Growth. Growth Theory. Mark Huggett 1. 1 Georgetown. January 26, 2018

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1 Growth Theory Mark Huggett 1 1 Georgetown January 26, 2018

2 Growth Theory: The Agenda 1. Facts motivating theory 2. Basic Solow model 3. Model properties 4. How to use the model 5. Full Solow model 6. Use the model to interpret data

3 Kaldor s Growth Facts 1. Output per capita grows over time 2. Capital per capita grows over time 3. Capital-Output ratio is approx constant over time 4. Capital and Labor s share is approx constant over time 5. Return to Capital has no trend 6. Output per capita varies widely across countries at a point in time

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5 Solow Growth Theory Key Assumptions: 1. Agg. Production Function w/ Diminishing MPK 2. Can Accumulate Physical Capital 3. Technology Grows Exogenously 4. Constant Saving/Investment Rate.

6 Basic Solow Model 1. C t + I t = Y t = F (K t, L t ) 2. I t = sf (K t, L t ) - investment 3. K t+1 = K t (1 δ) + I t - Capital Accumulation 4. L t = L Implication: K t+1 = K t (1 δ) + sf (K t, L t ) Steady State: δk = sf (K, L)

7 Restate Model in Terms of k K/L c t + i t = y t = F (k t, 1) - CRS i t = sf (k t, 1) - investment k t+1 = k t (1 δ) + i t - Capital Accumulation Implication: k t+1 = k t (1 δ) + sf (k t, 1) Steady State: δk = sf (k, 1)

8 Solow Model Steady State dk y* y=f(k,1) i* i=sf(k,1) 0 k* k

9 Main Properties of the Basic Solow Model Define k = K L 1. One positive capital steady state capital-labor ratio k. 2. Higher savings rate s implies a higher steady state value k. 3. The economy converges over time to the steady state k. 4. There is a max feasible steady state capital-labor ratio k where the production function intersects the depreciation line.

10 How to Use the Model 1. Use the graph to get qualitative insights to two types of experiments: (1) exogenous one-time changes in capital or labor (e.g. war or disease) and (2) permanent changes in model parameters (e.g. change the savings rate s). 2. One can get insight into how factor prices move from the production function if one adopts competitive theory of factor prices. 3. Make assumptions on the parametric form of the production function and choose all parameter values. Use the model for quantitative insights (e.g. how much does increasing the saving rate increase steady state output?).

11 Example: Recovery from WWII 1. War begins and ends in period Capital-labor ratio falls as, we assume, more capital is destroyed than people. 3. After the war ends, the model mechanism again operate. 4. After you determine what happens to the capital-labor ratio over time, then time variation in all other variables are easy to determine.

12 Solow Model: War Recovery Time Capital labor Output labor Wage

13 Example: Plague 1. The Black Death killed a large fraction of the English population in the 1300 s. 2. This increased the capital-labor ratio suddenly and dramatically for exogenous reasons. 3. Assume that the plague in the model permanently decreases the total labor in the model economy. 4. Determine implications of the model. Compare to Clark s data on real wages.

14 Solow Model: Black Death Time Capital labor Output labor wage Gross interest rate

15 0.5 England: Black Death Time Farm Wage Nonfarm Wage Population (millions/10)

16 Does the Basic Model Produce Kaldor s Facts? 1. In steady state y and k do not grow! 2. Thus, the only possibility to explain sustained growth in y = Y/L and k = K/L (Kaldor s Facts 1-2) is to have all countries be BELOW steady state. 3. This is unsatisfactory as then growth should be slowing down over time with a fixed saving rate. The data say that growth rates of Y/L are increasing over the last two hundred years in advanced countries! 4. This motivates the full Solow model.

17 Full Solow Model: (Use k t K t L t A t ) C t + I t = Y t = F (K t, L t A t ) I t = sf (K t, L t A t ) K t+1 = K t (1 δ) + I t L t+1 = L t (1 + n) A t+1 = A t (1 + g) Implication: K t+1 = K t (1 δ) + sf (K t, L t A t ) k t+1 (1 + n)(1 + g) = k t (1 δ) + sf (k t, 1) Steady State: k[(1 + n)(1 + g) (1 δ)] = sf (k, 1)

18 Figure 1: Solow Model Steady State k(d+n+g+ng) y* y=f(k,1) i* i=sf(k,1) k* 0

19 Interpretation of Steady State: k t K t L t A t 1. (k t, y t, i t, c t ) constant BUT (K t /L t, Y t /L t, I t /L t,...) grow 2. (Y t /L t, K t /L t ) grow at rate g and (Y t, K t ) grow (approx) at rate n + g. 3. Y t = W t L t + R t K t implies (in steady state) W t grows at rate g and R t is constant. 4. In steady state, labor and capital s share of output are EXACTLY constant. 5. Note: Points 1-4 are Kaldor s Facts 1-5!

20 What Are the Effects of Increasing the Savings Rate? Distinguish between the steady state or long-run effects and the effects in transition. Steady State: increasing savings rate s does NOT change the long-run growth rate of Y/L Transition: increasing savings rate s increases the growth rate of output for any finite time period

21 Assessing the Solow Model: Cross-Country Differences 1. Are countries with high capital-labor ratios rich? 2. Are countries with high savings rates rich? 3. Are countries with high population growth rates poor? 4. Do observed differences in savings rates across countries imply large GDP per worker differences within the Solow model? 5. Is the technology level the same across countries? We will give a first-pass answer to all of these questions.

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25 Assessing the Importance of Savings Rate Differences Two countries: same technology: y = F (k, 1) = Ak β savings rate differs: s H =.30, s L =.05 y H y L = F (k H,1) = kβ H F (k L = ( k,1) k β H kl ) β L In steady state: sak β = k(n + g + ng + δ) implies y H y L sa k = ( n + g + ng + δ )1/(1 β) = ( k H kl ) β = ( s H sl ) β 1 β = ( ) β 1 β and y H y L. = 2.15 if β =.3!

26 Assessing the Importance of Technology Differences Assume the production function is Cobb-Douglas Y i = A i F (K i, L i ) = A i K β i L1 β i Y i = A i F (K i, L i ) A i = Y i /F (K i, L i ) = Y i /(K i ) β (L i ) 1 β Assume (Y i, K i, L i ) can be measured (with some error) across countries and set β to the US value. Standard Finding: low Y i /L i countries have very low implied A i levels. Labor quality differences across countries are perhaps poorly measured.

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