A Simple Algebraic Model of the Simple (Keynesian) Multiplier
If we treat the expression as an accounting identity, we expect that the equality will always hold. The proper measurement of aggregate expenditure in any given period will always equate with the sum of all elements in the Output-Expenditure model.
If we treat the Output-Expenditure model as a statement of the equilibrium relationship among all the elements of the model, then we have to make a distinction between planned expenditures and actual expenditures. The accounting identity describes the relationship among actual expenditures. The equilibrium model specifies the necessary and sufficient conditions for equilibrium in the simple income determination model. Equilibrium obtains only when planned equals actual.
The Simple Multiplier offers an explanation of the process by which initial differences between planned and actual produce a change in the equilibrium level of real GDP.
We begin with a basic Consumption Function: , where , the marginal propensity to consume out of . Substituting for C in the Output-Expenditure model, we get . We use the zero subscript to identify variables that are exogenous, determined outside the model and treat by the model as constants. C is now endogenous, determined within the model; C is now a function of . The elements of autonomous expenditure in our rewritten basic model are , , and .
Rearranging the terms in the equilibrium equation, we get . Factoring out and dividing through by , we get . We can rewrite the equation for as .
We can now asked what would happen to the equilibrium level of or if there were a change in one of the elements of autonomous expenditure. For example, if the level of planned Investment were to change, what would happen to the equilibrium level of or ? [Remember, because , , and are exogenous, the value of each is determined outside the model and changes in value of inside the model do not influence the values for planned autonomous expenditure.]
Consider the following simple example, leaving out net exports.
Assume, , , and. . If we posit an increase in planned I from 100 to 200, we get a new equilibrium level of that is larger than the old equilibrium level of and the difference between the two is more than 100, the increase in planned I.
From the simple multiplier, we have or . The new value for is 800. The change in the value of is a multiple of the change in the planned expenditure on an element of autonomous expenditure.
Consider this step-by-step story of the change in resulting from a change in autonomous expenditure.
Planned I increases by $50 billion, and .
If we sum up all the incremental changes that have resulted from the initial changes in planned , we get $250 billion. The process is simple and straightforward.
1. A change in leads to a change in . Since is an element of , increases by the change in .
2. A change in leads to a change in . Since is an element of , increases by the change in .
3. A change in leads to a change in . Since is an element of , increases by the change in .
4. A change in leads to a change in . Since is an element of , increases by the change in .
5. And so on, until becomes infinitesimally small.
An interesting feature of this process is the change that takes place in the equilibrium level of . We see that has increased by $50 billion, that has increased by $200 billion, and thathas increased by $50 billion. The increase in planned has resulted in a corresponding change in actual .
We can now rethink both the accounting identity for and the equilibrium conditions for in our simple, closed-economy model. We use to identify the way output is divided among end users. We can also say that , which identifies how households divided up income among alternative uses. If the household can pay taxes, consume or save, then expenditures on output should equal the sum of the income allocated to each of the alternatives uses available to households. If we assume that is exogenous and does not change in our simple model with the changes in the level of income, the when the new equilibrium is reached.
We have $50 billion of the increase in not consumed by households and an additional $50 billion in output committed to expenditures on planned by businesses.
Our macroeconomic households pay taxes and use the remaining income (personal disposable income) to consume or to save. The accounting identity tells us that either is consumed by household, invested by businesses or used by government. If we make the simplifying assumption that households, being the ultimate owners of all factors of production, make the basic decision about the dividing disposable income between and , then holds both as our basic accounting identity and as a statement of conditions necessary for equilibrium in our simple model of income determination.
Y = C + I + G + (X-M) = GDP
Consumption Function: C= a + (MPC)*Y
Y = a + (MPC)*Y + I + G + (X-M)
Y (MPC)*Y = a + I + G + (X-M)
Y*(1 MPC) = a + I + G + (X-M)
a + I + G + (X-M) = Autonomous E
Y = (Autonomous E)/(1-MPC)
DY = D(Auto E)/(1-MPC)
1/(1-MCP) = Simple Multiplier