Recursive macroeconomic theory, Thomas Sargent 2nd Ed - Chapter 12 ppsx

Chapter 12 Recursive competitive equilibria 12.1 Endogenous aggregate state variable For pure endowment stochastic economies, chapter described two types of competitive equilibria, one in the style of Arrow and Debreu with markets that convene at time and trade a complete set of history-contingent securities, another with markets that meet each period and trade a complete set of one-period ahead state-contingent securities called Arrow securities Though their price systems and trading protocols differ, both types of equilibria support identical equilibrium allocations Chapter described how to transform the Arrow-Debreu price system into one for pricing Arrow securities The key step in transforming an equilibrium with time- trading into one with sequential trading was to account for how individuals’ wealth evolve as time passes in a time- trading economy In a time- trading economy, individuals not make any other trades than those executed in period but the present value of those portfolios change as time passes and as uncertainty gets resolved So in period t after some history st , we used the Arrow-Debreu prices to compute the value of an individual’s purchased claims to current and future goods net of his outstanding liabilities We could then show that these wealth levels (and the associated consumption choices) could also be attained in a sequential-trading economy where there are only markets in one-period Arrow securities which reopen in each period In chapter we also demonstrated how to obtain a recursive formulation of the equilibrium with sequential trading This required us to assume that individuals’ endowments were governed by a Markov process Under that assumption we could identify a state vector in terms of which the Arrow securities could be cast This (aggregate) state vector then became a component of the state vector for each individual’s problem This transformation of price systems is easy in the pure exchange economies of chapter because in equilibrium the relevant state variable, wealth, is a function solely of the current realization of the exogenous Markov state variable The transformation is more subtle in economies in which part of the aggregate state is endogenous in the sense that – 360 – The growth model 361 it emerges from the history of equilibrium interactions of agents’ decisions In this chapter, we use the basic stochastic growth model (sometimes also called the real business cycle model) as a laboratory for moving from an equilibrium with time- trading to a sequential equilibrium with trades of Arrow securities We also formulate a recursive competitive equilibrium with trading in Arrow securities by using a version of the ‘Big K , little k ’ trick that is often used in macroeconomics 12.2 The growth model Here we spell out the basic ingredients of the growth model; preferences, endowment, technology, and information The environment is the same as in chapter 11 except for that we now allow for a stochastic technology level In each period t ≥ , there is a realization of a stochastic event st ∈ S Let the history of events up and until time t be denoted st = [st , st−1 , , s0 ] The unconditional probability of observing a particular sequence of events st is given by a probability measure πt (st ) We write conditional probabilities as πτ (sτ |st ) which is the probability of observing sτ conditional upon the realization of st In this chapter, we assume that the state s0 in period is nonstochastic and hence π0 (s0 ) = for a particular s0 ∈ S We use st as a commodity space in which goods are differentiated by histories A representative household has preferences over nonnegative streams of consumption ct (st ) and leisure t (st ) that are ordered by ∞ β t u[ct (st ), t (st )]πt (st ) t=0 (12.2.1) st where β ∈ (0, 1) and u is strictly increasing in its two arguments, twice continuously differentiable, strictly concave and satisfies the Inada conditions lim uc (c, ) = lim u (c, ) = ∞ →0 c→0 In each period, the representative household is endowed with one unit of time that can be devoted to leisure t (st ) or labor nt (st ); 1= t (s t ) + nt (st ) (12.2.2) The stochastic growth model was formulated and fully analyzed by Brock and Mirman (1972) It is a work horse for studying macroeconomic fluctuations 362 Recursive competitive equilibria The only other endowment is a capital stock k0 at the beginning of period The technology is ct (st ) + xt (st ) ≤ At (st )F (kt (st−1 ), nt (st )), kt+1 (s ) = (1 − δ)kt (s t t−1 t ) + xt (s ), (12.2.3a) (12.2.3b) where F is a twice continuously differentiable, constant returns to scale production function with inputs capital kt (st−1 ) and labor nt (st ), and At (st ) is a stochastic process of Harrod-neutral technology shocks Outputs are the consumption good ct (st ) and the investment good xt (st ) In (12.2.3b ), the investment good augments a capital stock that is depreciating at the rate δ Negative values of xt (st ) are permissible, which means that the capital stock can be reconverted into the consumption good We assume that the production function satisfies standard assumptions of positive but diminishing marginal products, Fi (k, n) > 0, Fii (k, n) < 0, for i = k, n; and the Inada conditions, lim Fk (k, n) = lim Fn (k, n) = ∞, k→0 n→0 lim Fk (k, n) = lim Fn (k, n) = k→∞ n→∞ Since the production function has constant returns to scale, we can define ˆ F (k, n) ≡ nf (k) where ˆ k k≡ n (12.2.4) Another property of a linearly homogeneous function F (k, n) is that its first derivatives are homogeneous of degree and thus the first derivatives are funcˆ tions only of the ratio k In particular, we have Fk (k, n) = ∂ nt f (k/n) ˆ = f (k), ∂k (12.2.5a) Fn (k, n) = ∂ nt f (k/n) ˆ ˆ ˆ = f (k) − f (k)k ∂n (12.2.5b) Time-0 trading: Arrow-Debreu securities 363 12.3 Lagrangian formulation of the planning problem The social planner chooses an allocation {ct (st ), t (st ), xt (st ), nt (st ), kt+1 (st )}∞ t=0 to maximize (12.2.1 ) subject to (12.2.2 ), (12.2.3 ), the initial capital stock k0 and the stochastic process for the technology level At (st ) To solve this planning problem, we form the Lagrangian ∞ L= + β t πt (st ){u(ct (st ), − nt (st )) t=0 st µt (st )[At (st )F (kt (st−1 ), nt (st )) + (1 − δ)kt (st−1 ) − ct (st ) − kt+1 (st )]} where µt (st ) is a process of Lagrange multipliers on the technology constraint First-order conditions with respect to ct (st ) , nt (st ), and kt+1 (st ), respectively, are uc st = µt (st ), u t s t t (12.3.1a) t t = uc s At (s )Fn s , t uc s πt (s ) = β uc s t+1 πt+1 s (12.3.1b) t+1 st+1 |st At+1 st+1 Fk st+1 + (1 − δ) , (12.3.1c) where the summation over st+1 |st means that we sum over all possible histories st+1 such that st = st ˜ ˜ 12.4 Time- trading: Arrow-Debreu securities In the style of Arrow and Debreu, we can support the allocation that solves the planning problem by a competitive equilibrium with time trading of a complete set of date– and history–contingent securities Trades occur among a representative household and two types of representative firms 2 One can also support the allocation that solves the planning problem with a less decentralized setting with only the first of our two types of firms, and in which the decision for making physical investments is assigned to the household We assign that decision to a second type of firm because we want to price more items, in particular, the capital stock 364 Recursive competitive equilibria We let [q , w0 , r0 , pk0 ] be a price system, where pk0 is the price of a unit of the initial capital stock, and each of q , w0 and r0 is a stochastic process of prices for output and for renting labor and capital, respectively, and the time t component of each is indexed by the history st A representative household purchases consumption goods from a type I firm and sells labor services to the type I firm that operates the production technology (12.2.3a) The household owns the initial capital stock k0 and at date sells it to a type II firm The type II firm operates the capital-storage technology (12.2.3b ), purchases new investment goods xt from a type I firm, and rents stocks of capital back to the type I firm We now describe the problems of the representative household and the two types of firms in the economy with time- trading 12.4.1 Household The household maximizes β t u ct (st ), − nt (st ) πt (st ) t (12.4.1) st subject to ∞ ∞ qt (st )ct (st ) ≤ t=0 st wt (st )nt (st ) + pk0 k0 t=0 (12.4.2) st First-order conditions with respect to ct (st ) and nt (st ) , respectively, are β t uc st πt (st ) = ηqt (st ), t βu t t s πt (s ) = ηwt (st ), where η > is a multiplier on the budget constraint (12.4.2 ) (12.4.3a) (12.4.3b) Time-0 trading: Arrow-Debreu securities 365 12.4.2 Firm of type I The representative firm of type I operates the production technology (12.2.3a) with capital and labor that it rents at market prices For each period t and each realization of history st , the firm enters into state-contingent contracts at I time to rent capital kt (st ) and labor services nt (st ) The type I firm seeks to maximize ∞ 0 I qt (st ) ct (st ) + xt (st ) − rt (st )kt st − wt (st )nt (st ) (12.4.4) I ct (st ) + xt (st ) ≤ At (st )F kt st , nt (st ) (12.4.5) t=0 st subject to After substituting (12.4.5 ) into (12.4.4 ) and invoking (12.2.4 ), the firm’s objective function can be expressed alternatively as ∞ ˆI nt (st ) qt (st )At (st )f kt st 0 ˆI − rt (st )kt st − wt (st ) (12.4.6) t=0 st and the maximization problem can then be decomposed into two parts First, conditional upon operating the production technology in period t and history I st , the firm solves for the profit-maximizing capital-labor ratio, denoted kt (st ) I t Second, given that capital-labor ratio kt (s ), the firm determines the profitmaximizing level of its operation by solving for the optimal employment level, denoted nt (st ) The firm finds the profit-maximizing capital-labor ratio by maximizing the expression in curly brackets in (12.4.6 ) The first-order condition with respect ˆI to kt (st ) is ˆ q (st )At (st )f k I st − r0 (st ) = (12.4.7) t t t ˆI At the optimal capital-labor ratio kt (st ) that satisfies (12.4.7 ), the firm evaluates the expression in curly brackets in (12.4.6 ) in order to determine the optimal level of employment nt (st ) In particular, nt (st ) is optimally set equal to zero or infinity if the expression in curly brackets in (12.4.6 ) is strictly negative or strictly positive, respectively However, if the expression in curly brackets is zero in some period t and history st , the firm would be indifferent to the level of nt (st ) since profits are then equal to zero for all levels of operation in that 366 Recursive competitive equilibria period and state Here, we summarize the optimal employment decision by us0 ing equation (12.4.7 ) to eliminate rt (st ) in the expression in curly brackets in (12.4.6 ); 0 ˆI ˆI ˆI qt (st )At (st ) f kt st − f kt st kt st − wt (st )  t  < 0, then nt (s ) = 0; (12.4.8) = 0, then nt (st ) is indeterminate;  t > 0, then nt (s ) = ∞ if I In an equilibrium, both kt (st ) and nt (st ) are strictly positive and finite so expressions (12.4.7 ) and (12.4.8 ) imply the following equilibrium prices: 0 qt (st )At (st )Fk st = rt (st ) (12.4.9a) qt (st )At (st )Fn (12.4.9b) s t = wt (st ) where we have invoked (12.2.5 ) 12.4.3 Firm of type II The representative firm of type II operates technology (12.2.3b ) to transform output into capital The type II firm purchases capital at time from the household sector and thereafter invests in new capital, earning revenues by renting capital to the type I firm It maximizes ∞ II rt (st )kt st−1 − qt (st )xt (st ) (12.4.10) II II kt+1 st = (1 − δ) kt st−1 + xt st (12.4.11) II −pk0 k0 + t=0 st subject to II Note that the firm’s capital stock in period , k0 , is bought without any uncertainty about the rental price in that period while the investment in capital II for a future period t, kt (st−1 ), is conditioned upon the realized states up and until the preceding period, i.e., history st−1 Thus, the type II firm manages the risk associated with technology constraint (12.2.3b ) that states that capital must be assemblied one period prior to becoming an input for production In contrast, the type I firm of the previous subsection can decide upon how much Time-0 trading: Arrow-Debreu securities 367 I capital kt (st ) to rent in period t conditioned upon all realized shocks up and until period t, i.e., history st After substituting (12.4.11 ) into (12.4.10 ) and rearranging, the type II firm’s objective function can be written as ∞ II 0 k0 −pk0 + r0 (s0 ) + q0 (s0 ) (1 − δ) + II kt+1 st t=0 · −qt st + st 0 rt+1 st+1 + qt+1 st+1 (1 − δ) , (12.4.12) st+1 |st where the firm’s profit is a linear function of investments in capital The profitII maximizing level of the capital stock kt+1 (st ) in expression (12.4.12 ) is equal to zero or infinity if the associated multiplicative term in curly brackets is strictly negative or strictly positive, respectively However, for any expression in curly brackets in (12.4.12 ) that is zero, the firm would be indifferent to the level of II kt+1 (st ) since profits are then equal to zero for all levels of investment In an II II equilibrium, k0 and kt+1 (st ) are strictly positive and finite so each expression in curly brackets in (12.4.12 ) must equal zero and hence equilibrium prices must satisfy 0 pk0 = r0 (s0 ) + q0 (s0 ) (1 − δ) , 0 rt+1 st+1 + qt+1 st+1 (1 − δ) qt st = (12.4.13a) (12.4.13b) st+1 |st 12.4.4 Equilibrium prices and quantities According to equilibrium conditions (12.4.9 ), each input in the production technology is paid its marginal product and hence profit maximization of the type I firm ensures an efficient allocation of labor services and capital But nothing is said about the equilibrium quantities of labor and capital Profit maximization of the type II firm imposes no-arbitrage restrictions (12.4.13 ) across prices pk0 0 and {rt (st ), qt (st )} But nothing is said about the specific equilibrium value of an individual price To solve for equilibrium prices and quantities, we turn to the representative household’s first-order conditions (12.4.3 ) 368 Recursive competitive equilibria After substituting (12.4.9b ) into household’s first-order condition (12.4.3b ), we obtain βtu st πt (st ) = ηqt st At st Fn st ; (12.4.14a) and then by substituting (12.4.13b) and (12.4.9a) into (12.4.3a), 0 rt+1 st+1 + qt+1 st+1 (1 − δ) β t uc st πt (st ) = η st+1 |st qt+1 =η st+1 At+1 st+1 Fk st+1 + (1 − δ) (12.4.14b) st+1 |st Next, we use qt (st ) = β t uc (st )πt (st )/η as given by household’s first-order condi0 tion (12.4.3a) and the corresponding expression for qt+1 (st+1 ) to substitute into (12.4.14a) and (12.4.14b ), respectively This step produces expressions identical to the planner’s first-order conditions (12.3.1b ) and (12.3.1c), respectively In this way, we have verified that the allocation in the competitive equilibrium with time trading is the same as the allocation that solves the planning problem Given the equivalence of allocations, it is standard to compute the competitive equilibrium allocation by solving the planning problem since the latter problem is a simpler one We can compute equilibrium prices by substituting the allocation from the planning problem into the household’s and firms’ firstorder conditions All relative prices are then determined and in order to pin down absolute prices, we would also have to pick a numeraire Any such normalization of prices is tantamount to setting the multiplier η on the household’s present value budget constraint equal to an arbitrary positive number For example, if we set η = , we are measuring prices in units of marginal utility of the time consumption good Alternatively, we can set q0 (s0 ) = by setting t t η = (uc (s0 )) We can compute qt (s ) from (12.4.3a), wt (s ) from (12.4.3b ), and rt (st ) from (12.4.9a) Finally, we can compute pk0 from (12.4.13a) to get 0 pk0 = r0 (s0 ) + q0 (s0 )(1 − δ) Time-0 trading: Arrow-Debreu securities 369 12.4.5 Implied wealth dynamics Even though trades are only executed at time in the Arrow-Debreu market structure, we can study how the representative household’s wealth evolves over time For that purpose, after a given history st , we convert all prices, wages and rental rates that are associated with current and future deliveries so that they are expressed in terms of time-t, history-st consumption goods, i.e., we change the numeraire; u [cτ (sτ )] qτ (sτ ) = β τ −t πτ sτ |st , qt (st ) u [ct (st )] wτ (sτ ) t wτ (sτ ) ≡ t , qt (s ) rτ (sτ ) t rτ (sτ ) ≡ t qt (s ) t qτ (sτ ) ≡ (12.4.15a) (12.4.15b) (12.4.15c) In chapter we asked the question: what is the implied wealth of a household at time t after history st when excluding the endowment stream? Here we ask the some question except for that we now instead of endowments exclude the value of labor For example, the household’s net claim to delivery of goods in a future period τ ≥ t, contingent on history sτ , is given by t t [qτ (sτ )cτ (sτ ) − wτ (sτ )nτ (sτ )], as expressed in terms of time- t, history-st consumption goods Thus, the household’s wealth, or the value of all its current and future net claims, expressed in terms of the date-t, history-st consumption good is ∞ Υt (st ) ≡ t t qτ (sτ )cτ (sτ ) − wτ (sτ )nτ (sτ ) τ =t sτ |st ∞ = t qτ (sτ ) Aτ (sτ )F (kτ (sτ −1 ), nτ (sτ )) τ =t sτ |st t + (1 − δ)kτ (sτ −1 ) − kτ +1 (sτ ) − wτ (sτ )nτ (sτ ) ∞ = t qτ (sτ ) Aτ (sτ ) Fk (sτ )kτ (sτ −1 ) + Fn (sτ )nτ (sτ ) τ =t sτ |st t + (1 − δ)kτ (sτ −1 ) − kτ +1 (sτ ) − wτ (sτ )nτ (sτ ) ∞ = τ =t sτ |st t t rτ (sτ )kτ (sτ −1 ) + qτ (sτ ) (1 − δ)kτ (sτ −1 ) − kτ +1 (sτ ) Sequential trading: Arrow securities 371 12.5.1 Household At each date t ≥ after history st , the representative household buys consumption goods ct (st ), sells labor services nt (st ) and trades claims to date ˜ ˜ t + consumption, whose payment is contingent on the realization of st+1 Let at (st ) denote the claims to time t consumption that the household brings into ˜ time t in history st Thus, the household faces a sequence of budget constraints for t ≥ , where the time- t, history-st budget constraint is ˜ at+1 (st+1 , st )Qt (st+1 |st ) ≤ wt (st )˜ t (st ) + at (st ), ˜ ˜ n ˜ ct (st ) + ˜ (12.5.1) st+1 where {˜t+1 (st+1 , st )} , is a vector of claims on time– t + consumption, one a element of the vector for each value of the time–t + realization of st+1 To rule out Ponzi schemes, we must impose borrowing constraints on the household’s asset position We could follow the approach of chapter and compute state-contingent natural debt limits where the counterpart to the earlier present value of the household’s endowment stream would be the present value of the household’s time endowment Alternatively, we here just impose that the household’s indebtedness in any state next period, −˜t+1 (st+1 , st ), is bounded a by some arbitrarily large constant Such an arbitrary debt limit works well for the following reason As long as the household is constrained so that it cannot run a true Ponzi scheme with an unbounded budget constraint, equilibrium forces will ensure that the representative household willingly holds the market portfolio In the present setting, we can for example set that arbitrary debt limit equal to zero, as will become clear as we go along Let ηt (st ) and νt (st ; st+1 ) be the nonnegative Lagrange multipliers on the budget constraint (12.5.1 ) and the borrowing constraint with an arbitrary debt limit of zero, respectively, for time t and history st The Lagrangian can then be formed as ∞ β t u(˜t (st ), − nt (st )) πt (st ) c ˜ L= t=0 st ˜ at+1 (st+1 , st )Qt (st+1 |st ) ˜ ˜ n ˜ ˜ + ηt (st ) wt (st )˜ t (st ) + at (st ) − ct (st ) − st+1 a + νt (st ; st+1 )˜t+1 (st+1 ) , for a given initial wealth level a0 In an equilibrium, the representative house˜ ˜ hold will choose interior solutions for {˜t (st ), nt (st )}∞ because of the assumed c t=0 372 Recursive competitive equilibria Inada conditions The Inada conditions on the utility function ensure that the household will neither set ct (st ) nor t (st ) equal to zero, i.e., nt (st ) < The ˜ ˜ Inada conditions on the production function guarantee that the household will always find it desirable to supply some labor, nt (st ) > Given these interior ˜ solutions, the first-order conditions for maximizing L with respect to ct (st ), ˜ t t nt (s ) and {˜t+1 (st+1 , s )}st+1 are ˜ a β t uc (˜t (st ), − nt (st )) πt (st ) − ηt (st ) = , c ˜ (12.5.2a) c ˜ ˜ −β t u (˜t (st ), − nt (st )) πt (st ) + ηt (st )wt (st ) = , (12.5.2b) ˜ −ηt (st )Qt (st+1 |st ) + νt (st ; st+1 ) + ηt+1 (st+1 , st ) = , (12.5.2c) for all st+1 , t, st Next, we proceed under the conjecture that the arbitrary debt limit of zero will not be binding and hence, the Lagrange multipliers νt (st ; st+1 ) are all equal to zero After setting those multipliers equal to zero in equation (12.5.2c), the first-order conditions imply the following conditions on the optimal choices of consumption and labor, wt (st ) = ˜ c ˜ u (˜t (st ), − nt (st )) , t ), − n (st )) uc (˜t (s c ˜t (12.5.3a) c ˜ uc (˜t+1 (st+1 ), − nt+1 (st+1 )) ˜ Qt (st+1 |st ) = β πt (st+1 |st ), (12.5.3b) uc (˜t (st ), − nt (st )) c ˜ for all t, st and st+1 12.5.2 Firm of type I At each date t ≥ after history st , a type I firm is a production firm that ˜I chooses a quadruple {˜t (st ), xt (st ), kt (st ), nt (st )} to solve a static optimum c ˜ ˜ problem: ˜I max ct (st ) + xt (st ) − rt (st )kt (st ) − wt (st )˜ t (st ) ˜ ˜ ˜ ˜ n (12.5.4) ˜I ct (st ) + xt (st ) ≤ At (st )F (kt (st ), nt (st )) ˜ ˜ ˜ (12.5.5) subject to The zero-profit conditions are rt (st ) = At (st )Fk (st ), ˜ t t t wt (s ) = At (s )Fn (s ) ˜ (12.5.6a) (12.5.6b) Sequential trading: Arrow securities 373 If conditions (12.5.6 ) are violated, the type I firm either makes infinite profits by hiring infinite capital and labor, or else it makes negative profits for any positive output level, and therefore shuts down If conditions (12.5.6 ) are satisfied, the firm makes zero profits and its size is indeterminate The firm of type I is willing to produce any quantities of ct (st ) and xt (st ) that the market demands, ˜ ˜ provided that conditions (12.5.6 ) are satisfied 12.5.3 Firm of type II A type II firm transforms output into capital, stores capital, and earns its revenues by renting capital to the type I firm Because of the technological assumption that capital can be converted back into the consumption good, we can without loss of generality consider a two-period optimization problems where a type ˜ II II firm decides how much capital kt+1 (st ) to store at the end of period t after ˜II history st , in order to earn a stochastic rental revenue rt+1 (st+1 ) kt+1 (st ) and ˜ II t ˜ liquidation value (1− δ) kt+1 (s ) in the following period The firm finances itself by issuing state contingent debt to the households, so future income streams can ˜ be expressed in today’s values by using prices Qt (st+1 |st ) Thus, at each date t ˜ II t ≥ after history s , a type II firm chooses kt+1 (st ) to solve the optimum problem ˜II max kt+1 (st ) −1 + ˜ Qt (st+1 |st ) rt+1 (st+1 ) + (1 − δ) ˜ (12.5.7) st+1 The zero-profit condition is ˜ Qt (st+1 |st ) rt+1 (st+1 ) + (1 − δ) ˜ 1= (12.5.8) st+1 The size of the type II firm is indeterminate So long as condition (12.5.8 ) is ˜ II satisfied, the firm breaks even at any level of kt+1 (st ) If condition (12.5.8 ) is not ˜II satisfied, either it can earn infinite profits by setting kt+1 (st ) to be arbitrarily large (when the right side exceeds the left), or it earns negative profits for any positive level of capital (when the right side falls short of the left), and so chooses to shut down 374 Recursive competitive equilibria 12.5.4 Equilibrium prices and quantities We leave it to the reader to follow the approach taken in chapter to show the equivalence of allocations attained in the sequential equilibrium and the time- ˜ ˜ ˜ equilibrium; {˜t (st ), ˜t (st ), xt (st ), nt (st ), kt+1 (st )}∞ = {ct (st ), t (st ), xt (st ), nt (st ), kt+1 (st )}∞ c t=0 t=0 The trick is to guess that the prices in the sequential equilibrium satisfy t ˜ Qt (st+1 |st ) = qt+1 (st+1 ), t wt (s ) = ˜ t rt (s ) = ˜ t wt (st ), t rt (st ) (12.5.9a) (12.5.9b) (12.5.9c) The other set of guesses is that the representative household chooses asset portfolios given by at+1 (st+1 , st ) = Υt+1 (st+1 ) for all st+1 When showing that the ˜ household can afford these asset portfolios together with the prescribed quantities of consumption and leisure, we will find that the required initial wealth is equal to a0 = [r0 (s0 ) + (1 − δ)]k0 = pk0 k0 , ˜ i.e., the household in the sequential equilibrium starts out at the beginning of period owning the initial capital stock which is then sold to a type II firm at the same competitive price as in the time- trading equilibrium 12.5.5 Financing a type II firm ˜ II A type II firm finances purchases of kt+1 (st ) units of capital in period t after t history s by issuing one-period state-contingent claims that promise to pay ˜ II rt+1 (st+1 ) + (1 − δ) ·kt+1 (st ) consumption goods tomorrow in state st+1 In ˜ units of today’s time- t consumption good, these payouts are worth t t+1 ˜ II ˜ ˜ ) + (1 − δ) ·kt+1 (st ) (by virtue of (12.5.8 )) The st+1 Qt (st+1 |s ) rt+1 (s firm breaks even by issuing these claims Thus, the firm of type II is entirely owned by its creditor, the household, and it earns zero profits ˜ II Note that the economy’s end-of-period wealth as embodied in kt+1 (st ) in period t after history st , is willingly held by the representative household This follows immediately from fact that the household’s desired beginning-of-period wealth next period is given by at+1 (st+1 ) and is equal to Υt+1 (st+1 ), as given ˜ by (12.4.16 ) Thus, the equilibrium prices entice the representative household to enter each future period with a strictly positive net asset level that is equal Recursive formulation 375 to the value of the type II firm We have then confirmed the correctness of our earlier conjecture that the arbitrary debt limit of zero is not binding in the household’s optimization problem 12.6 Recursive formulation Following the approach taken in chapter 8, we have established that the equilibrium allocations are the same in the Arrow-Debreu economy with complete markets at time , and a sequential-trading economy with complete one-period Arrow securities This finding holds for an arbitrary technology process At (st ), defined as a measurable function of the history of events st which in turn are governed by some arbitrary probability measure πt (st ) At this level of general˜ ity, all prices {Qt (st+1 |st ), wt (st ), rt (st )} and the capital stock kt+1 (st ) in the ˜ ˜ sequential-trading economy depend on the history st That is, these objects are time varying functions of all past events {sτ }t =0 τ In order to obtain a recursive formulation and solution to both the social planning problem and the sequential trading equilibrium, we make the following specialization of the exogenous forcing process for the technology level 12.6.1 Technology is governed by a Markov process Let the stochastic event st be governed by a Markov process, [s ∈ S, π(s |s), π0 (s0 )] We keep our earlier assumption that the state s0 in period is nonstochastic and hence π0 (s0 ) = for a particular s0 ∈ S The sequences of probability measures πt (st ) on histories st are induced by the Markov process via the recursions πt (st ) = π(st |st−1 )π(st−1 |st−2 ) π(s1 |s0 )π0 (s0 ) Next, we assume that the aggregate technology level At (st ) in period t is a time-invariant measurable function of its level in the last period and the current stochastic event st , i.e., At (st ) = A At−1 (st−1 ), st For example, here we will proceed with the multiplicative version At (st ) = st At−1 (st−1 ) = s0 s1 · · · st A−1 , given the initial value A−1 376 Recursive competitive equilibria 12.6.2 Aggregate state of the economy The specialization of the technology process enables us to adapt the recursive construction of chapter to incorporate additional components of the state of the economy Besides information about the current value of the stochastic event s, we need to know last period’s technology level, denoted A, in order to determine current technology level, s A, and to forecast future technology levels This additional element A in the aggregate state vector does not constitute any conceptual change from what we did in chapter We are merely including one more state variable that is a direct mapping from exogenous stochastic events and it does not depend upon any endogenous outcomes But we need also to expand the aggregate state vector with an endogenous component of the state of the economy, namely, the beginning-of-period capital stock K Given the new state vector X ≡ [K A s] , we are ready to explore recursive formulations of both the planning problem and the sequential trading equilibrium This state vector is a complete summary of the economy’s current position It is all that is needed for a planner to compute an optimal allocation and it is all that is needed for the “invisible hand” to call out prices and implement the first-best allocation as a competitive equilibrium We proceed as follows First, we display the Bellman equation associated with a recursive formulation of the planning problem Second, we use the same state vector X for the planner’s problem as a state vector in which to cast the Arrow securities in a competitive economy with sequential trading Then we define a competitive equilibrium and show how the prices for the sequential equilibrium are embedded in the decision rules and the value function of the planning problem Recursive formulation of the planning problem 377 12.7 Recursive formulation of the planning problem We use capital letters C, N, K to denote objects in the planning problem that correspond to c, n, k , respectively, in the household and firms’ problems We shall eventually equate them, but not until we have derived an appropriate formulation of the household’s and firms’ problems in a recursive competitive equilibrium The Bellman equation for the planning problem is v(K, A, s) = max C,N,K π(s |s)v(K , A , s ) u(C, − N ) + β (12.7.1) s subject to K + C ≤ AsF (K, N ) + (1 − δ)K, A = As (12.7.2a) (12.7.2b) Using the definition of the state vector X = [K A s], we denote the optimal policy functions as C = ΩC (X), (12.7.3a) N (12.7.3b) K (12.7.3c) N = Ω (X), K = Ω (X) Equations (12.7.2b ), (12.7.3c), and the Markov transition density π(s |s) induce a transition density Π(X |X) on the state X For convenience, define the functions Uc (X) ≡ uc (ΩC (X), − ΩN (X)), (12.7.4a) U (X) ≡ u (Ω (X), − Ω (X)), (12.7.4b) Fk (X) ≡ Fk (K, Ω (X)), (12.7.4c) Fn (X) ≡ Fn (K, ΩN (X)) (12.7.4d) C N N The first-order conditions for the planner’s problem can be represented as U (X) = Uc (X)AsFn (X), Uc (X ) 1=β [A s FK (X ) + (1 − δ)] Π(X |X) Uc (X) (12.7.5a) (12.7.5b) X We are using the envelope condition v (K, A, s) = U (X)[AsF (X) + (1 − K c k δ)] 378 Recursive competitive equilibria 12.8 Recursive formulation of sequential trading We seek a competitive equilibrium with sequential trading of one-period ahead state contingent securities (i.e., Arrow securities) To this, we must use the ‘Big K , little k ’ trick 12.8.1 The ‘Big K , little k ’ trick Relative to the setup described in 8, we have augmented the time-t state of the economy by both last period’s technology level At−1 and the current aggregate value of the endogenous state variable Kt We assume that decision makers act as if their decisions not affect current or future prices In a sequential market setting, prices depend on the state, of which Kt is part Of course, in the aggregate, decision makers choose the motion of Kt , so that we require a device that makes them ignore this fact when they solve their decision problems (we want them to behave as perfectly competitive price takers, not monopolists) This consideration induces us to carry long both ‘big K ’ and ‘little k ’ in our computations Big K is an endogenous state variable that is used to index prices Big K is a component of the state that agents regard as beyond their control when solving their optimum problems Values of little k are chosen by firms and consumers While we distinguish k and K when posing the decision problems of the household and firms, to impose equilibrium we set K = k after firms and consumers have optimized More generally, big K can be a vector of endogenous state variables that impinge on equilibrium prices Recursive formulation of sequential trading 379 12.8.2 Price system To decentralize the economy in terms of one-period Arrow securities, we need a description of the aggregate state in terms of which one-period state-contingent payoffs are defined We proceed by guessing that the appropriate description of the state is the same vector X that constitutes the state for the planning problem We temporarily forget about the optimal policy functions for the planning problem and focus on a decentralized economy with sequential trading and one-period prices that depend on X We specify price functions r(X), w(X), Q(X |X), that represent, respectively, the rental price of capital, the wage rate for labor, and the price of a claim to one unit of consumption next period when next period’s state is X and this period’s state is X (Forgive us for recycling the notation for r and w from the previous sections on the formulation of history-dependent competitive equilibria with commodity space st ) The prices are all measured in units of this period’s consumption good We also take as given an arbitrary candidate for the law of motion for K : K = G(X) (12.8.1) Equation (12.8.1 ) together with (12.7.2b ) and a given subjective transition denˆ sity π (s |s) induce a subjective transition density Π(X |X) for the state X For ˆ now, G and π (s |s) are arbitrary We wait until later to impose other equilibˆ rium conditions including rational expectations in the form of some restrictions on G and π ˆ 12.8.3 Household problem ˆ The perceived law of motion (12.8.1 ) for K and the induced transition Π(X |X) of the state describe the beliefs of a representative household The Bellman equation of the household is J(a, X) = max c,n,a(X ) ˆ J(a(X ), X )Π(X |X) u(c, − n) + β (12.8.2) X subject to Q(X |X)a(X ) ≤ w(X)n + a c+ X (12.8.3) 380 Recursive competitive equilibria Here a represents the wealth of the household denominated in units of current consumption goods and a(X ) represents next period’s wealth denominated in units of next period’s consumption good Denote the household’s optimal policy functions as c = σ c (a, X), (12.8.4a) n = σ n (a, X), (12.8.4b) a a(X ) = σ (a, X; X ) (12.8.4c) uc (a, X) ≡ uc (σ c (a, X), − σ n (a, X)), (12.8.5a) u (a, X) ≡ u (σ (a, X), − σ (a, X)) (12.8.5b) Let c n Then we can represent the first-order conditions for the household’s problem as u (a, X) = uc (a, X)w(X), Q(X |X) = β uc (σ a (a, X; X ), X ) ˆ Π(X |X) uc (a, X) (12.8.6a) (12.8.6b) 12.8.4 Firm of type I Recall from subsection 12.5.2 the static optimum problem of a type I firm in a sequential equilibrium In the recursive formulation of that equilibrium, the optimum problem of a type I firm can be written as max {c + x − r(X)k − w(X)n} (12.8.7) c + x ≤ AsF (k, n) (12.8.8) r(X) = AsFk (k, n) (12.8.9a) w(X) = AsFn (k, n) (12.8.9b) c,x,k,n subject to The zero-profit conditions are Recursive competitive equilibrium 381 12.8.5 Firm of type II Recall from subsection 12.5.3 the optimum problem of a type II firm in a sequential equilibrium In the recursive formulation of that equilibrium, the optimum problem of a type II firm can be written as max k k Q(X |X) [r(X ) + (1 − δ)] (12.8.10) Q(X |X) [r(X ) + (1 − δ)] −1 + (12.8.11) X The zero-profit condition is 1= X 12.9 Recursive competitive equilibrium So far, we have taken the price functions r(X), w(X), Q(X |X) and the perceived law of motion (12.8.1 ) for K and the associated induced state transition ˆ probability Π(X |X) as given arbitrarily We now impose equilibrium conditions on these objects and make them outcomes of the analysis When solving their optimum problems, the household and firms take the endogenous state variable K as given However, we want K to be determined by the equilibrium interactions of households and firms Therefore, we impose K = k after solving the optimum problems of the household and the two types of firms Imposing equality afterwards makes the household and the firms be price takers An important function of the rational expectations hypothesis is to remove ˆ agents’ expectations in the form of π and Π from the list of free parameters of ˆ the model 382 Recursive competitive equilibria 12.9.1 Equilibrium restrictions across decision rules We shall soon define an equilibrium as a set of pricing functions, a perceived law ˆ of motion for the K , and an associated Π(X |X) such that when the firms and the household take these as given, the household and firms’ decision rules imply the law of motion for K (12.8.1 ) after substituting k = K and other market clearing conditions We shall remove the arbitrary nature of both G and π and ˆ ˆ therefore also Π and thereby impose rational expectations We now proceed to find the restrictions that this notion of equilibrium imposes across agents decision rules, the pricing functions, and the perceived law of motion (12.8.1 ) If the state-contingent debt issued by the type II firm is to match that demanded by the household, we must have a(X ) = [r(X ) + (1 − δ)]K , (12.9.1a) and consequently beginning-of-period assets in a household’s budget constraint (12.8.3 ) have to satisfy a = [r(X) + (1 − δ)]K (12.9.1b) By substituting equations (12.9.1 ) into household’s budget constraint (12.8.3 ), we get Q(X |X)[r(X ) + (1 − δ)]K X = [r(X) + (1 − δ)]K + w(X)n − c (12.9.2) Next, by recalling equilibrium condition (12.8.11 ) and the fact that K is a predetermined variable when entering next period, it follows that the left-hand side of (12.9.2 ) is equal to K After also substituting equilibrium prices (12.8.9 ) into the right-hand side of (12.9.2 ), we obtain K = [AsFk (k, n) + (1 − δ)] K + AsFn (k, n)n − c = AsF (K, σ n (a, X)) + (1 − δ)K − σ c (a, X), (12.9.3) where the second equality invokes Euler’s theorem on linearly homogeneous functions and equilibrium conditions K = k , N = n = σ n (a, X) and C = c = σ c (a, X) To express the right-hand side of equation (12.9.3 ) solely as a Recursive competitive equilibrium 383 function of the current aggregate state X = [K A s] , we also impose equilibrium condition (12.9.1b ) K =AsF (K, σ n ([r(X) + (1 − δ)]K, X)) + (1 − δ)K − σ c ([r(X) + (1 − δ)]K, X) (12.9.4) Given the arbitrary perceived law of motion (12.8.1 ) for K that underlies the household’s optimum problem, the right side of (12.9.4 ) is the actual law of motion for K that is implied by household and firms’ optimal decisions In equilibrium, we want G in (12.8.1 ) not to be arbitrary but to be an outcome We want to find an equilibrium perceived law of motion (12.8.1 ) By way of imposing rational expectations, we require that the perceived and actual laws of motion are identical Equating the right sides of (12.9.4 ) and the perceived law of motion (12.8.1 ) gives G(X) =AsF (K, σ n ([r(X) + (1 − δ)]K, X)) + (1 − δ)K − σ c ([r(X) + (1 − δ)]K, X) (12.9.5) Please remember that the right side of this equation is itself implicitly a function of G, so that (12.9.5 ) is to be regarded as instructing us to find a fixed point equation of a mapping from a perceived G and a price system to an actual G This functional equation requires that the perceived law of motion for the capital stock G(X) equals the actual law of motion for the capital stock that is determined jointly by the decisions of the household and the firms in a competitive equilibrium Definition: A recursive competitive equilibrium with Arrow securities is a price system r(X), w(X), Q(X |X), a perceived law of motion K = G(X) and ˆ associated induced transition density Π(X |X), and a household value function c J(a, X) and decision rules σ (a, X), σ n (a, x), σ a (a, X; X ) such that: ˆ Given r(X), w(X), Q(X |X), Π(X |X), the functions σ c (a, X), σ n (a, X), σ a (a, X; X ) and the value function J(a, X) solve the household’s optimum problem For all X , r(X) = AFk (K, σ n ([r(X) + (1 − δ)]K, X), w(X) = AFn (K, σ n ([r(X) + (1 − δ)]K, X) Q(X |X) and r(X) satisfy (12.8.11 ) 384 Recursive competitive equilibria The functions G(X), r(X), σ c (a, X), σ n (a, X) satisfy (12.9.5 ) π = π ˆ Item enforces optimization by the household, given the prices it faces Item requires that the type I firm break even at every capital stock and at the labor supply chosen by the household Item requires that the type II firm break even Item requires that the perceived and actual laws of motion of capital are equal Item and the equality of the perceived and actual G imply ˆ that Π = Π Thus, items and impose rational expectations 12.9.2 Using the planning problem Rather than directly attacking the fixed point problem (12.9.5 ) that is the heart of the equilibrium definition, we’ll guess a candidate G and as well as a price system, then describe how to verify that they form an equilibrium As our candidate for G we choose the decision rule (12.7.3c) for K from the planning problem As sources of candidates for the pricing functions we again turn to the planning problem and choose: r(X) = AFk (X), (12.9.6a) w(X) = AFn (X), (12.9.6b) Q(X |X) = βΠ(X |X) Uc (X ) [A s FK (X ) + (1 − δ)] Uc(X) (12.9.6c) In an equilibrium it will turn out that the household’s decision rules for consumption and labor supply will match those chosen by the planner: ΩC (X) = σ c ([r(X) + (1 − δ)]K, X), (12.9.7a) Ω (X) = σ ([r(X) + (1 − δ)]K, X) (12.9.7b) N n The key to verifying these guesses is to show that the first-order conditions for both types of firms and the household are satisfied at these guesses We The two functional equations (12.9.7 ) state restrictions that a recursive competitive equilibrium imposes across the household’s decision rules σ and the planner’s decision rules Ω Concluding remarks 385 leave the details to an exercise Here we are exploiting some consequences of the welfare theorems, transported this time to a recursive setting with an endogenous aggregate state variable 12.10 Concluding remarks The notion of a recursive competitive equilibrium was introduced by Lucas and Prescott (1971) and Mehra and Prescott (1979) The application in this chapter is in the spirit of those papers but differs substantially in details In particular, neither of those papers worked with Arrow securities, while the focus of this chapter has been to manage an endogenous state vector in terms of which it is appropriate to cast Arrow securities ... budget constraint (12. 4.2 ) (12. 4.3a) (12. 4.3b) Time-0 trading: Arrow-Debreu securities 365 12. 4.2 Firm of type I The representative firm of type I operates the production technology (12. 2.3a) with... expressions (12. 4.7 ) and (12. 4.8 ) imply the following equilibrium prices: 0 qt (st )At (st )Fk st = rt (st ) (12. 4.9a) qt (st )At (st )Fn (12. 4.9b) s t = wt (st ) where we have invoked (12. 2.5 ) 12. 4.3... w(X)n} (12. 8.7) c + x ≤ AsF (k, n) (12. 8.8) r(X) = AsFk (k, n) (12. 8.9a) w(X) = AsFn (k, n) (12. 8.9b) c,x,k,n subject to The zero-profit conditions are Recursive competitive equilibrium 381 12. 8.5