Two major points to discuss today:
1. Equilibrium v. identity
2. Solving the index number problem
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I. The Basic Classical Model
Aggregate supply:
(1) Y  F ( K , L)
P
AS
(2) W / P Y / L  MPL F (K , L)/ L
• Endogenous variables: W, P, Y. This
defines the AS curve in figure 1.
• However, we are one variable short for
total equilibrium.
•Note on AS: It is vertical because have
perfectly flexible wages and prices.
Yp
AS curve
Y
Aggregate demand
The standard way of closing the classical model is through deriving an
aggregate demand curve.
The simplest approach is to rely upon the quantity theory of prices and
money. This holds that money supply is exogenous and money
demand is proportional to money output:
(3) M = Ms
(supply of money exogenous)
(4) Md = kPY
(demand for money)
which gives the AD function:
(5) Y = Ms /kP
Note: The quantity theory is not essential here. We need aggregate
demand as a function of fiscal, monetary, and other variables. (5) is
just a simple and convenient example.
This is the classical AS-AD
relationship.
Notes:
1. Output is always at
potential output.
2. No effect of M (or more
generally AD) on
output; AD only affects
prices (“Money is
neutral.”)
P
AS
AD
Yp
Y
II. Supply and Demand for Loanable Funds
Mankiw (chap 3) shows an alternative derivation of the classical model
using the loanable funds. This approach examines the sources and
uses of saving and investment. Do for simplest system:
Basics:
(1)
Y=C+I
(expenditure identity)
(2)
S=Y-C
(definition of private saving)
(3)
I = I(r)
(investment equation)
These imply :
(4)
S= I(r)
In this alternative approach, make sure you understand the difference
between the savings-investment identity and the savingsinvestment equilibrium.
III. Equilibrium of saving and investment
r
Note: For now, assume that
saving is interest-inelastic
You should understand the
effect of:
1. Shift in investment curve.
2. Increase in planned S.
S(r)
B
r*
I(r)
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IV. Example of disequilibrium for S and I
In savings example, assume we
have a “corn economy.”
Corn can either be eaten or planted.
People eat a certain amount of corn
(C), and that leaves the rest (S)
available for farmers to plant (I).
If there is excess demand for I =
AC, then the rate must rise to
equilibrate S and I.
This means that planned I at the
original interest rate (r0) was
excessive.
Note that equilibrating mechanism
will differ in different markets –
Keynesian mechanism is
quantity rather than price.
r
S(r)
B
r*
C
r0
A
I(r)
Let’s go back and ask:
“Just what is this ‘Y’?”
“Just how do we measure GDP and real GDP?”
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How to measure output growth?
Now take the following numerical example.
• Suppose good 1 is computers and good 2 is shoes.
• How would we measure total output and prices?
period 1
Real output
q1
q2
Prices
p1
p2
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Ratio:
period 2 to
period 2 period 1
1
1
100
1
100
1
1
1
0.010
1.00
0.010
1.00
The growth picture for index numbers:
the real numbers!
Output (109 2005 $)
Sector
1958
Computers
Non computers
2008
Rate per year Growth Factor
0.00002
157.03200
31.8%
8,049,116.8
2,578
13,155
3.3%
5.1
Source: Bureau of Economics Analysis
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Growth of sector
Some answers
• We want to construct a measure of real output, Q = f(q1,…, qn ;p1,…, pn)
• How do we aggregate the qi to get total real, GDP(Q)?
– Old fashioned fixed weights: Calculate output using the prices of a
given year, and then add up different sectors.
– New fangled chain weights: Use new “superlative” techniques
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Old fashioned price and output indexes
Laspeyres (1871): weights with prices of base year
Lt = ∑ wi,base year (Δq/q)i,t
Paasche (1874): use current (latest) prices as weights
Πt = ∑ wi,t (Δq/q)i,t
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Start with Laspeyres and Paasche
period 1
Real output
q1
q2
Prices
p1
p2
Nominal output
= ∑piqi
Quantity indexes
Laspeyres (early p)
Paasche (late p)
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Ratio:
period 2 to
period 2 period 1
1
1
100
1
100
1
1
1
0.010
1.00
0.010
1.00
2.0
2.0
1.0
2.000
1.010
101.000
2.000
50.50
1.98
HUGE
difference!
What to do?
Solution
Brilliant idea: Ask how utility of output differs across different bundles.
Let U(q1, q2) be the utility function. Assume have {qt} = {qt1, qt2}. Then growth
is:
g({qt}/{qt-1}) = U(qt)/U(qt-1).
For example, assume “Cobb-Douglas” utility function, Q = U = (q1)λ (q2) 1- λ
Also, define the (logarithmic) growth rate of xt as g(xt) = ln(xt/xt-1). Then
Qt / Qt-1 =[(qt1)λ (qt2) 1- λ]/[(qt-11)λ (qt-12) 1- λ]
g(Qt) = ln(Qt/Qt-1) = λ ln(qt1/qt-11) + (1-λ) ln(qt2/qt-12)
g(Qt) = λ g(qt1) + (1-λ) g(qt2)
The class of 2nd order approximations is called “superlative.”
This is a superlative index called the Törnqvist index.
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period 1
Real output
q1
q2
Prices
p1
p2
Ratio:
period 2 to
period 2 period 1
1
1
100
1
100
1
1
1
0.010
1.00
0.010
1.00
2.0
2.0
1.0
1.00
10.00
10.00
2.000
1.010
101.000
2.000
50.50
1.98
Nominal output
= ∑piqi
Utility = (q1*q2)^.5
Quantity indexes
Laspeyres (early p)
Paasche (late p)
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What do we find?
1. L > Util > P [that
is, Laspeyres
overstates growth
and Paasche
understates relative
to true.
Currently used “superlative” indexes
Fisher* Ideal (1922): geometric mean of L and P:
Ft = (Lt × Πt )½
Törnqvist (1936): average geometric growth rate:
(ΔQ/Q)t = ∑ si,T (Δq/q)i,t, where si,T =average nominal share of
industry in 2 periods
(*Irving Fisher (YC 1888), America’s greatest macroeconomist)
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period 1
Real output
q1
q2
Prices
p1
p2
Nominal output
= ∑piqi
Utility = (q1*q2)^.5
1
1
Ratio:
period 2 to
period 2 period 1
100
1
100
1
1
1
0.010
1.00
0.010
1.00
2.0
2.0
1.0
1.00
10.00
10.00
Quantity indexes
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Fisher (geo mean of L
and P)
1.421
14.213
10.00
Törnqvist (wt.
average growth rate)
1.000
10.000
10.00
Now we construct
new indexes as
above: Fisher and
Törnqvist
Superlatives (here
Fisher and
Törnqvist) are
exactly correct.
Usually very close to
true.
Current approaches
• Most national accounts used Laspeyres until recently
– Why Laspeyres? Primarily because the data requirements are
less stringent.
• CPI uses Laspeyres index.
• US moved to Fisher for national accounts in 1995
• BLS has constructed “chained CPI” using Törnqvist since 2002
• China still uses Laspeyres in its GDP.
– Who knows whether Chinese data are accurate???
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Who cares about GDP and CPI measurement?
Some examples where makes a big difference
• social security for grandma
• taxes for you
• estimated rate of productivity growth for budget
– and, therefore, Congress’s spending inclinations
• comparisons of military “power”
– overestimates of Soviet GDP in 1980s led Reagan administration
to large increase in military budget
• projections of emissions in global warming models
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