Economics 111.01

**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

The *Simple Multiplier *offers
an explanation of the process by which initial differences between ** planned** and

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

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

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

_{}_{}_{}_{}- And so on
.

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 that_{}has 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)**

** **

**D****Y = ****D****(Auto E)/(1-MPC)**

** **

**1/(1-MCP) =
Simple Multiplier**