To calculate equilibrium real GDP (or income), we need a starting point. Let's assume a very simple world where the price level is fixed, capital doesn't depreciate, there are no indirect business taxes, and all income earned today is received today. This makes GDP, Net Domestic Product, National Income and Personal Income all equal to one another (which means we can ignore national income accounts like NDP, NI and PI). If we deduct personal taxes from GDP, we have disposable income.

That said, now we need a set of equations which describe the economy:

(a) C = 0.8(DI) + 480
(b) I = 1,000
(c) G = 800
(d) T = 100
(e) X = 0
(f) M = 0
(g) DI = Y - T

(C = consumption spending, DI = disposable income, I = investment expenditure, G = government spending, T = tax revenue, X = exports, M = imports, Y = real GDP)

These equations tell us that consumer spending would be at $480 if consumers had no income at all, and that consumers spend 80¢ of every extra additional dollar they receive in disposable income. We also observe that $1,000 is spent on investment (purchase of machinery and equipment), and that the government spends $800, while receiving $100 in taxes. We note further that there is no foreign trade.

Step 1 - Recall missing equations
a. To solve for equilibrium real GDP, we start with three equations:

(g) DI = Y - T
(i) AE = C + I + G + (X - M)
(j) AE = Y

(Y and DI are defined above, but AE is aggregate expenditure, the sum of all expenditures)

The first two equations (g and i) are true by definition. The last equation, however, is only true at the equilibrium. When AE > Y or AE < Y, we have unintended changes in inventories that result - which means we wouldn't be at equilibrium.

Step 2 - Set up the problem
a. Substitute equation (g) into the (a) equation.

(a') C = 0.8(Y - T) + 480

b. Take equations (a'), (b), (c), (d), (e) and (f), and substitute them into the (h) equation.

AE = [0.8(Y - 100) + 480] + 1,000 + 800 + 0 - 0

c. Using the equation (i), AE = Y, replace the AE with a Y variable.

Y = [0.8(Y - 100) + 480] + 1,000 + 800 + 0 - 0

Step 3 - Solve for equilibrium real GDP (which we'll call Y*)
For some, the algebra here is simple. Rather than assume this to be the case for all, let's include as many of the steps as possible. If you feel that you have a good enough grasp of the algebra at this point, then feel free to skip to the solution below.

a. Remove all parentheses by multiplying where necessary.

Y = 0.8Y - 80 + 480 + 1,000 + 800 + 0 - 0

b. Simplify the RHS of the equation

Y = 0.8Y + 2,200

c. Rearrange the equation

Y - 0.8Y = 2,200

d. Factor out the variable Y

(1-0.8)Y = 2,200

e. Simplify

0.2Y = 2,200

f. Divide both sides by 0.2

Y = 2,200/0.2

g. Simplify again

Y* = 11,000

Equilibrium real GDP is $11,000 this period. Of course, now that we have a solution for equilibrium real GDP, or Y*, we can ask all sorts of interesting questions like:

• What happens if investment goes up by $200?
  
• What happens if the government raises total taxes by $100?