The Ramsey Model. (Lecture Note, Advanced Macroeconomics, Thomas Steger, SS 2013)

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1 The Ramsey Model (Lecture Note, Advanced Macroeconomics, Thomas Steger, SS 213) 1 Introduction The Ramsey model (or neoclassical growth model) is one of the prototype models in dynamic macroeconomics. Extended versions of this dynamic general equilibrium model have been used to investigate important macroeconomic questions, like one of the following: 1. How can one explain the huge international differences in per capita incomes (see also "level accounting") and the substantial international differences in the growth rates of per capita incomes (see also "growth accounting"). 2. How can endogenous technical change be explained within a general equilibrium context and what are the main determinants of technical change? 3. What are the consequences of fiscal policies (taxation, redistribution, and productive government spending) in a dynamic perspective? 4. What are the driving forces behind sectoral change (see the so-called "Kuznets facts" on sectoral change and economic growth)? 5. Assuming that there are stochastic technology shocks, can such a simple model (extended by endogenous labor supply) explain the stylized facts of business cycle fluctuations ("RBC theory")? 6. How do rising prices for raw materials (like oil) affect output, employment, and welfare in the short run and in the long run? For studying "Advanced Macroeconomics" it is absolutely necessary to possess a sound understanding of this basic model. This short note summarizes the main assumptions, the basic economic logic, and the major implications. Please notice that learning must proceed sequentially. This means that some aspects need not be fully grasped at this stage. For instance, it is not necessary to understand all the details behind the maximum principle (or dynamic programming). 2 The market economy 2.1 Households There is a large number of identical and infinitely lived households (interpreted as dynasties) of "mass one". 1 The typical household is endowed with 1 This means that housholds are indexed by a continuous index [ 1]. Average quantities hence coincide with aggregate quantities since 1 () = (). 1

2 units of time per period, which are inelastically supplied to the market, and possesses wealth, whichisrentedtofirms. Total income is () = () + ()(), where(): wage rate; (): interest rate; [ ]: time index. The typical household is assumed to maximize its intertemporal welfare () with. This amounts to solving the following = R dynamic optimization problem 2 Z 1 1 max {} 1 (1) = + () = where :=. There are several ways to solve such a dynamic problem (for instance, Silberberg, 199; Chiang, 2). We apply the so-called maximum principle, which requires to set up the (current-value) Hamiltonian function 3 = ( + ) 1 where () denotes the shadow price of wealth. Next we form the usual first-order conditions = = (2) = + = + (3) Equations (2), (3), = + together with () = and ( ) = (for the determination of see below) describe the optimal solution of the problem under study. 4 Equation (2) implies = Combining this equation with (3) gives the Keynes-Ramsey rule of optimal consumption = 1 ( ) The Keynes-Ramsey rule says, very briefly, that whenever (economic interpretation!) and that (economic interpretation!). ( ) = 1 2 To simplify the notation the time index is often supressed in what follows. 3 The (current-value) Hamiltonian can be viewed as net national product in utility terms (Solow, 2, p. 127). This can be seen more clearly by writing the Hamiltonian as = ()+ and noting that the shadow price has the dimension "utility per unit of numeraire good". 4 Hence, this is a well-defined boundary-value problem. Background information for advanced readers: The final-boundary condition ( ) = results from steady state conditions ( = =) together with the transversality condition =and the continuity requirement for the shadow price. These latter requirements exclude all border steady states. 2

3 2.2 Firms There is a large number (of mass one) of identical firms producing a homogenous final output good that can be used universally for consumption or investment. Final output is chosen as the numeraire good, i.e. its price is set equal to unity ( =1). The output technology of the representative firm is Cobb-Douglas = 1 (4) where 1, denotes physical capital and labor input. The firm maximizes profits in a perfectly competitive environment. This implies that it rents capital and hires labor up to a point such that =(1 ) =(1 ) (5) = 1 1 = (6) where is the capital depreciation rate. 2.3 Market equilibrium and national income Capital market equilibrium requires =. Labor market equilibrium has already been assumed since denotes (exogenous) labor supply in (1) and labor demand by firms in (5). Total income of the typical household equals national income. This is due to the normalization according to which there is a continuum of mass one of households; formally: R 1 () = () (since all households are identical, we can suppress the index ). By noting (5) and (6) we get = + =(1 ) + = National income hence equals aggregate production net of depreciation. 2.4 Economic dynamics in macroeconomic equilibrium The evolution of the economy in macroeconomic equilibrium (with = 1 1, =, and = 1 ) is then governed by the following dynamic system = 1 ( 1 1 ) (7) = 1 (8) with boundary conditions () = and ( ) =. The unique (nontrivial) steady state results from = µ 1 = + 3 =.From 1 1 =and (7) we get

4 Plugging this solution into (8) gives = 1 Since =, the saving rate in the steady state is zero, i.e. = =.5 3 The social planner s solution The social planner maximizes welfare of the representative household. associated problem reads as follows The Z 1 1 max {} 1 = 1 () = The (current-value) Hamiltonian function is given by = ( 1 ) 1 where () denotes the shadow price of capital. Thefirst-order conditions are = = (9) = + = (1) From (9) we get = and hence, by noting (1), we have = 1 ( 1 1 ) (11) The dynamic evolution of the socially controlled economy is determined by (11), = 1, () = and ( ) =. Thisisexactlythe same system as the one which governs the dynamics of the market economy. Important implication. The decentralized equilibrium coincides with the social planner s solution, implying that the market equilibrium represents the (unique) first-best solution. This result is, of course, driven by the fact that the neoclassical model under study represents a perfect economy, i.e. markets are complete, individuals are perfectly informed and make rational decisions, there are no externalities, and product and factor markets are perfectly competitive. 5 This implication changes once we allow for either population growth or technical change. 4

5 4 Final comments 1. The model can readily be extended to include (exogenous) technical change. In this case, growth in per capita income is driven by technical change (increases in total factor productivity) together with capital accumulation. Formally, an output technology of the form = 1 implies = ( := and := ). Growth in per capita income can then be expressed as ˆ = ˆ + ˆ (where ˆ := etc.).6 2. The neoclassical model represents a one-sector model. There is one finaloutput sector which produces a commodity that can be used for consumption or investment. This should be interpreted in the sense that inputs (capital and labor) are either used to produce consumption or investment goods. Indeed, if the following assumptions hold, the one-sector economy is equivalent to the two-sector economy with two separate sectors for consumption and capital goods: (i) both sectors use the same production technology; (ii) the production functions exhibits CRS; (iii) input factors can move costless from one sector to another. Under these assumptions, the transformation curve in the ()-plane is linear. 3. There are a lot of technical details which have not been discussed here (sufficiency conditions, stability of the steady state and the like). Some of these aspects will be considered in the course "Quantitative Dynamic Macroeconomics". References [1] Barro and Sala-i-Martin, 24, MIT Press, Chapter 2. [2] Chiang, A. C., Elements of Dynamic Optimization, McGraw Hill, 2, New York. [3] Silberberg, E., The Structure of Economics: A Mathematical Analysis, Mc Graw-Hill, 199, New York. [4] Solow, R. M., Growth theory: an exposition, Oxford University Press, Oxford, 2. 6 ˆ + ˆ has been termed the speed limit of economic growth. 5