Equilibrium GDP is that output level at which the total amount of goods produced, GDP, is just equal to the total amount of goods purchased. In a world with no government or foreign sector, the amount purchased is C + Ig. Additionally, GDP and disposable income (DI) are the same, so that our earlier relationship between consumption and DI holds for GDP as well: C = a + bDI = a + bGDP where a is autonomous consumption and b is the marginal propensity to consume.
It is more convenient and traditional to use "Y" to represent GDP. Equilibrium GDP is then found as the solution to the following: Y = C + Ig = a + bY + Ig.
Subtracting bY from both sides of the equation, we obtain Y -bY = a + Ig. Next, factor out the common Y term: Y(1 -b) = a + Ig. Since the MPC is less than one, we can divide both sides by the factor (1 -b) to obtain our result: Ye = x (a + Ig).
Following the example in the text, the consumption schedule is C = 97.5 + .75Y and Ig = 20 billion. Hence, equilibrium GDP is Ye = x (97.5 + 20) = 4 x 117.5 = 470 billion. We can check our results by computing consumption at a GDP of $470: C = 97.5 + .75 x 470 = 450. Adding this to planned investment of 20, we see that planned purchases of $450 and $20 do indeed total the planned output of $470.