WEEK 11 (LECTURES 28-30)
April 10-14, 2000

[This week we began work on aggregate demand and the multiplier model.  The lectures go with Case & Fair's Chapter 9.]

Mon., April 10, 2000

* Today:
I. Aggregate demand
II. The multiplier model (begin)


New concept:
PLANNED INVESTMENT (Ip): total business expenditures on plant and equipment (i.e., on capital goods), plus planned production of inventories (inventory investment)

INVESTMENT (I): total business expenditures on plant and equipment, plus (total) production of inventories.

Ip is not always equal to I

I - Ip = UNPLANNED INVENTORY INVESTMENT, which can be either positive (unintended inventory accumulation; you produced too much) or negative (unintended inventory decumulation; you produced too little and had to run down your inventories)

Recall: GDP (Y), by the product/expenditure approach is:

Y = C + I + G + EX - IM
New concept:
AGGREGATE DEMAND (AD): The total quantity of goods and services demanded (i.e., purchased).
AD = C + Iplanned + G + EX - IM
-- To repeat: Planned investment does not include "unintended inventory accumulation". If firms produce goods (say, Edsels) but can't sell them, those goods are counted in GDP, as (unintended) inventory investment (I), but they are not part of aggregate demand, because, plainly, nobody was demanding those goods.
-- The difference between AD and GDP is the difference between planned investment and total investment. In a word, that difference is inventories. Unintended inventory accumulation counts total gross investment (I) but not toward planned investment. Unintended inventory investment can also be negative (we would call it unintended inventory decumulation), if AD exceeds output and firms meet the excess demand by selling off goods that they'd been planning to keep as inventories for the future.
-- AD is often referred to as effective demand, notably by Keynes.

AGGREGATE EXPENDITURES MODEL: a model in which GDP is ultimately determined by aggregate demand, and equilibrium GDP is the level of GDP where aggregate expenditures (AD) equal aggregate production (output).
-- also known as the MULTIPLIER MODEL


The term multiplier refers to the way that an initial increase in aggregate expenditures (C, I, G, net EX) causes a ripple effect that leads to more and more spending and raises GDP by a multiple of that initial increase in spending. The main reason why this happens is because when you spend money, the person who receives that money from you as payment will turn around and spend some of it. And the same thing will happen when that person spends his money -- the person he paid the money to will turn around and spend some it, too. The chain of spending continues until there's nothing left to spend.

Key concept: the marginal propensity to consume (MPC) -- the fraction of an extra dollar of a person's disposable income that the person will spend on consumer goods.

How does this multiplier work? A hypothetical example:
-- First off, suppose everyone has the same MPC, 0.75
-- I withdraw $100 from my savings acct and spend it all on a leather jacket
-- Biff, the leather jacket salesman, since he has MPC = 0.75, spends $75 (on a hat)
-- Cheryl, the hat salesperson, spends 0.75*$75 = $56 (on a puppy)
-- Ralph, the dog breeder, spends (0.75)2*$75 = $42 (on a haircut)
-- Olga, the hairstylist, spends (0.75)3*$75 = $32 ...
-- and so on. Note that each subsequent amount spent is 75% of the previous amount. After many more iterations the amount spent will be so tiny (75% of a fractional cent) that we can forget about it. But by then the total increase in spending will have been quite large.

Numerically, let's keep track of the total, cumulative increase in spending that results from an injection of $100 into the spending stream. We have assumed MPC = 0.75 and that it's the same for everyone.
-- I spend $100 on a leather jacket. The leather jacket vendor spends $75 (.75*$100) on a hat, and so on...
->    Increase in equilibrium GDP
        = Increase in total spending
        =     $100
            + (.75)($100)
            + (.75)(.75)($100)
            + (.75)(.75)($100)
            + ...
        = $100 * (1 + .75 + .752 + .753 + ...)
                                  (GEOMETRIC SERIES -- converges to a finite number, according to a simple formula)

        = $100 * [1/(1-.75)]
        = $400

Note that in this example .75 is the MPC.

multiplier = 1/(1-MPC)
Note: the multiplier model is a Keynesian economic model -- that is, it was first proposed by John Maynard Keynes, in The General Theory. (The book devotes three whole chapters to the marginal propensity to consume and the multiplier.) The multiplier model is a model of output determination -- it tells you what the level of output (GDP) will be, based on the behavior of consumption, planned investment, and the other components of aggregate demand.

Consumption accounts for about two-thirds of both GDP and aggregate demand, so let's start by examining the behavior of consumption. First, some notation:
Y  = GDP
C  = consumption
S  = saving
DI = disposable income (i.e., after-tax income)

All of a person's disposable income goes toward consumption and saving, i.e.,
DI = C + S

For now, we will assume there are no taxes, so Y = DI and
C = Y - S

Although in the real world there are several factors that determine a household's or a society's consumption, this model focuses on just two -- (1) everyone's basic subsistence needs (food, clothing, shelter, etc.), which each of us would somehow provide for even if we had no income, by borrowing or living off our past savings; and (2) income (people consume more when they have more income to spend). We break those two types of consumption down into (1) autonomous consumption and (2) induced consumption.

Thus we divide people's consumption into two parts:

Consumption = autonomous consumption + induced consumption
                            --------------------------------     ----------------------------
                                          |                                                |
                               a constant;                                         rises as income rises
                               unaffected by changes in income

or symbolically:

                    C = a + bY
                                  b = marginal propensity to consume (MPC)
                                  b = slope of the consumption function when we graph it

(a is autonomous consumption; bY is induced consumption.)

We assume that:
          a > 0 (people's autonomous consumption is some positive number)
    0 < b < 1 (the MPC is positive but less than 100% of people's income)

The above equation is a consumption function -- a simple linear (straight-line) equation that depicts consumption as a function of disposable income. Since we're assuming no taxes for now, the consumption function shows consumption as a function of total income, or GDP (Y).
-- [In class I drew a generic consumption function, with a slope of b and intersecting the vertical axis at a.  A nearly identical picture appears in Case & Fair's Figure 9.4, on page 180.]

Equivalently, we can use the two above equations (Y = C+S and C=a+bY) to derive a saving function:

S = Y - C
   = Y - (a+bY)
   = -a + (1-b)Y
        |        |
        |      1 - b = marginal propensity to save (MPS) = the slope of the saving function
      -a = autonomous saving

Note: MPC + MPS = 1
(just as consumption + saving = income. There is a mathematical connection between the two equations.)

-- [Technical note:  For those of you who've had calculus, the connection is this:  If you take the derivatives of both sides of the equation

C + S = Y
with respect to Y, you'll get MPC + MPS = 1.]

-- Verify: b + (1 - b) = b + 1 - b = 1

----> Since the multiplier = 1/(1-MPC) and MPS + MPC = 1 (--> MPS = 1 - MPC), then it's also true that the
          multiplier = 1/MPS

-- [I also drew the saving function on the board, with a slope of 1-b and intersecting the vertical axis at -a.  See the bottom graph of Case & Fair's Figure 9.6 (page 183) for a similar diagram.]

(Just why we bother to derive a saving function will become clearer in the next lecture.)


Wed., April 12, 2000


* Pick up solutions to Problem Set 7 (graphs only; rest of solution sheet is posted on the Internet)

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


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 are unknown. In macroeconomics, the multiplier model is most straightforwardly an algebraic model, where C represents consumption spending, S represents savings, Y 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 vertical axis (the y-axis). We would say that such a graph is in (x, y) space, since any point on the graph can be written as (x, y), based on the values of x and y at that point.

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

y = b + mx,

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)
                                                                            = Dy / Dx                                 (Recall: D = "change in")

-- [Refer to Case & Fair's 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 -- (x1 , y1 ) and (x2 , y2 ) -- on the line, then we can find the slope, m, of that line. The slope will be:

m = Dy / Dx = (y2 - y1 )/(x2 - x1 ) = ( y1 - y2 )/(x1 - x2 )
(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). Then we have everything we need to write the line in equation form (y = mx + b).

-- 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 = (3-7) / (2-5)
              = (-4) / (-3)
              = 4 / 3
              = 1.33

Knowing that the slope is 4/3, solve for b by plugging one of the given points (say, (2,3)) and the point (0, b) into the slope formula:

        4/3 = (3 - b) / (2 - 0) = (3 - b) / 2

  => 2 * (4/3) = 3 - b
  =>        8/3   = 3 - b
  => b = 3 - 8/3 = 9/3 - 8/3 = 1/3 = 0.33

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:

y = 0.33 + 1.33x

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

C = a + bY,

where C is the dependent variable (it goes on the y-axis), a is the vertical intercept (and autonomous consumption), b is the slope (DC/DY, also the marginal propensity to consume, MPC), and Y is real GDP (equivalently, output or real income). When the consumption function is given to us in equation form, it is easy to graph.

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

S = -a + (1-b)*Y,

where S is the dependent variable (it goes on the y-axis), -a is the vertical intercept (and autonomous saving), 1-b is the slope (DS/DY, also the marginal propensity to save, MPS), and Y is real GDP.

-- [We graphed a consumption function and a saving function in class.  See Case & Fair's Figure 9.6 (page 183) for similar examples.]


Total investment (I) is the sum of planned investment (Ip) and unplanned investment (Iu).
-- Unplanned investment is unintended inventory production. For example, if a company produces 100 cars, expecting to sell all of them this year but only sells 50 this year, then the remaining 50 are counted in GDP as unplanned inventory investment. (The first 50 are counted as consumption.)

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., Ip = 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).


Recall from last time:

Y = Aggregate output (GDP)
AD = C + Ip + G + EX - IM
(aggregate demand; same as effective demand, or planned aggregate expenditures)

Equilibrium occurs when aggregate output (Y) equals planned aggregate expenditure (AD), so the economy is in equilibrium when Y = C + Ip + G + EX - IM (Y = AD)
-- Note: Y = C + I + G (where I =actual investment, including unplanned inventory accumulation or decumulation), but Y = C + Ip + G holds only when the economy is in equilibrium.
----> I = Ip                  if, and only if, economy is in equilibrium
----> Iunplanned = 0     if, and only if, economy is in equilibrium

Since we're assuming G = 0 and EX - IM = 0, then aggregate demand is just consumption plus planned investment:

AD = C + Ip

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 = Y and the fact that Y = C + S:
-- If AD = Y then, subtracting C from both sides,
       AD - C = Y - C

===> C + Ip - C = S (substituting C + Ip for AD and S for Y-C)

===> Ip = S (or, as it's usually written, S = Ip)

An example of finding the equilibrium Y, with real numbers:

-- Assume no government (G=0) and no foreign sector (EX=IM=0) and that consumption and investment functions are as follows:
       C = 100 + 0.75Y
       Ip = 100

--> To find the equilibrium level of Y, plug those equations for C and Ip into the equation Y = C + Ip + G + EX - IM.

     C = 100 + 0.75Y
     Ip = 100
     G = 0
   EX = 0
 - IM = 0
      Y = 200 + 0.75Y

--> 0.25Y = 200
-->         Y= 800

--> C = 100 + 0.75(Y)
          = 100 + 600
          = 700

--> S = Y - C
          = 800 - 700
          = 100

       (S=Ip in equilibrium; and, indeed, Ip = 100, too)


Fri., April 14, 2000


I.   The multiplier and equilibrium GDP
II.  Disequilibrium: inflationary, recessionary gaps
III. Fiscal policy (begin)



Example from last time:
-- Find equilibrium Y, C, and S when:
    C = 100 + 0.75Y
    Ip = 100
    G = 0, T = 0, EX = 0, IM = 0
-- Equilibrium condition is: Y = C + Ip + G + EX - IM
-- Solve for Y, end up with:
       In equilibrium,
           Y = 800
           C = 700
           S = 100

We can represent this graphically, a la Case & Fair's Chapter 9, as follows:

Aggregate demand (AD) function:
       AD = C + Ip + G + EX - IM
              = (100 + 0.75Y) + 100 + 0 + 0 - 0
              = 200 + 0.75Y

Equilibrium condition:
       AD = Y (graphs as 45-degree line from the origin, because y-intercept is 0 & slope is 1.  On any graph where the vertical and horizontal axes are scaled the same, a line that extends out from the origin and has a slope of 1 will be a 45-degree line.)

-- [In class I drew the functions AD = C + Ip + G + EX - IM and AD=Y.  The point where they intersect corresponds to equilibrium GDP, because at that point AD=Y.  The lower graph in Case & Fair's Figure 9.8 (page 187) is very similar.]

In equilibrium, it's also true that S = Ip

We can represent that graphically, too. First, we must derive the saving function:
      S = Y - C
         = Y - (100 + 0.75Y)
         = 1.00Y - 100 - 0.75Y
         = -100 + 1.00Y - 0.75Y
         = -100 + 0.25Y

From before, Ip = 100.

-- [I drew graphs of the saving function S = -100 + 0.25Y and the planned investment function Ip = 100 and their intersection, which again corresponds to equilibrium GDP, at 800.  See Case & Fair's Figure 9.10 (page 189) for a similar diagram.]

-- Note that the ratio of equilibrium Y to "autonomous" spending is 800/200 = 4. The multiplier is also 4:

multiplier = 1/(1-MPC) = 1/(1-.75) = 1/.25 = 4
Four formulas for the multiplier:
(1) multiplier = 1/(1-MPC)
(2) multiplier = 1/MPS
(3) multiplier = (equilibrium GDP)/(total autonomous spending)
(4) multiplier = (change in equilibrium GDP) / (change in autonomous spending).

The multiplier is probably most easily calculated as 1/(1-MPC).
As long as you recall that in a consumption function, C = a + bY,b is the MPC, then the computation is straightforward:
-- Ex.: C = 100 + 0.75Y
                             MPC = 0.75

            multiplier = 1/(1-MPC) = 1/(1-0.75) = 1/0.25 = 4

More generally and more realistically, investment and import spending would also depend on the level of income, as might the government's spending, which historically has risen as GDP has risen. In that case we would also speak of a marginal propensity to invest, a marginal propensity to import, and the government's marginal propensity to spend. And the multiplier would be equal to

1 / (1 - MPC - Marginal Propensity to Invest - Government's Marginal Propensity to Spend + Marginal Propensity to Import)

(No, this extended multiplier will not be on the exam. But you should know the simple multiplier, 1/(1-MPC), inside and out.)

The quickest way to find the multiplier and equilibrium GDP:
(1) Compute the multiplier as 1/(1-MPC)
(2) Add up the total autonomous spending (= autonomous C + I + G + EX - IM)
(3) Yequil. = (total autonomous spending) * (multiplier)

Ex.: Suppose you are asked to find equilibrium GDP given the following consumption and investment functions, and that government spending, taxes, and net exports are all zero:

C = 100 + 0.9Y
Ip = 100

Step (1): multiplier = 1/(1-MPC) = 1/(1-0.9) = 1/0.1 = 10

Step (2): total autonomous spending     = Cautonomous + Ip + G + EX - IM

= 100 + 100 + 0 + 0
= 200
Step (3): Yequil. = 10 * 200 = 2000