Aggregation: A Brief Overview January 2011 () Aggregation January - - PowerPoint PPT Presentation

Aggregation: A Brief Overview

January 2011

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Macroeconomic Aggregates

Consumption, investment, real GDP, labour productivity, TFP, physical capital, human capital (quantity and quality), price level, in‡ation Sources include: Statistics Canada, US NIPA, Penn World Tables What do these indices mean ? How are they computed ? Why should we care?

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Aggregation — Apples and Oranges

Nominal value of GDP (households, h 2 f1, 2, ..., Hg): Vt = pat

H

∑

h=1

ah

t + pot H

∑

h=1

• h

t = H

∑

h=1

mh

t

= patAt + potOt = Mt want a measure of how much better o¤ society is as a result of changes in production, not in‡ation neoclassical theory ?

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Continuous Time Case

Household utility function: ut = ut(at, ot) change in utility over time: ˙ u = ua ˙ a + uo ˙

• ,

where ua = ∂u ∂a and uo = ∂u ∂o BUT we can’t measure marginal utility directly, so how is this helpful?

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Neoclassical theory ) at the optimum uo(a, o) ua(a, o) = po pa .

• r marginal utility is proportional to price for each good:

ua = λpa uo = λpo change in utility at given prices is given by ˙ u = λpa ˙ a + λpo ˙

• We can write the growth in household utility as

˙ u u = λm u paa m . ˙ a a + poo m . ˙

• ()

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If λm

u

is approximately constant, then utility growth is proportional to paa m ˙ a a + poo m ˙

• = ˙

y y This is exactly true only if the indirect utility function is a power function of m: v(m, pa, po) = B(pa, po)mα where α is a constant. , ! then the envelope theorem implies λ = vm = αBmα1 and so λm u = αBmα1.m Bmα = α

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Aggregate Index of Change in Real GDP

Suppose we normalize so that at a point in time y = m. Then we might write the aggregate change in GDP as ˙ Y =

H

∑

h=1

˙ yh = pa ˙ A + po ˙ O Is this a valid index of the change in aggregate welfare ? assume a utilitarian aggregate welfare function, U. Then: ˙ U =

H

∑

h=1

˙ uh =

H

∑

h=1

λh ˙ yh requires that marginal utility of income λh is “approximately” equal across households , ! unlikely to be true: we usually think of diminishing marginal utility , ! BUT if income is log–normally distributed and utility is a power function, it may be a reasonable index to use (see Assignment)

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GDP Growth (continuous time)

In any case, this is basic index used. In the intial (base) period let Y = M. Then ˙ Y Y = 1 M

• pa ˙

A + po ˙ O = paA M ˙ A A + poO M ˙ O O Index of real GDP growth: ˙ Y Y = γ ˙ A A + (1 γ) ˙ O O where γ = paA M

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Implicit price–de‡ator (continuous time)

De…ne P as Pt = Vt Yt , ! growth in price de‡ator ˙ P P = ˙ V V ˙ Y Y But ˙ V V = γ ˙ a a + ˙ pa pa

• + (1 γ)

˙

• + ˙

po po

• ,

! and so ˙ P P = γ ˙ pa pa + (1 γ) ˙ po po . Alternative way to compute real GDP growth: ˙ Y Y = ˙ V V ˙ P P .

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Generalization to n–good economy

Real GDP growth: ˙ Yt Yt =

n

∑

i=1

γit ˙ xit xit Implicit de‡ator is ˙ Pt Pt =

n

∑

i=1

γit ˙ pit pit . Inclusion of investment goods ˙ Yt Yt = ωt ˙ It It + (1 ωt) ˙ Ct Ct , where ωt = investment’s share of nominal output It = investment good sub–index Ct = consumption good sub–index

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Realistic Discrete Time Case

Problem: prices and quantities are measured at discrete dates (say t = 0, 1) ) no such thing as a change at a “point in time” Should we use prices at t = 1: g = ∑n

i=1 p1 i ∆xi

∑n

i=1 p1 i x0 i

= ∑n

i=1 p1 i x1 i

∑n

i=1 p1 i x0 i

1 ?

• r prices at t = 0:

g = ∑n

i=1 p0 i ∆xit

∑n

i=1 p0 i x0 i

= ∑n

i=1 p0 i x1 i

∑n

i=1 p0 i x0 i

1 ?

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x0 x1 xa xo

Figure: E¤ect of Decrease in Price of Oranges

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x0 x1 xa xo

Figure: Index of Welfare Change measured at New Prices

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x0 x1 xa xo QP

Figure: Paasche Quantity Index

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x0 x1 xa xo

Figure: E¤ect of Decrease in Price of Oranges

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x0 x1 xa xo

Figure: Index of Welfare Change Measured at Old Prices

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x0 x1 xa xo

QL Figure: Laspeyres Quantity Index

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Alternative Quantity Indices

Paasche Quantity index: QP = ∑n

i=1 p1 i x1 i

∑n

i=1 p1 i x0 i

The Laspeyres Quantity Index: QL = ∑n

i=1 p0 i x1 i

∑n

i=1 p0 i x0 i

True change in utility is between these two. , ! in current practice, the Fisher–Ideal or chain index is used: QF =

• QP 1

2

QL 1

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Implicit price indices

Laspeyres Price index: PL = ∑n

i=1 p1 i x0 i

∑n

i=1 p0 i x0 i

Paasche Price index: PP = ∑n

i=1 p1 i x1 i

∑n

i=1 p0 i x1 i

Fisher–Ideal or chain Price index: PF =

• PP 1

2

PL 1

2

) gross nominal income (expenditure) change: ∑n

i=1 p1 i x1 i

∑n

i=1 p0 i x0 i

= PLQP = PPQL = PF QF .

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International Comparisons: The Penn World Tables

Market exchange rates re‡ect prices of traded goods and capital ‡ows Large fraction of goods consumed by LDCs are non-traded Capital ‡ows are volatile Conversion into US dollars uses purchasing power parity (PPP) exchange rate PPP exchange rate for country A = Cost of representative basket of goods in US Cost of same basket of goods in country A Sometimes use an international basket of goods

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