Ranjit Dighe
Oct. 25-29, 1999

[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.]

Mon., Oct. 25, 1999


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 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 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 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 -- (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 = 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:

y = 0.333 + 1.333x

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

C = a + bQ,
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/DQ, also the marginal propensity to consume, MPC), and Q is real GDP (equivalently, output or real income). When the consumption function is given to us in equation form, it is easy to graph. [We graphed a consumption and a saving function in class. Someday those graphs will be in these web notes.]

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

S = -a + (1-b)*Q,
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/DQ, also the marginal propensity to save, MPS), and Q is real GDP.


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 week:
        Q = Aggregate output (GDP)
     AD = C + Ip + G + Xnet
     (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 + Ip + G + Xnet (Q = AD)
-- Note: Q = C + I + G (where I =actual investment, including unplanned inventory accumulation or decum.ulation), 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 "

Since we're assuming G = 0 and Xnet = 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 = Q and the fact that Q = C + S:
--          If AD = Q then, subtracting C from both sides,
          AD - C = Q - C
=>  C + Ip - C = S (substituting C + Ip for AD and S for Q-C)
=>               Ip = S (or, as it's usually written, S = Ip)

[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.]


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 -- but the cost of living the way that the typical American consumer is used to. That way of living, since America is one of the richest countries in the world, involves purchases of hundreds of different things, many of them luxuries. The "representative basket of goods" that the government uses in computing the Consumer Price Index (CPI) includes several hundred different goods and services.
-- 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 economists tend to think that the inflation rate overstates the actual rate of increase in the cost of living, is that the CPI cannot adequately account for improvements in the quality of goods and services. Today's cars, for example, cost a lot more than 1959's cars, but they're also much better cars -- in terms of "bang per buck," they're almost surely much better values, which is something that the inflation rate doesn't really address.
---- [To illustrate this point, we went through an exercise called the "Sears-Roebuck paradox." Suppose I give each of you a $1,000 gift certificate to be spent on merchandise in either the 1999 Sears-Roebuck catalog or their 1979 catalog. You can only buy things out of one catalog, not both. Prices have roughly doubled since 1979, so your $1,000 buys nearly twice as much stuff from the 1979 catalog. With this in mind, I asked the class which catalog they'd choose to buy from. In both sections, all but one student in each section chose the 1999 catalog. The reason: there's better stuff in the 1999 catalog, including many things that weren't even invented in 1979 and others that are vastly superior to their 1979 models (computers, TV's, VCR's, etc.). So it would seem that while the price level has doubled since 1979, the range and quality of products has more than doubled.]

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.


Fri., Oct. 29, 1999


* 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



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 = Iplanned).  For now, we are assuming G = 0 and Xnet = 0, so AD = C + Ip + G + Xnet becomes AD = C + Ip .

-- 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:  Qequil. = 800.)

We may also want to know the equilibrium levels and consumption and saving, as well as the multiplier.  Since we already know that Qequil. = 800, finding Cequil. and Sequil. is now easy:

      Cequil. = 100 + 0.75(Qequil.)              (plugging Qequil. into the consumption function)
                = 100 + 600
                = 700

Since S = Q - C always, we can now compute Sequil. as:

     Sequil. = Qequil. - Cequil. = 800 - 700 = 100

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

We could also have solved for equilibrium Q by using that equivalent equilibrium condition, S = Ip .  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 Ip and solve for Q:
                         S = Ip = 100
=>  -100 + 0.25Q = 100
=>              0.25Q = 200
=>                     Q = 800              (i.e., Qequil. = 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 + Ip , 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.]
    [We can also graph the saving and investment functions on a grid that has Q on the horizontal axis and S and I on the vertical axis, so as to show the equilibrium as the point where those two functions, S = -100 + 0.25Q and Ip = 100, intersect.  And, indeed, the point where they intersect is the point where Q = 800 and S = Ip = 100.]

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 Ip), and then we could just compute:
                                           multiplier = 800/200 = 4.