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 f 1 , 2 , ..., H g ): H H H ∑ a h ∑ o h ∑ m h V t = p at t + p ot t = t h = 1 h = 1 h = 1 = p at A t + p ot O t = M t 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 ˙ a + u o ˙ o , ˙ 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: = λ p a u a u o = λ p o change in utility at given prices is given by u = λ p a ˙ a + λ p o ˙ o ˙ We can write the growth in household utility as � p a a � u u = λ m a a + p o o o ˙ m . ˙ m . ˙ u o () Aggregation January 2011 5 / 20
If λ m is approximately constant, then utility growth is proportional to u � p a a � ˙ � p o o � ˙ a o y o = ˙ a + m m 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 = α Bm α � 1 and so = α Bm α � 1 . m λ m = α Bm α u () 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 H y h = p a ˙ ˙ ∑ A + p o ˙ Y = O ˙ h = 1 Is this a valid index of the change in aggregate welfare ? assume a utilitarian aggregate welfare function, U . Then: H H u h = λ h ˙ y h ˙ ∑ ∑ U = ˙ 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 � p a A � ˙ � p o O � ˙ � � = ˙ Y = 1 Y A O p a ˙ A + p o ˙ O A + M M M O Index of real GDP growth: ˙ ˙ ˙ Y A O Y = γ A + ( 1 � γ ) O where γ = p a A M () Aggregation January 2011 8 / 20
Implicit price–de‡ator (continuous time) De…ne P as P t = V t Y t , ! growth in price de‡ator ˙ ˙ ˙ P V Y P = V � Y But � ˙ � � ˙ � ˙ V a p a o p o a + ˙ o + ˙ V = γ + ( 1 � γ ) p a p o , ! and so ˙ P p a p o P = γ ˙ + ( 1 � γ ) ˙ . p a p o Alternative way to compute real GDP growth: ˙ ˙ ˙ Y V P Y = V � P . () Aggregation January 2011 9 / 20
Generalization to n–good economy Real GDP growth: n ˙ Y t x it ˙ ∑ = γ it Y t x it i = 1 Implicit de‡ator is n ˙ P t p it ˙ ∑ = γ it . P t p it i = 1 Inclusion of investment goods ˙ ˙ ˙ Y t I t C t = ω t + ( 1 � ω t ) , Y t I t C t where 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: g � = ∑ n i = 1 p 1 = ∑ n i = 1 p 1 i x 1 i ∆ x i i � 1 ? ∑ n ∑ n i = 1 p 1 i x 0 i = 1 p 1 i x 0 i i or prices at t = 0: g �� = ∑ n = ∑ n i = 1 p 0 i = 1 p 0 i x 1 i ∆ x it i � 1 ? ∑ n ∑ n i = 1 p 0 i x 0 i = 1 p 0 i x 0 i i () Aggregation January 2011 11 / 20
x a x 0 x 1 x o Figure: E¤ect of Decrease in Price of Oranges () Aggregation January 2011 12 / 20
x a x 0 x 1 x o Figure: Index of Welfare Change measured at New Prices () 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 x o Figure: E¤ect of Decrease in Price of Oranges () Aggregation January 2011 15 / 20
x a x 0 x 1 x o Figure: Index of Welfare Change Measured at Old Prices () 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: Q P = ∑ n i = 1 p 1 i x 1 i ∑ n i = 1 p 1 i x 0 i The Laspeyres Quantity Index: Q L = ∑ n i = 1 p 0 i x 1 i ∑ n i = 1 p 0 i x 0 i True change in utility is between these two. , ! in current practice, the Fisher–Ideal or chain index is used: � Q P � 1 2 � Q L � 1 2 . Q F = () Aggregation January 2011 18 / 20
Implicit price indices Laspeyres Price index: P L = ∑ n i = 1 p 1 i x 0 i ∑ n i = 1 p 0 i x 0 i Paasche Price index: P P = ∑ n i = 1 p 1 i x 1 i ∑ n i = 1 p 0 i x 1 i Fisher–Ideal or chain Price index: � P P � 1 2 � P L � 1 P F = 2 ) gross nominal income (expenditure) change: ∑ n i = 1 p 1 i x 1 = P L Q P = P P Q L = P F Q F . i ∑ n i = 1 p 0 i x 0 i () 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 = Cost of representative basket of goods in US PPP exchange rate for country A Cost of same basket of goods in country A Sometimes use an international basket of goods () Aggregation January 2011 20 / 20
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