Lecture 2: The Neoclassical Growth Model

Transcription

1 Lecture 2: The Neoclassical Growth Model Florian Scheuer 1 Plan Introduce production technology, storage multiple goods 2 The Neoclassical Model Three goods: Final output Capital Labor 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 labor in each period Technology Capital and labor can be combined to produce final output according to production function Y = f (K, L) 1

2 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 depreciates at rate δ, so K t+1 = (1 δ) K t + I t 3 Competitive Equilibrium Markets: For output at time t. Price p t. Output at time zero is numeraire (p 0 0) For labor 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 r t p t. This means that the price of capital services in terms of time-t output is r t Equivalently (sequential formulation): at each time t there is a market for Output next period at price q t+1 (or interest rate R t+1 = 1 q t+1 ) Labor at price w t Capital services at price r t 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, r t, p t as given and has access to the productive technology, but it doesn t own anything. 2

3 It rents capital and hires labor and combines them to produce output seeking to maximize profits, which it then pays out to the household that owns it. (Under constant returns to scale, which we assume just below, profits are zero in equilibrium) The household gets income from selling its labor, 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: Household problem: π = max p t [y t w t l t r t k t ] {y t,k t,l t } y t = f (k t, l t ) max β t u (c t ) {c t,i t,k t+1 } k t+1 = (1 δ) k t + i t p t (c t + i t ) p t [w t + r t k t ] + π k 0 given (1) (2) Definition 1. A competitive equilibrium is given by an allocation {Y t, C t, K t, L t } and prices {p t, w t, r t } such that: 1. Y t, K t, L t solve the firm s problem (1), taking prices as given 2. C t, I t, K t solve the household s problem (2), taking prices as given 3. Markets clear: (a) Y t = C t + I t (b) L t = 1 4 Characterizing the Equilibrium 4.1 Firm s problem Static problem: maximize profits period by period 3

4 Simplifies to: max k,l f (k, l) r K k wl FOC: f K (k, l) r = 0 (3) f L (k, l) w = 0 (4) Assumption 1. F( ) is homogeneous degree 1 and differentiable Definition 2. f (K, L) is h.d.1 if f (λk, λl) = λ f (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, L) k + f L (K, L) L = f (K, L) Proof. We know that f (λk, λl) = λ f (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 = = = 0 p t [ f (k t, l t ) r t 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 ] where the first step follows from the firm s FOC and the second from Proposition 1. Some hidden assumptions in the proof. We have used the first order conditions (3) and (4), but these are only valid if the firm s problem has an interior solution. 4

5 The logic of the proof is that the market clearing condition requires that the firm s problem have an interior solution; otherwise the labor or capital markets wouldn t clear Household Problem Household FOC: β t u (c t ) λp t = 0 This leads to a difference equation: where 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 ) (5) R t+1 p t = 1 p t+1 q t+1 is the gross interest rate between periods t and t + 1. Equation (5) is called an Euler equation No arbitrage requires R t+1 = r t δ Two ways of transfering resources from one period to the next (or vice versa) Trading goods of different dates in the market Converting goods into capital and renting it out Also follows from FOCs w.r.t. i t, k t Equilibrium From household: From firm: Combining: u (c t ) = β [r t δ] u (c t+1 ) f K (k, 1) = r u (c t ) = β [ f K (k t+1, 1) + 1 δ] u (c t+1 ) (6) 5

6 Plus the resource constraint k t+1 = f (k t, 1) c t + (1 δ) k t (7) (6) and (7) 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? 4.4 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 (6) and (7), c t+1 = c ss and k t+1 = k ss From (6): β [ f K (k ss, 1) + 1 δ] = 1 (8) From (7): k ss = f (k ss, 1) c ss + (1 δ) k ss c ss = f (k ss, 1) δk ss (9) Interpretation Phase diagram: 6

7 Does the system converge to steady state? 5 The Social Planner s Problem Single household, so no need to specify weights in a welfare function Planner solves max β t u (c t ) {c t,i t,k t+1 } 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 ) 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 class looking at this mathematical problem 7