The consumption function, also known as the Keynesian consumption function tells us the relationship between total output and consumption. In equilibrium it tells us how much all agents within the economy are consuming.
Suppose that we get the following consumption function:
C = 100 + 0.5Y
The value $100 represents autonomous spending. This is spending the consumers spend regardless of the size of their income. Things like food, shelter, healthcare could all be considered autonomous spending.
The letter Y stands for income (this is essentially synonymous with GDP) , which means the value 0.5Y is telling us that for each extra $1 that we make, we will spend an extra $0.5.
Now suppose that we were told than investment was $100 and that government spending was $50 and there were no taxes. Using our national income identity, we could write:
|Y = C + I + G|
And then we could replace C, I and G with the values that we have been given:
|Y = 100 + 0.5Y + 100 + 50|
Now what you do is you move the 0.5Y and solve for Y.
|Y - 0.5Y = 250
(1 - 0.5) Y = 250
Y = 250/0.5
Y = 500
Our equilibrium is simple how much output that we would produce with the consumption function and given level of investment and government spending.