PRINCIPLES OF MACROECONOMICS
Ranjit Dighe
WEEK 10 (LECTURES 25 & 26)
Nov. 1-5, 1999

[Last revised on Sun., Nov. 14, at 6:45 pm.]

LECTURE 25
Mon., Nov. 1, 1999

Today: A last look at the multiplier model
I.    Deriving the multiplier; finding Qequil. the quick way
II.  Government and taxes in the multiplier model
III. Net exports in the multiplier model
IV. Disequilibrium: recessionary, inflationary gaps
 

I.   DERIVING THE MULTIPLIER; FINDING Qequil. THE QUICK WAY

Where we left off: Three formulas for the multiplier:

(1) multiplier = (change in equilibrium GDP) / (change in autonomous spending).

(2) multiplier = 1/(1-MPC)

(3) multiplier = 1/MPS

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

-- Ex.: C = 100 + 0.75Q
                               |
                           MPC = 0.75

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

Q: Why is the multiplier = 1/(1-MPC)? How does this multiplier work?

A: Sticking with that same numerical example, and my chain-of-consumption-spending example from Fri., Oct. 22 (the leather jacket, the hat, the dog, the haircut, ...), 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)]
                                                              -----------
                                                                      |
                                                                     4
                                              = $400

(Note that in this example .75 is the MPC; also, since MPC+MPS = 1, then 1-.75 = 1-MPC = MPS.)

In this example, an increase in autonomous spending of $100 gives rise to a $400 increase in equilibrium income.  Thus the multiplier is 4 and we got it through the formula
                                                         multiplier = 1/(1-MPC).

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 + Xnet)
(3) Qequil. = (total autonomous spending) * (multiplier)

[We went through several examples in class. I will include just one here.]

Ex.: Suppose you are given the following consumption and investment functions, and that government spending, taxes, and net exports are all zero:
    C = 100 + 0.9Q
    Ip = 100
Step (1): multiplier = 1/(1-MPC) = 1/(1-0.9) = 1/0.1 = 10
Step (2): total autonomous spending  = Cautonomous + Ip + G + Xnet
                                                           = 100 + 100 + 0 + 0
                                                           = 200
Step (3): Qequil. = 10 * 200 = 2000
 

II.  GOVERNMENT AND TAXES IN THE MULTIPLIER MODEL

... complicate things only a little bit.  A positive level of government spending (e.g., G = $100) gets added onto aggregate demand (AD), which one could then use to find equilibrium Q by setting AD=Q and solving for Q.  Or if one is using the quickest way, shown above, to find equilibrium output, then all that changes is that we're adding a positive number, and not just zero, for G when we total up autonomous spending.  When we add in government spending (i.e., go from G = $0 to some positive value for G), equilibrium GDP goes up, because aggregate demand and autonomous spending have gone up.

Taxes (T) complicate things a bit further -- in particular, the consumption function will now be different because it will no longer be true that disposable income (DI, or after-tax income) equals GDP (Q). Instead,
                                          DI = Q - T
and the consumption function (which is really a function of DI, because you can't consume or save the part of your salary that gets taxed, since you never see that money) becomes
                                            C = a + b(DI)
                                                = a + b(Q-T),
which will necessitate a couple more algebraic steps when we solve for the multiplier and the equilibrium level of Q.

In this simplified model, we typically assume that taxes are levied in one of two ways:
(1) a fixed, lump-sum tax (e.g., T = $100), which does not depend on income.  Everyone pays the same dollar amount and it adds up to T.  A lump-sum tax is not exactly fair (should millionaires and paupers really pay the same dollar amount?), but it is very easy to deal with.
(2) a flat-rate income tax (e.g., T = 0.2Q), in which every dollar of income is taxed at the same rate.

In both cases, going from no taxes (T = $0) to some positive level of taxes will lower equilibrium GDP, because it will lower consumption.  With less of their income available to spend or save, people will consume less, thus lowering aggregate demand and lowering GDP.
 

III.  NET EXPORTS IN THE MULTIPLIER MODEL

... complicate things hardly at all, because so far we're not assuming a marginal propensity to import (or to export.  Realistically, if there is a marginal propensity to consume, there should be a marginal propensity to import, because some of our consumption spending is on imports.  But we're keeping it simple for now.  Exports do not really depend on the level of U.S. GDP, but on the GDP's of other countries that would buy our exports, so there's no marginal propensity to export).  Instead, we just assume that net exports are fixed and do not depend on the level of income (Q), so net exports are just another form of autonomous spending.  Once again,
                 total autonomous spending = Cautonomous + Ip + G + Xnet
 

IV. DISEQUILIBRIUM: RECESSIONARY, INFLATIONARY GAPS

Q: What happens when AD is not equal to Q?
A: The economy is in disequilibrium -- it will either be in a recession or it will be in an inflationary boom.

If AD < Q: The economy is in a recession or "recessionary gap" -- goods will be piling up on shelves (unintended inventory accumulation)
--> What will happen? Firms will cut back on their production, try to sell off those inventories. Eventually, equilibrium will be reached at a lower level. Production will adjust to that lower-than-expected level of aggregate demand.

If AD > Q: The economy has an "inflationary gap" -- since demand outstrips production, firms will sell off much of their inventories of goods that they had been planning to sell later. With a large excess demand for goods, prices of those goods will be bid up, generating inflation. Firms will expand their production to meet the higher-than-expected demand, and eventually equilibrium will be reached at a higher level.

***

Wed., Nov. 3, 1999 -- SECOND MIDTERM EXAM

***

PRINCIPLES OF MACROECONOMICS
WEEK 10, LECTURE 26
Fri., Nov. 5, 1999

Today: The Aggregate Demand-Aggregate Supply (AD-AS) Model (begin)
I.    Re-introducing the price level (P)
II.  Aggregate demand (begin)
 

I.    REINTRODUCING THE PRICE LEVEL (P)

The multiplier model is a fixed-price model -- it never mentions prices, and implicitly assumes the price level (P) is fixed at all times.

The most widely used macro model, at the introductory-course level, is the aggregate demand-aggregate supply (AD-AS) model, in which aggregate demand (AD) and aggregate supply (AS) curves are plotted in (Q,P) space. That is, Q (real GDP) is on the horizontal (x) axis and P (the price level) is on the vertical (y) axis.
-- [Refer to the left-hand graph at the top of McConnell's page 233 to see a typical AD-AS diagram.]
-- The AD curve slopes downward, just like a micro demand curve; and the AS curve slopes upward, just like a micro supply curve. The similarity to micro supply-and-demand models ends there, however, because the factors driving the shapes and shifts of these curves are different. First, note that P represents the aggregate price level -- not, as in micro supply-and-demand diagrams, the price of a specific commodity (say, bananas) with all other prices held constant. The AD curve (or "schedule") tell us how the aggregate quantity demanded of all goods and services (by households, firms, government, and foreigners) is affected by the price level. The AS curve (or "schedule") tell us how the aggregate quantity supplied of all goods and services (by firms) is affected by the price level.
-- The AD-AS model, unlike the multiplier model, is a variable-price model, because it realistically assumes that the price level can and does change from time to time.
 

II. AGGREGATE DEMAND (AD) AND THE PRICE LEVEL

Aggregate demand (AD) has an inverse relationship with the price level (P)
-- A lower price level is good for AD (i.e., increases the sum of C, Iplanned , G, and Xnet)
-- A higher price level is bad for AD (i.e., decreases the sum of C, Iplanned , G, and Xnet)
-- The AD curve slopes downward.
-- [See Figure 11-1 on McConnell's page 222 to see a typical AD curve.]

There are three reasons why the AD curve slopes downward (or, equivalently, why a lower price level raises AD):

(1) Real-wealth effect
-- A lower price level raises the real value of the money in people's pockets and bank accounts, and raises real wealth in general. (This is the flip side of how inflation -- a higher price level -- lowers people's real wealth.) Wealthier people consume more, so the increase in aggregate real wealth raises household consumption.

* (2) Interest-rate effect
-- (This one has the asterisk {*} next to it because it's the most important of the three reasons.)
-- A lower price level will reduce interest rates, which will stimulate durable-goods consumption and business investment (both of which are types of interest-sensitive spending), through the following channel:
---- P falls --> in the money market, less money is needed for a given level of transactions --> money demand falls --> the price of money (which is the interest rate, i) falls.
------ (Alternatively and equivalently, one could say the drop in P raises the real money supply, causing the real-money-supply curve to shift out while the real-money-demand curve stays put, giving rise a new money-market equilibrium at a lower interest rate. {If this last sequence is confusing to you, don't worry -- I didn't do it in class and won't test you on it.)

(3) Foreign purchases effect (real-exchange-rate effect)
-- A lower domestic price level, relative to the price levels of other countries, means the home country's products become cheaper relative to foreign products (and hence the real foreign exchange rate rises). If the American price level falls, relative to the world price level, then foreigners will buy more American exports and Americans will buy fewer foreign imports, and American net exports increase.

Q: Hey, wait a minute: What about the fact that a lower price level is bad for debtors by raising the real burden of debt? Think back to our unit on inflation; we call this phenomenon the debt-deflation effect, and it has severe, adverse effects on AD, because an increased debt burden forces indebted consumers and firms to retrench (the households' consumption falls, the firms' investment falls) and, in many cases, declare bankruptcy or default on their debts, in which case their creditors take a hit, too.
A: Based on the empirical evidence, it seems that the combined positive effects -- (1), (2), and (3) -- of a lower price level on AD outweigh the negative debt-deflation effect, so the net effect of lower prices on AD is still positive. But the debt-deflation effects on debtors and many creditors are sufficiently severe that U.S. economic policymakers have made sure to keep the U.S. out of deflation ever since the 1940s.

A summing up

A lower price level raises aggregate demand (AD = C + Iplanned + G + Xnet ) as follows:
 
AD = C + Iplanned + G + Xnet
increases through (1) real wealth effect and (2) interest-rate effect on durable-goods consumption increases through (2) interest-rate effect increases through (3) foreign-purchases effect