Introductory Macroeconomics Lecture 16: The Solow-Swan Growth Model II PDF

Introductory Macroeconomics Lecture 16: The Solow-Swan Growth Model II Jonathan Thong Daniel Minutillo 2nd Semester 2024 1  This Lecture The Solow-Swan Growth Model II (1) productivity growth and employment...

Introductory Macroeconomics Lecture 16: The Solow-Swan Growth Model II Jonathan Thong Daniel Minutillo 2nd Semester 2024 1  This Lecture The Solow-Swan Growth Model II (1) productivity growth and employment growth (2) convergence hypothesis (3) growth accounting BOFAH Chapter 15 2 Productivity Growth and Employment Growth 3  (1) Productivity and Employment Growth So far, we have defined a steady-state level of output – But output actually grows! – Add underlying trend growth to the Solow-Swan model Total factor productivity grows at rate gA per period At+1 = 1 + gA At Labour force grows at rate gL per period Lt+1 = 1 + gL Lt 4  Labour-Augmenting Productivity More convenient to work with labour-augmenting productivity Yt = F (Kt , At Lt ) Example: For Cobb-Douglas production function et Ktα L1−α , Yt = Ktα (At Lt )1−α = A et ≡ A1−α A t t (i.e., redefine the productivity term to make the maths simple) Write output, capital etc per effective worker Yt Kt yt = , kt = At Lt At Lt 5  Intensive Form of the Production Function Using constant returns to scale, we can write     Yt F (Kt , At Lt ) Kt At Lt Kt = = F , = F ,1 At Lt At Lt At Lt At Lt At L t In short, we have the intensive form of the production function yt = f (kt ), where f (kt ) ≡ F (kt , 1) To use this, need to put rest of the model in intensive form too 6  Capital Accumulation Capital accumulation is again given by Kt+1 − Kt = sYt − δKt In terms of capital per effective worker kt this becomes (1 + gA + gL )(kt+1 − kt ) = sf (kt ) − (δ + gA + gL )kt (see Appendix A for tedious algebra - not examinable!) 7  Solow-Swan Diagram Revisited Qualitatively, dynamics of kt same as basic Solow-Swan model (next slide) The dynamics depend on the size of effective investment vs. effective depreciation. If (i) effective investment > effective depreciation, capital per effective worker is increasing, kt+1 > kt (ii) effective investment < effective depreciation, capital per effective worker is decreasing, kt+1 < kt (iii) effective investment = effective depreciation, capital per effective worker is not changing, kt+1 = kt 8 Solow-Swan Diagram Revisited 9  Steady-State Steady state capital per effective worker k ∗ satisfies sf (k ∗ ) = (δ + gA + gL )k ∗ Graphically, this is where investment per effective worker curve (= sf (kt )) intersects effective depreciation line (= (δ + gA + gL )kt ) Steady state output and consumption per effective worker are then y ∗ = k ∗α , and c∗ = (1 − s)y ∗ 10 Solow-Swan Diagram Revisited 11  Balanced Growth Steady state capital and output per effective worker are constant Kt k∗ = ⇔ Kt = k ∗ At Lt At Lt Yt y∗ = ⇔ Yt = y ∗ At Lt At Lt Steady state therefore implies that capital and output both grow at the same rate as effective labour, i.e. a balanced growth path gK = gY = gA + gL The long run growth rate of output per worker gY − gL = gA Fully determined by growth rate gA of total factor productivity 12  Permanent Increase in Savings Rate Consider a permanent increase in the savings rate from s to s′ > s – shifts up investment per effective worker curve along an unchanged depreciation per effective worker line * steady state capital per effective worker increases from k∗ to k∗′ * steady state output per effective worker increases from y ∗ to y ∗′ On impact, capital accumulates and growth is faster than trend But over time, diminishing returns set in and, in the long run, growth slows back down to trend 13 Increase in Savings Rate 14 Increase in Savings Rate 15 Increase in Savings Rate 16 Convergence Hypothesis 17  (2) Convergence Hypothesis Basic prediction of Solow-Swan model: A country should grow faster when it is far below its steady state and slower when it is close to its steady state. Does this mean – poor countries should grow faster than rich countries? – in long run, poor countries should catch up with rich countries? 18 Convergence Hypothesis in the Data 19 Convergence Hypothesis in the Data 20  Conditional vs. Unconditional Convergence Solow-Swan model predicts conditional convergence – conditional on parameters s, δ, gA , gL , etc – but independent of initial capital per effective worker k0 In other words, predicts that ’clubs’ of countries with same parameters will have same long run output per worker Does not predict unconditional convergence, that countries will have same long run output per worker regardless of parameters. 21 Growth Accounting 22  (3) Growth Accounting Tool for decomposing observed growth into growth in inputs and ’productivity’ residual [the ’Solow residual’] Consider Cobb-Douglas production function Yt = At Ktα L1−α t So in growth rates (see Appendix B) gY = gA + αgK + (1 − α)gL Infer productivity from observed input and output growth gA = gY − αgK − (1 − α)gL , given α 23  Numerical Example Data: US economy 1948-2014 had observed growth gY − gL = 2.4% per year gK − gL = 2.7% per year And suppose α = 1/3. Question: What was productivity growth over this period? Which was more important, productivity growth or capital accumulation? Answer: Inferred productivity growth gA = (gY −gL )−α(gK −gL ) = 2.4−2.7/3 = 1.5% per year Productivity growth accounts for more than half, 1.5/2.4 = 0.625. 24  Learning Outcomes 1 Mathematically derive the equations required to convert the standard Solow-Swan model into the Solow-Swan model with trend growth in productivity and employment. 2 Understand and explain disequilibrium dynamics of the Solow-Swan model with trend growth, paying careful attention to the subtle differences from the standard Solow-Swan model. 3 Understand and explain the convergence hypothesis, distinguishing between conditional and unconditional convergence. 4 Understand and explain growth accounting. 5 Mathematically derive factors shares. Understand and explain how α in the Cobb-Douglas production function relates to factors shows, including key assumptions and mathematical derivations. 25  New Formula(s) and Notation Steady-state condition for model with trend sf (k ∗ ) = (δ + gA + gL ) k ∗ Growth Accounting gY = gA + αgK + (1 − α) gL Note: these represent key results c consumption per worker or per effective worker k capital per worker or per effective worker y output per worker or per effective worker f (·) a reduced-form function 26  Next Lecture International Trade I – direction of trade - ’small open economy’ model – winners and losers from trade – effect of protection policies: tariffs BOFAH Chapter 16 27 Appendices: Not Examinable 28  Appendix A: Trend Growth Algebra Start with capital accumulation Kt+1 − Kt = sYt − δKt Divide both sides by effective labour At Lt Kt+1 Kt Yt Kt − = s −δ At Lt At L t At Lt At Lt Multiply and divide first term on the left by At+1 Lt+1 At+1 Lt+1 Kt+1 Kt Yt Kt − = s −δ At Lt At+1 Lt+1 At Lt At Lt At Lt or    At+1 Lt+1 Kt+1 Kt Yt Kt − = s −δ At Lt At+1 Lt+1 At Lt At L t At L t 29  And using growth rates for productivity and employment Kt+1 Kt Yt Kt (1 + gA )(1 + gL ) − = s −δ At+1 Lt+1 At Lt At Lt At Lt Per effective worker kt ≡ Kt /At Lt and kt+1 ≡ Kt+1 /At+1 Lt+1 etc and using intensive version of production function yt = f (kt ) (1 + gA )(1 + gL )kt+1 − kt = sf (kt ) − δkt Subtract (1 + gA )(1 + gL )kt from both sides and rearrange (1 + gA )(1 + gL )(kt+1 − kt ) = sf (kt ) + [1 − (1 + gA )(1 + gL ) − δ]kt 30  Now using standard growth rate approximation (1 + gA )(1 + gL ) ≈ 1 + gA + gL (neglects second order term gA gL ) Hence 1 − (1 + gA )(1 + gL ) − δ ≈ −(gA + gL + δ) Leaves us with formula given in lecture (1 + gA + gL )(kt+1 − kt ) = sf (kt ) − (gA + gL + δ)kt 31 Appendix B: Maths approximating Growth Rates The growth gX in some variable Xt from period t to t + 1 is approximated as: Xt+1 − Xt gX = ≈ ln Xt+1 − ln Xt Xt To see this, write Xt+1 Xt+1 gX = − 1 ⇔ 1 + gX = Xt Xt A property of natural logs is that for gX small, we have that ln(1 + gX ) ≈ gX hence ln(1 + gX ) = ln Xt+1 − ln Xt ≈ gX 32  Growth Accounting Across Time For production function Yt = At Ktα L1−α t , take the natural log ln Yt = ln At + α ln Kt + (1 − α) ln Lt ⇒ ln Yt+1 = ln At+1 + α ln Kt+1 + (1 − α) ln Lt+1 Subtract the first equation from the second to get Yt+1 −Yt = ln At+1 −ln At +α(ln Kt+1 −ln Kt )+(1−α)(ln Lt+1 −ln Lt ) Apply the approximation for growth rates gY = gA + αgK + (1 − α)gL 33