*[Last revised on 2-November-1999, at 4:30 pm. The original
example of computing a line's slope and intercept from two points had a
couple mistakes in it, which have since been corrected.]*

**LECTURE 22**
**Mon., Oct. 25, 1999**

**O. IMPEDIMENTA**

**Today: The multiplier model (continued)**
**I. High-school algebra and the consumption and saving
functions**
**II. Investment**
**III. Equilibrium GDP in the multiplier model**

* POP QUIZ

**I. HIGH SCHOOL ALGEBRA AND THE CONSUMPTION AND SAVING FUNCTIONS**

**A quick review of some high-school algebra: **Algebra is just the
use of letters (like *x *and *y*) to represent numbers, especially
numbers whose values can vary (we call such numbers *variables*) or
whose values are unknown. In macroeconomics, the multiplier model is most
straightforwardly an algebraic model, where C represents consumption spending,
S represents savings, Q represents real GDP, etc.

Algebra and geometry naturally go together. In geometry, we often draw
two-dimensional graphs, with a
**horizontal axis (which we call the x-axis)**
and a

Any straight line can be written algebraically as the ** equation
of a line**:

where:
**x** **is the independent variable** (*x* does not
depend on *y*);
**y** **is the dependent variable** (*y* depends
on *x*);
**b** **is the vertical intercept** (or "*y*-intercept"
-- it is **the value of y at the point where the line crosses the vertical
axis **-- when *x *= 0, *y *= *b*);
**m** **is the slope*** *(the change in y that is
associated with a one-unit change in x).

-- Computationally, when we compare any two points on the
line,
** slope = (change in y)
/ (change in x)**

*[Refer to the appendix to Chapter 1 appendix if this isn't instantly
familiar to you.]*

*[I drew a generic graph and line with equation *y = b + mx*
in class.]*

**If we are given a line and** we know the values of any **two points
-- ( x_{1} , y_{1} ) and (x_{2}
, y_{2} ) **-- on the line, then we can find the
slope,

* m* = Dy
/ Dx = (*y*_{2} - *y*_{1
})/(*x*_{2}
- *x*_{1 }) = ( *y*_{1} - *y*_{2 })/(*x*_{1}
- *x*_{2} )

(It doesn't matter which point comes first; the answer will come out the same.)

**We can then use the value of the slope and the values (or "coordinates")
of x and y at either of those points to find b, by writing out the
slope equation with one of the given points and with the point (0, b),
which corresponds to the vertical intercept** (see example below).

-- Ex.: Suppose we have a line that includes the points (2,3) and (5,7)*.
[I drew the line on the board. Someday I'll include a graph of it in these
notes on the web.]* The slope of this line is:

* m* = Dy /
Dx

= (3-7) / (2-5)
* = *(-4) / (-3)

= 4 / 3

= 1.333

Knowing that the slope is 1.333, solve for *b* by plugging one
of the given points and the point (0, *b*) into the slope formula:

1.333 = (3 - *b*) / (2 - 0) = (3 - *b*) / 2

=> 2 * 1.333 = 3 - *b*

=> 2.667 = 3 - *b*

=> *b*
= 3 - 2.667 = 0.333

So now we have an equation, of the form *y = b + mx*, for the line
that runs through the two points (2,3) and (5,7). It is:

The consumption function, from last week, can be expressed as the equation of a line:

The saving function can also be expressed as the equation of a line:

**II. INVESTMENT**

**Total investment (I) is the sum of planned investment (I _{p})
and unplanned investment (I_{u}).**

--

In real life, planned investment is a function of many factors, including
real interest rates, expectations of future profitability (which would
make firms want to expand production), and the current level of production
(the more you produce, the more you wear out the physical capital stock
and need to replace it). **At its simplest,** however, **the multiplier
model assumes planned investment is fixed** at some constant level (e.g.,
I_{p} = 100). Since investment, as a constant, does not depend
on the level of GDP, we say it is a type of *autonomous *spending,
just like autonomous consumption (and, in later lectures, government spending
and exports and imports).

The level of government purchases of goods and services (**G**) also
depends on many factors, but for now, to keep things simple, we will **assume
it's fixed at zero** (anarchy!). We will **assume imports and exports,
and hence net exports, are zero **(autarky; no foreign trade).

**III. EQUILIBRIUM GDP IN THE MULTIPLIER MODEL**

Recall from last week:

Q = Aggregate output (GDP)

AD = C + I_{p} + G + X_{net}

(aggregate demand; same as *effective demand*,
or planned *aggregate expenditures)*

Equilibrium occurs when aggregate output (Q) equals planned aggregate
expenditure (AD), so **the economy is in equilibrium when** Q = C +
I_{p} + G + X_{net} (**Q = AD**)

-- Note: Q = C + I_{ }+ G (where I_{ }=actual investment,
including unplanned inventory accumulation or decum.ulation), but Y = C
+ I_{p} + G holds only when the economy is in equilibrium.

--> I = I_{p} if, and only if, economy is in equilibrium

--> I_{unplanned} = 0 "

Since we're assuming G = 0 and X_{net} = 0, then aggregate demand
is just consumption plus planned investment:

AD = C + I_{p}

**When the economy is in equilibrium, it will also be the case that
saving equals planned investment**. This follows from the original equilibrium
condition, AD = Q and the fact that Q = C + S:

-- If AD = Q
then, subtracting C from both sides,

AD - C = Q -
C

=> C + I_{p} - C = S (substituting C + I_{p}
for AD and S for Q-C)

=>
I_{p} = S (or, as it's usually written, **S = I _{p}**)

[I started on an example with real numbers, where we are given a consumption function and an investment level and then compute equilibrium GDP, but since we finished it up on Friday, I've moved the entire example to Friday's notes.]

*[This lecture finished up, and Monday's lecture started with, a demonstration
of how to do the first question on Problem Set 6 (Chapter 9, #6). Refer
to the solution sheet for Problem Set 6.]*

***

**PRINCIPLES OF MACROECONOMICS**
**WEEK 9, LECTURE 23**
**Wed., Oct. 27, 1999**

**Today: Problem set questions, past and present**

*[This class was devoted to better understanding two questions on
the last two problem sets and to going over the first question on Problem
Set 6. Again, all the work I did in class on that PS6 problem can be found
on the solution sheet to PS6.]*

A comment on the last question on PS4:

-- The question asked if the **inflation rate**, as computed in
this problem, accurately reflected changes in the cost of living. Many
of your answers took the expression "cost of living" much too literally
-- in common usage, **"cost of living" does not mean the cost of surviving**
-- i.e., of purchasing bread, water, and other basic necessities --

-- As for the problems with using the inflation rate to measure changes in the cost of living, refer to the solution sheet to PS4 for three key problems. A fourth problem, and a big reason why

----

The other old problem set question that I addressed was the quote about "voluntary" unemployment by Robert Solow, on PS5 -- the heavy sarcasm in that quote (he doesn't really think unemployment is voluntary, nor does he think Vietnam combat deaths were suicides) seems to have eluded many students, and therefore so did his main point. Solow was saying that unemployment, especially for skilled or educated people, involves painful dilemmas, like choosing between continued unemployment or "slumming" in a fast-food or other low-end job for which one is grossly overqualified. Refer to PS5 and its solution sheet for the original quote and more commentary.

*******

**PRINCIPLES OF MACROECONOMICS**
**WEEK 9, LECTURE 24**
**Fri., Oct. 29, 1999**

**O. IMPEDIMENTA**

*** Today: The multiplier and equilibrium GDP**

* **Exam will be on WED. (Nov. 3), not Mon.**
*** PS6 -- don't turn in; pick up PS6.SOL **and go over with fine-toothed
comb

* QUIZ

**I. THE MULTIPLIER AND EQUILIBRIUM GDP**

Let us start over with Monday's example of finding the equilibrium level of GDP, or Q, using real numbers:

First, recall that in equilibrium AD = Q (or S = I_{planned}).
For now, we are assuming G = 0 and X_{net }= 0, so AD = C + I_{p}
+ G + X_{net} becomes AD = C + I_{p} .

-- Assume the consumption function is C = 100 + 0.75Q and planned investment
(I _{p}) is 100. Aggregate demand (AD) is their sum:

C = 100 + 0.75Q

I = 100

_______________

AD = 200 + 0.75Q

In equilibrium, AD = Q, so the equilibrium condition here is:

Q = 200 + 0.75Q. We can solve for Q as follows:

--> 0.25Q = 200

--> Q = 800
(Thus, we can compactly write: Q_{equil.} = 800.)

We may also want to know the equilibrium levels and consumption and
saving, as well as the multiplier. Since we already know that Q_{equil.}
= 800, finding C_{equil.} and S_{equil.} is now easy:

C_{equil.} = 100 + 0.75(Q_{equil.})
(plugging Q_{equil.} into the consumption function)

= 100 + 600

= 700

Since S = Q - C always, we can now compute S_{equil.} as:

S_{equil.} = Q_{equil.} - C_{equil.}
= 800 - 700 = 100

Recalling from Monday that in equilibrium S = I_{p }, we can
verify our work by checking if this condition in fact holds. Checking,
we see immediately above that S_{equil.} = 100, and it was given
that I_{p} = 100, so indeed, our work looks correct.

We could also have solved for equilibrium Q by using that equivalent
equilibrium condition, S = I_{p }. That would involve first
deriving the saving function from the consumption function, as follows:

S = Q - C

= Q - (100 + 0.75Q)

= -100 + 0.25Q

Then we would set S equal to I_{p }and solve for Q:

S = I_{p }= 100

=> -100 + 0.25Q = 100

=>
0.25Q = 200

=>
Q = 800
(i.e., Q_{equil.} = 800)

We can also portray this equilibrium graphically *[see Figures 9-9
and 9-10 in McConnell's textbook, which are similar to the ones I drew
on the blackboard. The first graph I drew, with Q on the horizontal
axis and aggregate demand (AD) on the vertical axis showed an AD function,
equal to C + I _{p} , which in turn was equal to 200 + 0.75Q.
The equilibrium level of GDP, at which AD = Q, is given by the point where
that AD function, AD = 200 + 0.75Q, intersects the line AD = Q, which is
given by a 45-degree line from the origin (since a 45-degree line has a
slope of 1 and a vertical intercept term of 0, just like the generic line
y=x). The lines AD = 200 + 0.75Q and AD = Q intersect at the point
(800,800), since in this example both AD and Q are 800 when the economy
is in equilibrium.]*

Note that autonomous spending ($100 of autonomous consumption plus $100
of investment) was only $200, yet equilibrium Q is $800. This is
possible because each dollar of autonomous spending gets "multiplied,"
because it touches off a whole chain of consumption, whereby the person
receiving the dollar spends 75 cents, and the person receiving that 75
cents spends 75% of that, and so on. This is the *multiplier process*.

We define the multiplier as follows:
**MULTIPLIER: the change in equilibrium GDP that eventually
results from a $1 increase in autonomous spending;**
**multiplier = D{equilibrium GDP} / D{autonomous
spending}**

It is also the change in *total spending* (which is the sum of
autonomous spending plus induced consumption) that eventually results from
a $1 increase in autonomous spending.

-- **We can compute the multiplier in one of two ways:**

**1. (The easy way:) multiplier
= 1/(1-MPC)**
**
**-- (Granted, I have not yet shown you *why* the multiplier = 1/(1-MPC),

but I will on Mon., Nov. 1.)

-- So in this example, since the MPC is 0.75,

the multiplier is 1/(1-.75) = 1/(.25) = 4.

**2. multiplier = (equilibrium GDP)/(total
autonomous spending)**

-- In this example, once we've computed equilibrium GDP, we would

then note that autonomous spending is $200 ($100 in autonomous C

plus $100 in I_{p}), and then we could just compute:

multiplier = 800/200 = 4.