Applications of Growth Theory I: Growth Accounting

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1 (2b)-P.1 Applications of Growth Theory I: Growth Accounting Objective: Use empirical data on output, capital stocks, and labor supply, to interpret history ( accounting ), to compare across countries, or to make projections. (Y t, K t, L t ) at discrete dates t. - Productivity index A is not directly observable must be inferred. - Labels: Y/L = labor productivity vs. A = total/multi factor productivity - Often work with growth rates over intervals: g x = 1 t 1 t 0 [ln(x(t 1 ) ln(x(t 0 )] - Even when capital and labor shares vary, use Cobb-Douglas as approximation. growth theory applies to international data: - Traditional: apply growth theory to each country separately. - Alternative: integrated world economy, possibly with frictions on capital mobility.

2 (2b)-P.2 Growth Accounting with Cobb Douglas Production Y = K α (AL) 1 α = R K α L 1 α with R = A 1 α as Solow Residual - lny = α ln K + (1 α)ln L + ln R - Take time differences over some interval [t 1, t 2 ]: => Δ lny = α Δ ln K + (1 α) Δ ln L + Δ ln R => g Y = α g K + (1 α) g L + g R = α g K + (1 α) g L + (1 α) g A Apply to data: start with (g Y,g K,g L ) for various time periods; estimate α. - Use Cobb-Douglas to infer g R = g Y α g K (1 α) g L or R = Y (K α L 1 α ) - Use R = A 1 α to infer TFP: g A = g R /(1 α) or A = R 1/(1 α )

3 (2b)-P.3 Growth Accounting in per-capita units Growth accounting for per-capita variables [Math: Growth of ratio = difference of rates] - Income: g Y / L = g Y g L = α g K α g L + g R = α g K / L + g R - Growth in K/L called capital deepening. Conclude: Growth of per-capita income = Capital deepening and productivity growth. - Growth accounting means decomposition of growth rates into components. - Sometimes decompose further: multiple types of capital and labor, each weighted by its factor share. Multi-factor productivity is the residual a measure of ignorance. Common findings: - Growth rates g L,g R,g A vary over time. - Solow model assumes constant g L = n and g A = g: only an approximation - Steady state has predictive power if (n,g) are stable; otherwise moving target. - Diagnostic for balanced growth: should find g Y = g K = g+ n and g Y / L = g K / L = g

4 (2b)-P.4 Numerical Example - Output: Y(1980)=100, Y(2000)= Capital: K(1980)=100, K(2000)=180 - Labor force: L(1980)=100, L(2000)= Compute growth rates g L = 20 1 [ln(150) ln(100)] = 2.03% g Y = 20 1 [ln(200) ln(100)] = 3.47%g Y / L =1.44%. g K = 1 20 [ln(180) ln(100)] = 2.97%g K / L = 0.94%. - Note: Output grew faster than capital => Economy is not in steady state. -2: Use the average capital share to calibrate αα=1/3. -3: Infer total productivity growth: g R = g Y α g K (1 α) g L = 3.47% % % =1.13% - For GDP growth: TFP 1.13%, capital 0.99% = %, labor 1.37% = %. - For per-capita income: TFP 1.13%, capital deepening 0.31% %

5 (2b)-P.5 - Y = f (k) AL = f (e x ) AL with x = ln(k) = ln( K AL ) => lny = ln f (e x ) + ln A + ln L Note that d [ ln f (e x ] ) = f '(ex ) dt => f (e x ) d dt lny = α (k) d ln k k dt e x = f '(k) f (k) k = α k (k) + d dt ln A + d dt ln L = α k (k) d dt ln K + (1 α k (k))( d dt ln A + d dt ln L) Conclude: Growth accounting relationships always hold for instantaneous growth rates. - - Cobb-Douglas approximates α k (k) α,

6 (2b)-P si/y - Growth Accounting: Compare TFP levels and growth rates across countries. Specific questions separate field: Development (Also miracles & disasters)

7 (2b)-P.7 R = f '(k) δ = αk (1 α ) δ -- - k = K / L A. If A i < A w, then (K /L) i < (K /L) w is consistent with k i = k w. => Observed cross-country differences in

8 (2b)-P.8 δ.. a- - - [Optional: Acemoglu 3.3.] --c - -

9 (2b)-P [

10 (2b)-P.10 Figure 1: World Demographic Projections Figure 2: The Actual and Effective World Labor Force Figure 3: The Solow Model Implications for Capital and Returns A. Capital-Output Ratio B. Return on Capital % % % % % Projected Demographics Cohort size UP +0.5%/year Projected Demographics Cohort size UP +0.5%/year Cohort size DOWN -0.5%/year Cohort size DOWN -0.5%/year

11 (2b)-P.11 Application III: Natural Resources -Douglas approach (Romer 1.8, here modified) - Land T (fixed) and energy E (exhaustible resources) as factor of production, weights β and γ: Y = K α (AL) 1 α β γ T β E γ R(t) = Resource stock at time t, starting with fixed value R(0) - Resource use: E = dr / dt = R = Rate at which the resource is extracted - Define s E = E / R = Share of the resource used in the current period, assumed constant => R and E are declining over time: R / R = E / E = s E [Differs from Romer: here E is productive. But R and E proportional -> same qualitative implications as Romer] for per-capita income: y = Y /(AL) = k α (T /AL) β (E /AL) γ - Growth accounting => g y = α g k + β ( n g) + γ (g E n g) - On a balanced growth path g y = g k,g E = s E implies 1 α β γ g Y /L = g y + g = g γ 1 α 1 α s E γ + β 1 α n 1. Fixed resources and land reduce output growth. 2. Per-capita income growth is still possible if g is high enough - Nordhaus: % growth reduction - Caveat: Share of energy in factor income has been declining

12 Learning Objectives Problem sets for practice. - (2b)-P.12