Aggregation: A Brief Overview January 2011 () Aggregation January 2011 1 / 20
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? () Aggregation January 2011 2 / 20
Aggregation Apples and Oranges Nominal value of GDP (households, h 2 f1, 2,..., Hg): H V t = p at a h H t + p ot ot h = h=1 h=1 = p at A t + p ot O t = M t H mt h h=1 want a measure of how much better o society is as a result of changes in production, not in ation neoclassical theory? () Aggregation January 2011 3 / 20
Continuous Time Case Household utility function: u t = u t (a t, o t ) change in utility over time: u = u a ȧ + u o ȯ, where u a = u a and u o = u o BUT we can t measure marginal utility directly, so how is this helpful? () Aggregation January 2011 4 / 20
Neoclassical theory ) at the optimum u o (a, o ) u a (a, o ) = p o p a. or marginal utility is proportional to price for each good: u a = λp a u o = λp o change in utility at given prices is given by u = λp a ȧ + λp o ȯ We can write the growth in household utility as u u = λm pa a u m.ȧ a + p oo m.ȯ o () Aggregation January 2011 5 / 20
If λm u is approximately constant, then utility growth is proportional to pa a ȧ m a + po o ȯ m o = ẏ y This is exactly true only if the indirect utility function is a power function of m: v(m, p a, p o ) = B(p a, p o )m α where α is a constant.,! then the envelope theorem implies λ = v m = αbm α 1 and so λm u = αbmα 1.m Bm α = α () Aggregation January 2011 6 / 20
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 Ẏ = H ẏ h = p a Ȧ + p o Ȯ h=1 Is this a valid index of the change in aggregate welfare? assume a utilitarian aggregate welfare function, U. Then: U = H u h H = λ h ẏ h h=1 h=1 requires that marginal utility of income λ h 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) () Aggregation January 2011 7 / 20
GDP Growth (continuous time) In any case, this is basic index used. In the intial (base) period let Y = M. Then Ẏ Y = 1 M Index of real GDP growth: pa Ȧ + p o Ȯ pa A Ȧ = M A + po O Ȯ M O Ẏ Y = γȧ A + (1 γ)ȯ O where γ = p aa M () Aggregation January 2011 8 / 20
Implicit price de ator (continuous time) De ne P as,! growth in price de ator But,! and so P t = V t Y t Ṗ P = V V Ẏ Y ȧ ȯ V V = γ a + ṗa + (1 γ) p a o + ṗo p o Ṗ P = γṗa p a + (1 γ)ṗo p o. Alternative way to compute real GDP growth: Ẏ Y = V V Ṗ P. () Aggregation January 2011 9 / 20
Generalization to n good economy Real GDP growth: n Ẏ t ẋ it = Y t γ it i=1 x it Implicit de ator is n Ṗ t ṗ it = P t γ it. i=1 p it Inclusion of investment goods where Ẏ t Y t = ω t İ t I t + (1 ω t )Ċt C t, ω t = investment s share of nominal output I t = investment good sub index C t = consumption good sub index () Aggregation January 2011 10 / 20
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: or prices at t = 0: g = n i=1 p 1 i x i n i=1 p 1 i x 0 i g = n i=1 p 0 i x it n i=1 p 0 i x 0 i = n i=1 pi 1x i 1 n i=1 pi 1x i 0 = n i=1 pi 0x i 1 n i=1 pi 0x i 0 1? 1? () Aggregation January 2011 11 / 20
x a x 0 x 1 Figure: E ect of Decrease in Price of Oranges x o () Aggregation January 2011 12 / 20
x a x 0 x 1 Figure: Index of Welfare Change measured at New Prices x o () Aggregation January 2011 13 / 20
x a x 0 x 1 Q P x o Figure: Paasche Quantity Index () Aggregation January 2011 14 / 20
x a x 0 x 1 Figure: E ect of Decrease in Price of Oranges x o () Aggregation January 2011 15 / 20
x a x 0 x 1 Figure: Index of Welfare Change Measured at Old Prices x o () Aggregation January 2011 16 / 20
x a x 0 x 1 Q L x o Figure: Laspeyres Quantity Index () Aggregation January 2011 17 / 20
Alternative Quantity Indices Paasche Quantity index: The Laspeyres Quantity Index: Q P = n i=1 pi 1x i 1 n i=1 pi 1x i 0 Q L = n i=1 pi 0x i 1 n i=1 pi 0x i 0 True change in utility is between these two.,! in current practice, the Fisher Ideal or chain index is used: Q F = Q P 1 2 Q L 1 2. () Aggregation January 2011 18 / 20
Implicit price indices Laspeyres Price index: P L = n i=1 pi 1x i 0 n i=1 pi 0x i 0 Paasche Price index: P P = n i=1 pi 1x i 1 n i=1 pi 0x i 1 Fisher Ideal or chain Price index: 1 P F = P P 2 P L 1 2 ) gross nominal income (expenditure) change: n i=1 p 1 i x 1 i n i=1 p 0 i x 0 i = P L Q P = P P Q L = P F Q F. () Aggregation January 2011 19 / 20
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 () Aggregation January 2011 20 / 20