Macroeconomics I University of Tokyo Lecture 12 The Neo-Classical Growth Model: Prelude to LS Chapter 11. Julen Esteban-Pretel National Graduate Institute for Policy Studies
The Cass-Koopmans Model: Environment Time is discrete time: t = 0,1,2,... Continuum of homogeneous agents of mass one: Representative agent. Endowment: Households have 1 ut of labor (per period) and initial capital k0. Preferences: The representative household s objective function is: t u(c t ), (0, 1). (12.1) where we assume that u is strictly increasing, strictly concave, differentiable, and satisfies the Inada condition (lim u (c) = ). c 0 - Note: HHs obtain no utility from leisure, so supply of labor is nt = 1. Technology: Production function is F(kt,nt) = F(kt,1) f(kt). Assume that f( ) satisfies the usual: f(0) = 0, f > 0, f < 0, lim k f (k) = 0, lim k 0 f (k) =. Resource constraint (RC): c t + k t+1 F(k t, 1) + (1 )k t (t = 0, 1, 2,...), k 0 > 0 given. (12.2) A feasible allocation is a non-negative sequence {c t } such that there exists a non-negative sequence {k t+1 } satisfying the resource constraint. 2
Social Planner s Problem The Social Planner s problem is max t u(c t ) {c t,k t+1 } (12.1) This is the same as the SP s problem solved in the complete markets lectures where we had I agents (such problem is called a Negishi program), but now: u(c t ) We can write the Lagrangian as: s.t. c t + k t+1 f(k t ) + (1 )k t (t = 0, 1, 2,...) k 0 > 0 given. max c 1 t,c2 t,...,ci t I i=1 iu i (c i t) s.t. I i=1 c i t c t. (12.2) (12.3) L = t u(c t ) + t t [f(k t ) + (1 )k t c t k t+1 ]. where β t λt is the Lagrange multiplier associated with (12.2). Note: Do not confuse λt with the Negishi weights. (12.4) 3
Social Planner s Problem Optimal Conditions Restating the Lagrangian: FOCs: k t+1 : t = t+1 [f (k t+1 ) + (1 )]. Combining (12.5) and (12.6) we can obtain the Euler equation: The Euler equation (12.7) and the resource constraint (12.2) form a system of two first-order non-linear difference equations in (ct,kt). We can use the RC (12.2) into (12.7) to eliminate ct and write the Euler equation as a second-order non-linear difference equation in kt: (12.8) u (f(k t ) + (1 )k t k t+1 ) = [f (k t+1 ) + (1 )]u (f(k t+1 ) + (1 )k t+1 k t+2 ). To solve for the sequence {k t+1 } we need another condition besides k0. We use the transversality condition (TVC): lim t u [f(k t ) + (1 )k t k t+1 ][f (k t ) + (1 )]k t = 0. (12.9) t L = t u(c t ) + c t : u (c t ) = t, t t [f(k t ) + (1 )k t c t k t+1 ]. (12.5) (12.6) u (c t ) = [f (k t+1 ) + (1 )]u (c t+1 ). (12.7) 4
The Steady State A steady state is an equilibrium where variables are constant: Suppose that such a solution exists. That is, suppose that there exists a solution to the Euler equation (12.8) satisfying: - initial condition k0. - property that kt k * What is the steady-state solution? k t = k, c t = c } The SS capital stock k * can be found using the Euler equation (12.7) after imposing ct = ct+1 = c * : [f (k ) + (1 )] = 1 (12.11) or f 1 (k ) = + where (12.12) 1 + This is called the modified golden rule. The SS consumption level c * can be found using the RC (12.2): c = f(k ) k (12.13) t (12.10) This solution satisfies TVC, hence is a solution to the SP problem. 5
Competitive Equilibrium - Complete Markets Assume that markets are complete. Trades take place at date 0. Capital is owned by the household and rented out to firms. 3 physical commodities: The good. Capital services. Labor. Let {qt, rt, wt} be the Arrow-Debreu prices of these commodities. Note: We could allow firms to own the capital and issue shares, but the equilibrium allocations would be the same. 6
Firms Problem of the firm: max [q t F(k d {n d t,kd t } t, n d t ) r t k d t w t n d t ]. (12.14) The problem of the firm is in reality a static problem. The firm chooses labor and capital period by period to maximize current profits. max n d t,kd t [q t F(k d t, n d t ) r t k d t w t n d t ]. (12.15) FOCs (MP conditions): n d t : w t = q t F n (k d t, n d t ), k d t : r t = q t F k (k d t, n d t ). (12.16) (12.17) 7
Households The household s problem is: max {c t,k s t+1 } t u(c t ) (12.18) s.t. q t (c t + k s t+1 (1 )k s t) (r t k s t + w t n s t). (12.19) c t 0, k s t+1 0, and 0 n s t 1 (t = 0, 1, 2,...). k s 0 = k 0 > 0 given. As before, since agents get no utility from leisure: n s t = 1. Note: We do not include dividends from firms since profits are zero in equilibrium (due to the MP conditions and Euler s theorem). 8
Household s Problem Optimal Conditions Let µ be the Lagrange multiplier associated with the budget const. (12.19). We can write the Lagrangian as: FOCs: L = t u(c t ) + µ [(r t k s t + w t n s t) q t (c t + k s t+1 (1 )k s t)]. (12.20) c t : t u (c t ) = µq t (ct > 0 due to Inada cond.) (12.21) k t+1 : r t+1 + q t+1 (1 ) q t, " = " if k s t+1 > 0. (12.22) From (12.21) in t and t+1: u (c t ) = q t q t+1 u (c t+1 ) (12.23) 9
Equilibrium Def: A competitive equilibrium is a sequence of Arrow-Debreu prices {q t, r t, w t } and associated quantities {k d t, n d t, c t, k s t+1, n s t} with k s 0 = k 0 such that: (i) Optimization: Given 0 = k 0 and {q t, r t, w t } - {c t, k s t+1, n s t} solve the HH s problem. - {k d t, n d t } solve the firm s problem. (ii) Market clearing: k s - Goods: - Capital: - Labor: c t + k s t+1 (1 )k s t = F(k d t, n d t ), k d t = k s t, n d t = n s t (= 1). (12.24) (12.25) (12.26) 10
Equilibrium Conditions Imposing the factor market clearing conditions: k t = k s t = k d t, n t = n s t = n d t In equilibrium, kt+1 > 0 and nt = 1 for all t. As before, we write f(kt) F(kt,1). We can then write the equilibrium conditions as: (MP) r t = q t f (k t ), w t = q t (f(k t ) k t f (k t )), (12.16), (12.17) (HH FOCs) u (c t ) = q t q t+1 u (c t+1 ), q t = r t+1 + q t+1 (1 (RC) c t + k t+1 (1 )k t = f(k t ). ), (12.21), (12.22) (12.24) Note: The HH s BC is implied by the RC, MP an Euler s theorem. From the MP and HH s FOCs, we can derive the Euler equation: u (c t ) = [f (k t+1 ) + 1 ]u (c t+1 ). (12.27) Obtain (ct,kt) from RC and Euler eq. (12.27). Get (rt,wt) from the MP cond. 11
Transversality Condition The initial condition k0, Euler and RC are not enough to pin down {c t, k t }. We need the TVC. The TVC is implicit in the HH s budget constraint. We have assumed that the infinite sums such as are well defined. [r tk t + q t (1 )k t ] However, necessary condition for this infinite sum to converge is: - From the HH s FOC: lim [r tk t + q t t (1 )k t ] = 0. q t = 1 µ t u (c t ) (12.28) (12.29) - From the Firm s FOC: r t = q t f (k t ) (12.30) Using (12.29) and (12.30) into (12.28) we obtain the transversality cond. We have shown that the conditions determining the complete markets are the same as the ones characterizing the SP s equilibrium. Hence the CE in the Cass-Koopmans model is Pareto optimal. 12
Cass-Koopsmans with Government We introduce the government in the simplest way (non distortionary way). The environment is the same as before, expect now the government appropriates gt units of good. The government dumps it into the ocean, and hence has no effect on the representative agents utility. The resource constraint becomes: (RC) c t + k t+1 (f(k t ) g t ) + (1 )k t (t = 0, 1, 2,...). (12.31) 13
Social Planner s Problem The planner s problem is: max t u(c t ) {c t,k t+1 } s.t. c t + k t+1 + g t f(k t ) + (1 )k t (t = 0, 1, 2,...) k 0 > 0 given. As in the case without government, the set of necessary and sufficient conditions for optimality are: Initial condition k0, RC (12.31), Euler equation: u (c t ) = [f (k t+1 ) + (1 )]u (c t+1 ), TVC: lim t u [f(k t ) + (1 )k t k t+1 g t t ][f (k t ) + (1 )]k t = 0. (12.32) Hence, the only difference between the model with and without gov. is the gt term in the RC. - f(kt) has been replaced by f(kt) - gt. 14
Steady State Assume that the sequence of gov. expenditures is such that gt g *. In the steady state k and k * and c * t = k, c t = c t can be found using: (modified Golden rule) f (k ) = +, with 1 (RC in the SS) c = f(k ) k g. 1 +, (12.33) Changes in g * imply one-to-one changes in c *. 15
Competitive Equilibrium - Complete Markets Same ownership structure and trading arrangement as before. The government taxes (lump sum, τht) the household to finance gt. Def: A government policy consists of an expenditure plan {g t } plan { ht }. and a tax Def: A government policy is budget feasible if it satisfies the government budget constraint (GBC): q t g t q t ht. (12.34) Note: This assumes that the government, like households and firms, can issue Arrow-Debreu securities. 16
Households The household s problem is: max {c t,k s t+1 } t u(c t ) s.t. q t (c t + k s t+1 (1 )k s t) (r t k s t + w t n s t q t ht ). (12.35) c t 0, k s t+1 0, and 0 n s t 1 (t = 0, 1, 2,...). k s 0 = k 0 > 0 given. FOCs: c t : u (c t ) = q t q t+1 u (c t+1 ). k t+1 : r t+1 + q t+1 (1 ) q t, " = " if k s t+1 > 0. 17
Equilibrium Def: A competitive equilibrium is a sequence of Arrow-Debreu prices {q t, r t, w t } and associated quantities {k d t, n d t, c t, k s t+1, n s t} with k s 0 = k 0 and a government policy {g t, ht} such that: (i) Optimization: Given k s 0 = k 0, {q t, r t, w t } and { ht } - {c t, k s t+1, n s t} solve the HH s problem. - {k d t, n d t } solve the firm s problem. (ii) Market clearing: - Goods: - Capital: - Labor: (iii) GBC: c t + k s t+1 (1 )k s t + g t = F(k d t, n d t ), k d t = k s t, n d t = n s t (= 1). q t g t q t ht Sometimes the gov. s expenditure plan is considered exog. and the gov. adjusts taxes to meet the GBC. Hence, { ht } is an equil. object. 18
Ricardian Equivalence { ht } shows up twice in the definition of the equilibrium, in the HH s BC and the GBC, and both times as q t ht. Hence, for given {g t }, if {q t, r t, w t, ht} is an equilibrium price and taxes sequence, then {q t, r t, w t, ht} is an equilibrium if q t ht = The previous statement implies that the timing of taxes does not matter for the equilibrium sequences of prices and allocations. Hence, with lump-sum taxes, it is not restrictive to assume balance budget period by period: gt = τt q t ht. 19
Equilibrium Conditions It is easy to show that the conditions which characterize the complete markets equilibrium are: Initial condition, RC, Euler equation: u (c t ) = [f (k t+1 ) + (1 )]u (c t+1 ), TVC. The GBC is satisfied since the gov. chooses { ht } to satisfy it. The HH s BC is also satisfied given the above equil. conditions and the GBC. As in the case without government, the market equilibrium coincides with the SP s equilibrium. The optimality of the CE is due to the form of the taxes: lump-sum. With distortionary taxes, the market equilibrium cannot be computed by solving the SP s problem. The Euler equation needs to be solved. 20