Solow Model Introduction I The production function is useful in describing the state of the economy or compare economies at di erent time periods I It is also a useful starting place for accounting for the di erent factors that contribute to growth I However it does not provide an explanation for a number of important issues I What is the process of growth is it a ected by some primitive decisions parameters or policies I What happens beyond GDP How about consumption The Solow Growth Model provides answers to some of these questions 46 91 Solow Model Main Questions 1 What is the relationship between long run standards of living and fundamental factors such as saving rates population growth technological progress 2 How does growth evolve over time Will it stabilize Accelerate Stop 3 Are there economic forces that generate catch up In levels or growth rates 47 91 A few things to note From a learning perspective a few remarks I This model is our first encounter with dynamics or the evolution of the economy over time I Exogenous vs Endogenous variables I Some assumptions may seem unreasonable but we ll see how it helps simplifying the analysis 48 91 Assumptions and Notation I Nt is the number of workers population at time t I n population growth rate Nt 1 1 n Nt I Yt Ct It I I No Government expenditure G 0 Closed economy N X 0 I Capital depreciates at a constant rate 0 I We focus on per worker terms Ct t I y Yt ct N kt K t Nt Nt t it 1 It Nt I A standard production function I Individuals save and invest a constant fraction s of their income It sYt 49 91 Production I With constant returns we know that yt Yt At F Kt Nt At F Nt Nt Kt N t Nt Nt At f kt 1 3 I e g Cobb Douglas and 2 3 labor share of output yt At kt I The rest of the assumptions imply that the shape of this per worker production function is concave 50 91 Investment and Capital Accumulation I Capital K evolves over time Kt 1 1 Kt It I Some of the existing stock depreciates at rate I New capital is accumulated through investment I I What does this say about the conversion of consumption to investment goods 51 91 Investment and Capital Accumulation I Capital K evolves over time Kt 1 1 Kt It I Some of the existing stock depreciates at rate I New capital is accumulated through investment I I What does this say about the conversion of consumption to investment goods I Derive the evolution of kt Kt Nt 52 91 Investment and Capital Accumulation I Capital K evolves over time Kt 1 1 Kt It I Some of the existing stock depreciates at rate I New capital is accumulated through investment I I What does this say about the conversion of consumption to investment goods I Derive the evolution of kt Kt Nt kt 1 1 kt it 1 n 53 91 Steady State Definition A steady state is when kt yt ct and it are all constants i e don t change over time I Let the constants be k ss kt for all t css ct for all t etc I For a steady state to exist K Y C and I all grow at a constant rate pause kt 1 kt k ss Kt 1 Kt Nt 1 Nt Kt 1 Nt 1 1 n Kt Nt 1 2 3 54 91 Steady State Definition A steady state is when kt yt ct and it are all constants i e don t change over time I Let the constants be k ss kt for all t css ct for all t etc I For a steady state to exist K Y C and I all grow at a constant rate pause kt 1 kt k ss Kt 1 Kt Nt 1 Nt Kt 1 Nt 1 1 n Kt Nt I 1 2 3 Intuition steady state provides average levels adjusted for population growth of the economy in the very long run 55 91 Steady States are useful because I We can calculate the levels of c i k and y I c is important We ll use it to argue that increasing y is not necessarily the best thing to do why 56 91 Steady States are useful because I We can calculate the levels of c i k and y I c is important We ll use it to argue that increasing y is not necessarily the best thing to do why I We can ask if and how we expect the economy to actually reach a steady state I It is a useful departure point for analyzing the e ects of certain changes shocks that an economy may experience 57 91 Steady State Investment I We know that k K N should be constant and that Kt 1 1 n Kt I Derive the equation for steady state investment kt 1 1 kt it 1 n iss n k ss I In order to keep the capital labor ratio constant we must invest in order to exactly compensate for depreciation and population growth rates 58 91 Steady State Consumption I How to calculate start with the resource constraint Yt C t It css y ss iss Af k ss yt c t i t c t yt it n k ss I The easiest way to understand this is to plot this I Consumption is the di erence between the concave production function and the linear investment line 59 91 60 91 Finding a steady state The importance of k I I Recall the fixed savings rate 0 s 1 is given exogenously If we know k ss can characterize the state of the economy i e the rest of the variables y ss Af k ss iss sAf k ss css 1 I s Af k ss In order to find the steady state all we need is to find a solution for k that is consistent with a steady state I Required investment to compensate for depreciation and population growth is iss n k ss I Actual investment comes from s iss sy ss sAf k ss I In Steady state actual required sAf k n k 61 91 Finding a steady state 62 91 Finding a steady state example I Assume a Cobb Douglas production function y Af k Ak I In a steady state Derive k ss iss y ss and css I Recall the system of equations sA k ss n k ss y ss A k ss iss sy ss css 1 s y ss 63 91 Convergence to steady state A key question is whether the economy actually reaches the steady state I Assume an economy that is not in steady state will it get there if so what s the process I Assume an economy in a steady state and a shock happens Will it reach a new steady state what is the process of getting …