UIUC Econ 303 Section I Growth theory Chapter 7a Production function and Malthusian model Examples of production function Linear perfect substitution between inputs Leontief no substitution between inputs Cobb Douglas some substitution between inputs Intermediate Macro Zhao 1 UIUC Econ 303 Partial 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 MPK units If capital increases by 10 units output will increase approximately by units Graphical representation of marginal product of capital For a fixed amount of labor for example L 10 plot the production function Y 5 Intermediate Macro Zhao 10 2 UIUC Econ 303 Similarly Marginal product of labor Definition Measures the impact of changes of L alone on Y How to use the marginal product of labor MPL If labor increases by 1 unit output will increase approximately by MPL units 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 labor Intermediate Macro Zhao 3 UIUC Econ 303 Production and marginal product page 20 Production and marginal product page 21 Intermediate Macro Zhao 4 UIUC Econ 303 A production function must satisfy More input more output F is increasing in both K and L Marginal product MPK and MPL are 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 Y F K 10 MP MPK K K A 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 derivatives Intermediate Macro Zhao 5 UIUC Econ 303 Cobb 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 Intermediate Macro Zhao 6 UIUC Econ 303 Changes 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 net change gross change Stable population Eat 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 her If ct c population is stable If ct c population expands If ct c population shrinks Intermediate Macro Zhao 7 UIUC Econ 303 Consumption per person Total consumption at time t Ct Total output at time t Yt Goods market equilibrium no government 0 0 0 no capital closed economy Hence Land 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 then Population expands Population shrinks Intermediate Macro Zhao 8 UIUC Econ 303 Chain of the events Suppose technology z does not change Total amount of land D is fixed Production Reproduction Steady 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 Intermediate Macro Zhao 9 UIUC Econ 303 How 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 Intermediate Macro Zhao 10 UIUC Econ 303 What about living standard 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 Lucas on Industrial revolution Intermediate Macro Zhao 11 UIUC Econ 303 Long 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 land Graphic representation The fundamental difference equation 45 degree line Intermediate Macro Zhao 12 UIUC Econ 303 A larger economy and a more productive economy 45 degree line Steady 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 productivity Intermediate Macro Zhao 13