The Neoclassical Growth Model
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1 The Neoclassical Growth Model 1 Setup Three goods: Final output Capital Labour One household, with preferences β t u (c t ) (Later we will introduce preferences with respect to labour/leisure) Endowment Initial stock of capital k 0 One unit of labour in each period Technology Capital and labour can be combined to produce final output according to production function Y = f (K, L) Final output good can be consumed or invested. Investing converts output into capital one-for-one: Y t = I t + C t (we may or may not impose I t 0, depending on whether we want to allow capital to be converted back into final output; typically it doesn t matter whether we do) Capital depreciated at rate δ, so K t+1 = (1 δ) K t + I t 1
2 2 Competitive Equilibrium Markets: For output at time t. Price p t. Output at time zero is numeraire (p 0 0) For labour at time t. Price is w t p t. This means that the price of labour in terms of time-t output is w t. For capital services at time t. Price is rt K p t. This means that the price of capital services in terms of time-t output is rt K What does it mean to hire capital services? I own one unit of capital (a machine) I let you use it for one period You return it to me (depreciated) at the end of the period Production is carried out by a representative firm. The representative firm is owned by the household The firm takes prices w t, rt K, p t as given and has access to the productive technology, but it doesn t own anything. It hires capital and labour and combines them to produce output seeking to maximize profits, which it then pays out to the household that owns it The household gets income from selling its labour, from renting the capital that it owns and from the firm s profits. It chooses how much to consume each period and how much capital to build. Firm s problem: π = max {y t,k t,l t} p t y t p t w t L t p t rt K k t s.t. y t = f (k t, L t ) (1) 2
3 Household problem: max {c t,i t,k t+1 } s.t. β t u (c t ) k t+1 = (1 δ) k t + i t [ ] p t (c t + i t ) pt w t + p t rt K k t + π k 0 given (2) Note that implicit in the way we formulate the household problem is that there exist markets for trading goods of different periods at prices p t. Since there is a single household, in equilibrium they will not be trading these claims, but the prices will have to be such that this is an optimal choice. Definition 1. A competitive equilibrium is given by an allocation {y t, c t, k t, L t } { } pt, w t, rt K such that: and prices 1. The firm solves problem (1), taking prices as given 2. The household solve problem (2), taking prices as given 3. Markets clear: (a) y t = c t + i t (b) L t = 1 (c) k t chosen by the firm equals k t chosen by the household 3 Characterizing the Equilibrium 3.1 Firm s problem Static problem: maximize profits period by period Simplifies to: max f (k, L) k,l rk k wl FOC: f K (k, L) r K = 0 f L (k, L) w = 0 3
4 Assumption: f ( ) is homogenous degree 1 and differentiable. Definition 2. f (k, L) is h.d.1 if f (λk, λl) = λf (k, L) for any λ > 0 h.d.1 is just another way of saying constant returns to scale Proposition 1. (Euler s Theorem) If f (k, L) is h.d.1 and differentiable, then f K k + f L L = f (k, L) Proof. We know that f (λk, λl) = λf (k, L) Differentiate both sides w.r.t. λ: f K (λk, λl) k + f L (λk, λl) L = f (k, L) Evaluating at λ = 1 gives the result. Proposition 2. In equilibrium, firms make zero profits Proof. Profits are given by: π = = [ ] p t f (kt, L t ) rt K k t w t L t p t [f (k t, L t ) f K (k t, L t ) k t f L (k t, L t ) L t ] = 0 where the first step follows from the firm s FOC and the second from Proposition Household Problem Household FOC: β t u (c t ) λp t = 0 This leads to a difference equation: β t u (c t ) β t+1 u (c t+1 ) = λp t λp t+1 u (c t ) = β p t p t+1 u (c t+1 ) u (c t ) = βr t+1 u (c t+1 ) (3) 4
5 where R t+1 p t p t+1 is the gross interest rate between periods t and t + 1. Why? Give up one unit at time t obtain p t Use this to buy t + 1 goods obtain pt p t+1 Equation (3) is an Euler equation - we ve encountered it before No arbitrage requires R t+1 = r K t δ Two ways of transfering resources from one period to the next (or viceversa) Trading goods of differet dates in the market Converting goods into capital and renting it out 3.3 Equilibrium From household: From firm: Combining: u (c t ) = β [ r K t δ ] u (c t+1 ) f K (k, 1) = r K u (c t ) = β [f K (k t+1, 1) + 1 δ] u (c t+1 ) (4) Plus the resource constraint k t+1 = f (k t, 1) c t + (1 δ) k t (5) (4) and (5) define a system of two difference equations. We need two initial / terminal conditions to have the solution k 0 given is one of them What is the other? 5
6 3.4 The steady state and a phase diagram A steady state is defined as {c ss, k ss } such that if c t = c ss and k t = k ss, then according to (4) and (5), c t+1 = c ss and k t+1 = k ss From (4): β [f K (k ss, 1) + 1 δ] (6) From (5): k ss = f (k ss, 1) c ss + (1 δ) k ss c ss = f (k ss, 1) δk ss (7) Interpretation Phase diagram: 6
7 Does the system converge to steady state? 4 The Social Planner s Problem Single household, so no need to specify weights in a welfare function 7
8 Planner solves max {c t,i t,k t+1 } s.t. β t u (c t ) k t+1 = (1 δ) k t + i t c t + i i f (k t ) k 0 given This is a special case of a general problem of the form (SLP notation) sup {x t} β t F (x t, x t+1 ) s.t. x t+1 Γ (x t ) x 0 given t Mapping: x t k t F (x t, x t+1 ) u (f (k t ) + (1 δ) k t k t+1 ) Γ (x t ) = [0, f (k t ) + (1 δ) k t ] We ll spend the next few classes looking at this mathematical problem 8
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