1Intermediate Macro ZhaoUIUC Econ 303Section I: Growth theoryChapter 7a: Production function and Malthusian modelExamples of production function• Linear (perfect substitution between inputs)• Leontief (no substitution between inputs)• Cobb-Douglas (some substitution between inputs)2Intermediate Macro ZhaoUIUC Econ 303Partial derivatives• Definition• Measures the impact of changes of K alone on Y. • Marginal product of capital, MPK• How to use it?If capital increases by 1 unit, output will increase approximately by MPKunits. If capital increases by 10 units, output will increase approximately by units. Graphical representation of marginal product of capitalFor a fixed amount of labor, for example L=10, plot the production functionY5103Intermediate Macro ZhaoUIUC Econ 303Similarly, Marginal product of labor • Definition• Measures the impact of changes of L alone on Y. • How to use the marginal product of labor, MPLIf labor increases by 1 unit, output will increase approximately by MPLunits. If labor increases by 10 units, output will increase approximately by units. Cobb-Douglas production function• Production function• Marginal product of capital• Marginal product of labor4Intermediate Macro ZhaoUIUC Econ 303Production and marginal product (page 20)Production and marginal product (page 21)5Intermediate Macro ZhaoUIUC Econ 303A production function must satisfy• More input, more output– F is increasing in both K and L– Marginal product, MPKand MPLare both positive.• Diminishing marginal product– When the number of programmers is fixed, say 2, the additional codes produced when adding the first computer is large, while the additional codes produced when adding the tenth computer is small. – Marginal products are decreasing functions.– Production function is concave. (steep at the beginning and flat at the end)KYKF(K, 10)MPKMPA aggregate production function also satisfies• Constant return to scale– Doubling capital and labor input at the same time doubles output.– F(nK, nL)=nF(K, L). True for any positive number, n.• Complementarities between capital and labor– Increasing labor increases the marginal product of capital, and vice versa.– Positive cross partial derivatives6Intermediate Macro ZhaoUIUC Econ 303Cobb-Douglas production function• Constant return to scale• The exponents: share of income generate by that factor.Before industrial revolution• Production mainly uses land and labor, not much capital• Land is a resource that has fixed supply and cannot be increased. I will use D to denote the size of land input instead of L. • Production function:– F satisfies all the requirement of an aggregate production– N is size of population; we use N as a measure of labor input.– z is pulled out of F to represent productivity explicitly.7Intermediate Macro ZhaoUIUC Econ 303Changes over time – dynamic system • We use time subscripts to tell changes over time– Nt: population now– Nt+1: population next year– Nt-1: population a year ago•Let n denote the growth rate of population • Stable population net changegross changeEat and multiply• Population may change over time, depending on per person food intake, c.• Sustainable level of consumption, Enough for one person to stay alive and reproduce one to replace him/herIf ct=c*, population is stableIf ct>c*, population expandsIf ct<c*, population shrinks8Intermediate Macro ZhaoUIUC Econ 303Consumption per person• Total consumption at time t, Ct• Total output at time t, Yt• Goods market equilibrium• Hence=0closed economyno governmentno capital=0=0Land per person and population density• Land per person,• Population density, • To produce certain amount per person, we need certain size of land per person. •Let d*be amount of land needed per person so that c*units of output is produced per person•thenPopulation shrinksPopulation expands9Intermediate Macro ZhaoUIUC Econ 303Chain of the events• Suppose technology, z, does not change• Total amount of land, D, is fixedProductionReproductionSteady state of a dynamic system• Steady state– The point where the changing element stays put once it gets there. – In this model, zero population growth.– Constant population and population density.• Claim that Proof:10Intermediate Macro ZhaoUIUC Econ 303How to find the steady state of a Malthusian economy?• Economy has production function• The sustainable consumption, c*, is 200 units of consumption per person.• The land is 40 thousand acres. Solution:• How many units of land is needed to produce 200 units of consumption.• At the steady state, each person needs 4 acres of land. Population density is 0.25 person per acre. The total population is 40/4 = 10 thousand.Steady state of Malthusian world (page 22)11Intermediate Macro ZhaoUIUC Econ 303• Living standard is measured by GDP per capita • In the Malthusian model , consumption per person• At the steady state of Malthusian model, per capital consumption is at the sustainable level. • Then …What about living standard?Lucas on Industrial revolution12Intermediate Macro ZhaoUIUC Econ 303Long run behavior and comparative statics• Steady state = long run behavior = the state of affairs after a while. • Something changes in the society. When dust settles down, we move to a new steady state. What would change? – comparative statics– Technological progress over time• We can also use the same idea to compare two different economies.– A larger country– A country is endowed with more fertile landGraphic representationThe fundamental difference equation 45 degree line13Intermediate Macro ZhaoUIUC Econ 303A larger economy and a more productive economy45 degree lineSteady state is stable• Stable: after temporary changes, economy goes back to the original steady state.• A plague or major war leads to a temporary higher death rate then usual.• Good whether temporarily increase the