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Chapter 7 Class notes on Production Function, Malthusian Model, Capital accumulation, and Solow model

by: Mitchell Chin

Chapter 7 Class notes on Production Function, Malthusian Model, Capital accumulation, and Solow model Econ 303

Marketplace > University of Illinois at Urbana-Champaign > Economics > Econ 303 > Chapter 7 Class notes on Production Function Malthusian Model Capital accumulation and Solow model
Mitchell Chin

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About this Document

These are just class notes. If you have seen my previous notes, you can see that my class isn't going in order of the chapters. Just keep that in mind!
Rui Zhao
Class Notes
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This 6 page Class Notes was uploaded by Mitchell Chin on Friday September 9, 2016. The Class Notes belongs to Econ 303 at University of Illinois at Urbana-Champaign taught by Rui Zhao in Fall 2016. Since its upload, it has received 5 views. For similar materials see Macroeconomics in Economics at University of Illinois at Urbana-Champaign.


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Date Created: 09/09/16
Chapter 7a) Production Function and Malthusian Model Production Functions: Linear (perfect substitution between inputs): Y = K + L Leontief (no substitution between inputs): Y = min{K,L} Cobb Douglas (some substitution between inputs): Y = K  L a b K is capital, L is labor, z is total factor productivity (TFP). a b Partial Derivatives using Y = zK  L : Marginal product of Labor: MP  L bzK  L a b­1 Marginal product of Capital: MP  = azK  L a­1 b K Usually more input means more output. Both marginal products are positive. Diminishing Marginal Product: As we continue to increase one of the inputs, keeping the other constant, the output increases less and less. Aggregated Production Function satisfies: Constant Return to Scale: Doubling K and L at the same time gives double the output. F(nK, nL)=nF(K, L) stays true as long as n is positive. Complementarities between capital and labor:  Keeping L constant and doubling K will decrease MP  but wiKl increase MP  by less Lhan doubled. Keeping K constant and doubling L will decrease MP  but wiLl increase MP  by less Khan doubled. a b Constant Return to Scale has Y = zK  L  where a + b = 1. Capital Share a = Compensation for K/GDP Labor Share b = Compensation for L/GDP = 1­a Before Industrial Revolution:  Production mainly uses land and L, not much K.  Land is a resource that has fixed supply and cannot be increased.  Production function: Y = zF(D,N)  F satisfies all the requirement of an aggregate production   D is size of land input  N is size of population; we use N as a measure of labor input.   z is pulled out of F to represent productivity explicitly n is the population growth rate: net change n = (N  –t+1/ N t t N ts population now gross change 1 + n = N / N t+1 t N t+1 population next year stable population n = 0 N  = Nt+1 t N t­1 population last year Population may change over time, depends on per person food intake, c 1 + n = g(c) t * Sustainable level of consumption, c For one person to stay alive and reproduce N t+1N(1 + t) = N x t(c) t t g(c ) = 1 If ct= c , population is stable g’(c) < 0 if c > c , g(c) > 1 if ct> c , population expands * * if c < c , g(c) < 1 if ct= c , population shrinks Total consumption at time t, C t C t N x ct t Total output at time t, Y t Y t zF(D,N) t Goods market equilibrium C t + It + G t + NX t = Y t It= 0, no capital G t 0, no government NX = 0, closed economy t c = C/G = Y/ N = zF(D,N)/ N = N F(D/N,1)/N = zF(D/N,1) t t t t t t t t t t t Land per person and population density: Land per person, d = D/N t t Population density, 1/d = N/D t t To produce certain amount person, we need certain size of land per person: C =tzF(d,1) t dtincreases then c = incrtase d , amount of land needed per person so c  units of output is produced per person * * c  = zF(d ,1) then d > d , c > c , n > 0, population expands * * d < d , c < c , n < 0, population shrinks Chain of the events Suppose technology, z, doesn’t change Total amount of land, D, is fixed ct= zF(d,1) t 1 + n =tg(c) t N  d = D/N  c  1 + n    N  … N  = N x (1 + n) t t t t t t+1 t+1 t t Steady State: Point where the changing element stays put once there. * Zero population growth, thus  N  t+1 = N t Constant population and population density. * * Claim that, D/N  = d * * d , the amount of land per person, sufficient to produce c  units of consumption per person. EX: How to find the steady state of a Malthusian Economy? Economy has production function c=zF(d,1) = 100d 1 =100d 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 Y = 100D N   c = Y/N = 100D N /N = 100d Sustainable consumption, c , is 200 units of consumption per person. The land is 40,000 acres. How many units of land is needed to produce 200 units of consumption? 0.5 ANSWER: 200 = 100d d  = 2 d = 4 At steady state, each person needs 4 acres of land. Population density is 0.25 person per acre.  The total population is 40/4 = 10,000. Living Standard: … measured by GDP per capita Y/N In Malthusian model Y/N = c, consumption per person At steady state of Malthusian model, per capital consumption is at the sustainable level. * c = c Long run behavior and comparative statics: Steady state = long run behavior = the state of affairs after a while. With new steady state, comparative statics changes and technology progresses over time. Steady state is stable Stable: after temporary changes, economy goes back to the original steady state. Plague or major wars leads to temporary higher death rate than usual N t N   d > dt  c > c  tn > 0, N  > N t+1 t Good whether temporarily increase the productivity ct­1 c   n > 0  N > N  t d < d   c t c   n < t, N  < N t+1 t Chapter 7b) Capital accumulation and Solow model Industrial revolution: from Malthus to Solow Before: Y = zF(D,N) D can’t grow over time t After: Y = zt(K,N) Induttritl Revolution brought capitalism and it improved over time. Changes of capital over time: Investment Constant percentage of GDP is saved and invested every period s: saving rate = investment rate It= s x Y t Depreciation, consumption of fixed capital Every period a constant proportion of capital becomes useless d: depreciation rate Capital lost per period: d x K t Over time K  = K – d x K + s x Y t+1 t t t Living standard output per person y = Y/N capital per person k = K/N, capital intensity Assume no population growth, N = N for all t t K /t+1 K/N = st/N – dK/N t t k t+1k = ty – dk t nk t t Per capita production function: y = f(k) = zk a Aggregate production function: Y = F(K,N) = zK N a 1­a a k t+1k = t(zk ) – tk ­nk t t Steady state: constant capital intensity, k * * k t+1k = t , k  ­ t+1 0 t This happens when the newly added machines (investment) are just enough to replace the dying  out ones (depreciation) and to equip new workers (dilution). Sz(k )  = (d + n)k * * * a At steady state, output per person is y  = z(k ) Goods market equilibrium: C t + It + G t + NX t = Y t It= Y t G t 0, no government NX =t0, closed economy * * Consumption per person c  = (1 ­ s)y a­1 Marginal Product of Capital: MP  = f ’(k)K= azk Marginal Product of Labor: MP  = f(k) – f ’(k)k = (1­a)f(k) L Golden rule of investment: the investment rate that maximizes consumption Higher isn’t always better. Graphically, find the max or the peak of the parabola. * * c  = (1 – s)f(k (s)) How to find: * * * * * c  = f(k ) – sf(k ) = f(k ) – (d + n)k f ‘(k ) – (d + n) = 0


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