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TA SECTION 20Th/21St NOVEMBER 1997 the SOLOW GROWTH MODEL: a REVIEW, an EXAMPLE and the EFFECTS of POPULATION GROWTH

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TA SECTION 20Th/21St NOVEMBER 1997 the SOLOW GROWTH MODEL: a REVIEW, an EXAMPLE and the EFFECTS of POPULATION GROWTH C11 Macroeconomics, Fall 1997 TA: Paulo Santiago Prof. Christiano TA SECTION 20th/21st NOVEMBER 1997 THE SOLOW GROWTH MODEL: A REVIEW, AN EXAMPLE AND THE EFFECTS OF POPULATION GROWTH 1. OBJECTIVE In this section, we’re going to review the Solow growth model (started last week) with a new notation (the one of Prof. Christiano as opposed to the one of the book). We’ll illustrate the theory with an example taken from this week’s HW. Also, we’ll talk about the effects of population growth. ISSUES: · How can we explain growth over time ? · How can we explain differences in growth and GDP per capita ? · What factors might influence growth ? Savings (Capital accumulation), Population growth, Technological progress. 2. THE SOLOW GROWTH MODEL: THE BASIC FRAMEWORK Production Function (The supply side): Y = F(K,L) Y- Aggregate Output, K- capital, L- labor (note that in the book N denotes labor), F- Aggregate Production Function. Specific Example we’ll consider: Y = K1/4 L3/4 Assumptions: · Constant returns to scale (CRS) Our example: multiply by l all the inputs and check that production is exactly multiplied by l. · Diminishing Marginal Products dF dF MPK = MPL = dK dL MPK (MPL) decreases as K (L) increases. Our example: 2 ways to check diminishing marginal products · Technical way: Second derivative (and check that it is negative). · Plugging in numbers and check that MPK (MPL) decreases as K (L) increases. Transformation of Production Function: Write in per worker (or per capita as Prof. Christiano does). Using CRS, we can reach the following production function: Y K = F( ,1) L L Let y = Y/L (output per worker) and k = K/L (capital per worker). We then have: y = f (k) 1/4 3/4 Our example: Y = K L 1/4 Y æ K ö Dividing both sides by L, we get: = ç ÷ or y = k1/4 L è L ø dy Define MPk = = f ’(k), Marginal product of capital per worker: how output changes dk when capital per worker changes. It also decreases as k increases (diminishing marginal products). y f(k) k From now on, we only use lowercase letters (everything is per worker). Demand Side: We ignore government spending and taxes. y = c + i where, c - consumption per worker, i - investment per worker. You either consume or save. In our economy, since private savings = total savings (no government) and since we have a closed economy, then saving is equal to investment. Let s be the saving rate (and NOT saving per worker). We have: i = s y = s f(k) c = (1-s)y = (1-s) f(k) y,i f(k) sf(k) k Capital Accumulation: Changes in capital stock => Changes in output What causes changes in per worker capital stock (k) ? · (1) Investment · (2) Depreciation Let d be the depreciation rate (rate at which capital wears out). Evolution of Capital Stock: Change in capital stock = investment - depreciation Dk = i - dk Dk = sf(k) - dk This rule describes how investment is growing/shrinking over time. Steady State Capital Stock: Level of capital stock (kss) at which capital stock is not changing: Dk=0 (invst = depreciation) kss is obtained by assuming Dk=0 => s f(kss) = d kss Key: The steady state capital stock is the capital stock towards which the economy will tend over time. Once in steady state, without shocks, economy will be stationary. Graphical Depiction: y k At k1, sf(k1) > dk1 => capital stock is growing. At k2, sf(k2) < dk2 => capital stock is shrinking. At kss, sf(kss) > dkss => steady state. Our example: y = k1/4 1/4 SS computation: s kss = d kss, 4/3 Solving, we get : kss = (s/d) 1/4 1/3 Then, yss = kss = (s/d) 3. THE IMPACT OF THE SAVING RATE ON THE STEADY-STATE Key: For each saving rate, there is a different steady state capital stock to which the economy will converge over time. Example: Consider the case of two different saving rates (s2 > s1). y f(k) s2f(k) s1f(k) k Conclusion: Higher saving rate => higher steady state of capital stock and output. Our example: y = k1/4 4/3 1/3 SS for s1 : kss = (s1/d) and yss = (s1/d) Assume s1 = d = 0.1, then kss = yss =1. Now assume that s increases to s2 = 0.2. 4/3 1/3 The new SS will be kss = (s2/d) = 2.52 and yss = (s2/d) = 1.26 What is the path for k and y, from the old SS to the new SS ? We know that kt+1 - kt = sf(kt) - dkt 1/4 So, kt+1 = (1-d)kt + sf(kt) or (in our example) kt+1 = (1-d)kt + skt The path for k is given by this equation. To get the path for y consider that y = f(k)= k1/4. 4. GOLDEN RULE STEADY STATE Golden Rule capital stock: Capital stock which maximizes steady state consumption: Max css = f(k) - dk => kGR k Solution: taking derivative and setting equal to zero yields f ‘(kGR) = d This is the rule for selecting golden rule capital stock. This has a graphical interpretation. Golden Rule savings rate: Can be obtained using sGRf (kGR) = dkGR Our example: y = f(k) = k1/4 f ‘(k) = (1/4) k-3/4 -4/3 To get the kGR, we equate this to d=0.1, and get kGR = (0.4) = 3.39 3/4 3/4 Then, sGR = d kGR = 0.1 * 3.39 = 0.25 5. POPULATION GROWTH AND THE SOLOW GROWTH MODEL ISSUES: · How does population growth impact results of the model ? Capital accumulation, Steady state, Golden rule outcome. · How might population growth explain differences in wealth across countries ? Population Growth: Suppose population grows at constant rate of n per year. n = 0.01 => population is growing at 1% per year. Capital accumulation: Three factors impacting change in stock of capital per worker: 1. Investment 2. Depreciation 3. Population growth: as number of workers grows, amount of capital per worker falls. Evolution of capital stock: The expression is a new one that takes into consideration population growth: Dk = i - (d+n)k (Can give mathematical justification) nk = effect of population growth on capital per worker => capital being spread among growing population of workers. Dk = s f(k) - (d+n)k Steady state capital stock: Dk = 0 => kss => s f(kss) = (d+n)kss Investment balancing out loss of capital stock due to depreciation and need to provide capital to new workers. (d+n)k y,i f(k) sf(k) k Result: In Steady state, capital per worker and output per worker are not changing. But, total output and total capital are approximately growing at a rate of n per year. Growth rate (X.Y) » Growth rate of X + Growth rate of Y Since Y = y. L, growth rate (Y) = growth rate (y) + growth rate (L)= 0+n Same thing for K = k.L. Importance of population growth rate: Issue: How do differences in population growth impact steady state outcomes ? (d+n2)k (d+n1)k y,i f(k) sf(k) n2 > n1 k Higher population growth yields (1) Lower steady state capital per worker; (2) Lower steady state output per worker; Population growth may explain long run differences in per capita income across countries. Golden Rule outcome: The approach is the same as before, the final result being given by: (1) capital stock f ‘(kGR) = d+n (2) saving rate sGR f(kGR) = (d+n)kGR.
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