment function in which I = 400 - 40P

Government expenditures, G = 300
Lump sum taxes, T = 100

Hint: you want an equation in which Y is a function of P.

Ans: To find the aggregate demand curve solve for equilibrium Y (which will depend on the price level as follows:
Y = C + I + G
Y = 100 + .9(Y - 100) - 20P + 400 - 40P + 300
Y = 710 + .9Y - 60P
.1Y = 710 - 60P
Y = 7,100 - 600P, which is the equation for the aggregate demand curve.

### 2. Now that you have aggregate supply and demand curves, solve for the equilibrium level of Y and for the price level.

To find the equilibrium level of Y, substitute the AS equation for P in the AD equation and solve:

Y = 7100 - 600(1.41 + .0001Y)
Y = 7100 - 846 - .06Y
1.06Y = 6,254
Y = 6,254/1.06
Y = 5,900

To find the equilibrium price level, substitute 5,900 into the AS curve equation:

P = 1.41 + .0001(5,900) P = 2.0

### 3. Solve for the components of GNP, i.e., C, I and G (to check yourself they should sum to the level of GNP that you found in question #2.

From the consumption function:

C = 100 + .9(5,900 - 100) - 20(2.0)
C = 5,280

From the investment function:

I = 400 - 40(2.0)
I = 320

To check:

5,280 + 320 + 300 = 5,900

### 4. Now suppose that the government decides to stimulate the economy by raising its expenditures from 300 to 400. What will be the consequences for GNP? for the price level and on the various components of GNP? Will this expansion in government spending have caused any "crowding out" effects on investment? Will the change in the price level have caused the kind of change in consumption expenditures that the consumption function would seem to suggest? If not, what has offset this expected effect?

The rise in government expenditure will change the aggregate demand function:

Y = C + I + G
Y = 100 + .9(Y - 100) - 20P + 400 - 40P + 400
Y = 810 + .9Y - 60P
.1Y = 810 - 60P
Y = 8,100 - 600P
Because the intercept 8,100 is larger than it was before (7,100) we know the AD curve has shifted to the right by 1000.

To find the new level of GNP, we once again combine the aggregate demand and supply curves:

Y = 8,100 - 600(1.41 + .0001Y)
Y = 8,100 - 846 - .06Y
1.06Y = 72,254
Y = 7,254/1.06
Y = 6843.40
So, we see that the equilibrium level of GNP has risen from 5,900 to 6843.40

To find the new price level, substitute 6,843 into the AS curve:

P = 1.41 + .0001(6,843.40)
P = 2.094
So, we see that the government stimulus has raised the price level.

To solve for the components of GNP:

From the consumption function:

C = 100 + .9(6,843.40 - 100) - 20(2.094)
C = 6,127.18
So, we see consumption spending has increased from 5,280 to 6127.18.

From the investment function:

I = 400 - 40(2.094)
I = 316.24
So, we see investment expenditure has decreased from 320 to 316.24.

Therefore, we can see that yes, there has been a "crowding out" of investment. However, whereas the consumption function suggests that a rise in the price level will reduce consumer spending it has, in fact, increased. The negative influence of the rise in prices has been offset by the positive effect of the rise in income brought about by the increase in government expenditures.

## Question #2

Assume an aggregate supply curve of the sort:

P = 1 + .00125Y

### 1. Now, derive an aggregate demand function from the following data:

C = 50 + .8Yd - 10P
I = 200 - 30P
G = 200
T = 50

Ans: To derive the aggregate demand function, we proceed as before:

Y = C + I + G
Y = 50 + .8Yd - 10P + 200 - 30P + 200
Y = 50 + .8(Y - 50) - 10P + 200 - 30P + 200
Y = 410 + .8Y - 40P
.2Y = 410 - 40P
Y = 2,050 - 200P
which gives us the equation for the AD function.

### 2. Solve for equilibrium output and equilibrium price level.

Given:
AD curve = Y = 2,050
AS curve = P = 1 + .00125Y,
then
Y = 2,050 - 200(1 + .00125Y)
Y = 1,850 - .25Y
1.25Y = 1,850
Y = 1,480
which gives us the equilibrium level of GNP, and :

From the AS curve:

P = 1 + .00125(1,480)
P = 2.85 (br) which gives us the equilibrium level of prices.

### 3. Check the math by calculating C and I, and verifying that Y = C + I + G at equilibrium output.

C = 50 + .8(1,480 - 50) - 10(2.85)
C = 1165.5

I = 200 - 30(2.85)
I = 114.5

So: 1116.5 + 114.5 + 200 = 1,480 = Y, which checks

### 4. Suppose the government tries to deal with the current deficit by raising taxes from 50 to 100. What will be the effects on Y? on C? on I? Compare the change in consumption spending to the size of the tax increase. Which is greater? Explain this result briefly.

If the government raises taxes from 50 to 100, then the equation:

Y = 50 + .8(Y - 50) - 10P + 200 - 30P + 200
becomes:
Y = 50 + .8(Y - 100) - 10P + 200 - 30P + 200
Y = 370 + .8Y - 40P .2Y = 370 - 40P Y = 1,850 - 200P which is the new AD curve. Because the intercept has dropped we know the AD curve has been shifted to the left.

To find the effects on Y, we substitute as before and solve for Y:

Y = 1,850 - 200(1 + .00125Y)
Y = 1,650 - .25Y
1.25Y = 1,650
Y = 1,320
and we see that Y has dropped (as expected) due to an increase in taxes.

To find the effects on C and I we must find the effects on P, so we substitute the new Y into the AS curve:

P = 1 + .00125(1,320)_
P = 2.65(br) which we see has dropped.

From the consumption function:

C = 50 + .8(1,320 -100) - 10(2.65)
C = 999.5
so we see C has dropped.

From the investment function:

I = 200 - 30(2.65)
I = 120.5
and we see I has risen.

The drop in consumption is from 1165.5 to 999.5, or 166.0. This drop is greater than the tax increase of 50, because the tax increase causes GNP and income to decrease which further drops consumption.