Assuming that both exports and imports are zero, the demand of goods is the sum of consumption, investment, and government spending: $$Z = C + I +G$$
replacing and , from the Composition of GDP, we get that: $$ Z = c_0 + c_1(Y-T) + \bar I + G$$
The demand for goods depends on income , taxes , investment , and government spending .
Let's now turn to equilibrium in the goods market, and the relation between production and demand. To make everything a little bit more simple, we will consider inventory investment to always equal to zero, and equilibrium in the goods market requires that production to be equal to the demand for : $$ Y = Z$$This equation is called an equilibrium condition. Models include three types of equation: identities, behavioural equations, and equilibrium conditions.
Replacing by its decomposition we get that $$ Y = c_0 + c_1(Y-T) + \bar I + G$$
This equation represents: In equilibrium, production , is equal to demand. Demand in turn depends on income, , which is itself equal to production.
From the equation we can see that the equilibrium point is $$ Y= \frac1{1-c_1}(c_0 + \bar I+ G -c_1 T)$$
The term is that part of the demand that doesn't depend on output. For this reason it's called autonomous spending.
We can't make sure that the autonomous spending is always positive, but we can say it is very likely. The main thing is how to control the term . Suppose that the government is running a balanced budget, taxes equal government spending. If , then trivially get that , and so is autonomous spending. Only on the weird case that government were running a very large budget surplus, if taxes were much bigger than spending, could the autonomous spending can be negative.
We call the multiplier, because it multiplies the autonomous spending. We see that the multiplier is always greater than . This makes it so that any change of the autonomous spending is amplified.