News Driven Business Cycles in Heterogenous Agents Economies Paul Beaudry and Franck Portier DRAFT February 9 Abstract We present a new propagation mechanism for news shocks in dynamic general equilibrium models. The existing literature has considered representative agents models, in which news shock impact the economy through intertemporal substitution mechanisms. We consider different setups with heterogenous agents. We show that news-driven business cycles are possible in such environments, because such news create new current opportunities to trade between agents. In those environments, a Social Planner that would maximize a weighted sum of the different agents intertemporal utilities would choose allocations in which investment, consumption and hours are not all procyclical. In contrast, competitive allocations are displaying a business cycle pattern. Here we present three simple models with heterogenous agents that can display Expectation Driven Business Cycles (hereafter EDBC). The two first model have an OLG structure, while in the third one coexist intertemporal and static agents. In the two last models, we assume a two-sectors structure, while the first OLG model has a unique sector. The three models can be made very close to standard Representative Agent (hereafter RA) Business Cycle models. In sharp contrast with RA models, those models can predict a boom following a good news about future technology. An OLG Model with One Sector It is a one-sector model with two-periods overlapping generations. The particular assumption that is made is that agents consume only when (second period of life), and that they work Canada Research Chair at University of British Columbia and NBER. Toulouse School of Economics (Université de Toulouse and CEPR)
in both periods. we consider only a simple (almost) analytical version.. The Setup The final good is produced according to a CES constant return technology, using labor and capital services as input: Y t = C t + I t = ψ t K α t ( γ(l Y t ) σ + ( γ)(l O α t ) σ) σ () L Y can be only supplied by agents, L Y by agents, K is is the capital stock and ψ is a technological shift. σ belongs to the interval ], ] and < α <.. Aggregate capital accumulated with a one period time to build, according to K t+ = ( δ)k t + I t () The whole sequences of the shift parameters φ and ψ are assumed to be known, and we therefore study only perfect foresight dynamics of the model... Preferences In every period, one new agent is born, that represents the current generation (we assume no population growth). Each generation lived for two periods, works in both periods and consumes in its second period of life. Preferences of the representative agent of the generation born in period t are given by.. Markets Organization U(C O t+, L Y t, L t+) = (CO t+) χ χ θ η (L Y t ) η θ η (L O t+) η (3) There exist one market for the final good, that is assumed to be the numéraire. Labor markets exist separately for the labor supplied by the agents and for the one supplied by the agents, with respective wages w O and w Y. The rental rate of capital is denoted z. Firms in both sectors live for one period, and are owned by agents, who receive profits Π if any. We assume that all markets are competitive.. A First Illustrative Example We first consider a particular case in which some analytical results can be obtained.
.. Restrictions on Technology As for technology, we assume that capital fully depreciates in on period, so that equation () becomes K t = I t (4) The final good technology is assumed to be Cobb-Douglas (σ = ), so that () writes Y t = C t + I t = ψ t K α t (L Y t ) ( α)γ (L O t ) ( α)( γ) (5) We also assume that utility is linear in consumption, so that χ =... Competitive Allocations A typical generation t agent solves the following problem: max C O t+,l Y t,lo t+,it Ct+ O θ η (L Y t ) η θ η (L O t+) η s.t. I t wt Y L Y t (λ Y t ) Ct+ O wt+l O O t+ + z t+ I t + Π t (λ O t+) where the λs are Lagrange multipliers. One can write the problem first order conditions as λ O t+ = (6) θ (L Y t ) η θ (L O t ) η = λ Y t w Y t (7) = λ O t+w O t+ (8) λ Y t = z t+ (9) Firms optimal behavior implies the equalization of marginal productivity of factor with their prices, so that wt Y = ( α)γy t /L Y t () z t = αy t /I t () wt O = ( α)( α)y t /L O t () Profits are null. Optimality conditions, budget constraints and market equilibrium conditions characterize competitive equilibrium allocation. Proposition The example one-sector OLG model displays Expectation Driven Business Cycles (EDBC) 3
Proof : Using equations (6), (8) written in period t and (), one obtains a relations between L Y and L O : θ ( α)( γ) = Iα t (L O t ) (( α)( γ) ) (η ) (L Y t ) ( α)γ (3) as (( α)( γ) ) (η ) <, L Y and L O will always move in the same direction following a shock to expectations. We still need to prove that if the shock to expectations causes an increase in ivestment, it will also cause an increase in consumption and worked hours. The budget constraint when, together with equation (5) and (), imply I t = ( α)γy t Using the resource constraint Y t = C t + I t, we obtain C t = ( ( α)γ)y t I t = ( α)γy t Therefore, if investment increases, consumption will also increase, and therefore output, and both and labor (according to (3)). Q.E.D. In this simple setup, we therefore obtain a result that we have shown to be impossible to obtain in standard neoclassical representative agent setups, unless one modifies the production set in a way that allows for sectorial cost complementarities...3 A Social Planner Problem To contrast this model with a representative one, let us consider the case of a Social Planner that maximizes the geometrically discounted sum of utilities of the different generations, with discount rate δ S ], [: W = C O θ η (L O ) η + ( (δ S ) t Ct+ O θ (L Y t ) η θ ) (L O η η t+) η t= Note that such an objective would correspond to the intertemporal discounted utility of a (4) pseudo representative agent whose discount factor will be δ and instantaneous utility C O t δ S θ η (L Y t ) η θ η (L O t ) η and for I given.. The Social Planner chooses the sequence of L Y, L O, C and I in order to maximize W subject to an initial level of capital I and to the technological restrictions resource constraints C t + I t = ψ t K α t (L Y t ) ( α)γ (L O t ) ( α)( γ) 4
to which we associate the multiplier (δ S ) t λ t. The problem first order conditions are θ (L Y t ) ɛ = γ( α) Y t L Y t θ δ S (LO t ) ɛ = ( γ)( α) Y t λ t (5) L Y t λ t (6) λ t = δ S (7) λ t δ S λ t+ α Y t+ I t (8) plus the resource constraint and a transversality condition. Using those equation, we can prove the following proposition. Proposition In the example one-sector OLG model, allocations that maximize a geometrically discounted sum of generations utility do not display EDBC. Proof : From equations (5) to (8), one obtains Y t = αδ S I t. Therefore, the socially optimal allocations imply no current reaction of the economy to news, and does not exhibit EDBC. Q.E.D...4 Discussion In the next paragraph, we write down a more general version of this model, with incomplete depreciation, capital input in both sectors and consumption in both periods. The qualitative properties of the simple model are preserved in the more general setting..3 A More General Version of the Model Here we keep the general CES production function () and assume incomplete depreciation according to equation (). We cannot derive simple analytical results in this setting, and therefore solve numerically for the non-linear perfect foresight dynamics of the model. 5
.3. Competitive Allocations Househs programs are unchanged, and budget constraints and optimality conditions (6), (8), (7) and (9) still h. As for the firm, first order conditions are now given by z t = αy t /K t (9) w Y t = γ( α)k α t (L Y t ) σ (γ(l Y t ) σ + ( γ)(l O t ) σ ) ( α)/σ () w O t = ( γ)( α)k α t (L O t ) σ (γ(l Y t ) σ + ( γ)(l O t ) σ ) ( α)/σ ().3. Social Planner The objective of the Social Planner now writes W = C O θ L O + t= (δ S ) t ( (C O t+ ) χ chi θ (L Y t ) η θ ) (L O η η t+) η () The Social Planner maximizes W subject to the technological restrictions Y t = C t + I t = ψ t K α t (L Y t ) ( α)γ (L O t ) ( α)( γ) and K t+ = ( δ)k t + I t. First order conditions of this problem are given by ( (Ct O ) χ = δ S (Ct+) O χ α Y ) t+ + δ K t+ θ (L Y t ) η (3) = δ S (C O t ) χ γ( α)k α t (L Y t ) σ (γ(l Y t ) σ + ( γ)(l O t ) σ ) ( α)/σ (4) θ (L O t ) η = (C O t ) χ ( γ)( α)k α t (L O t ) σ (γ(l Y t ) σ + ( γ)(l O t ) σ ) ( α)/σ (5) Those optimality conditions plus the technological restrictions and a transversality condition for capital characterize those socially optimal allocations..3.3 Numerical results Here we do not aim at realism in choosing the numerical values of the parameters, but rather exhibit counter-examples to the theoretical result of Expectation Driven Business Cycles impossibility in competitive models with standard technology that we have obtained in Beaudry and Portier [7]. We restrict to quite linear (χ =.) utility functions for consumption, in 6
Table : Parameters Values for the One-Sector OLG Model Elasticity of production to capital input α.33 Parameter Ruling the elasticity of substitution between and labor σ -, or Weight parameter between and labor γ.5 Disutility of labour () θ and η. and Disutility of labour () θ and η. and Curvature of the utility for consumption function parameter χ. Capital depreciation δ. Social Planner discount factor δ S.99 TFP level ψ order to obtain a aggregate boom following a news shock. If we assume larger curvature, (higher χ), one also obtains EDBC, but with all variables going down on impact). Figure and displays the IRF for both the competitive allocations and for the socially optimal allocation we are looking at (with geometrically decreasing weights on future generations). We obtain EDBC in the competititive allocations, whereas it is newer the case for our infinitely lived problem. Figure : News Shock in the Competitive Equilibrium, One-sector OLG Model..5.5.5..5 σ= σ= σ= ψ.3... Investment..5..5 3 x 3 Agg. Consumption Agg. Labor.5..5 st Period Labor.5 nd Period Labor This figure displays the response of the competitive economy to the following news shock: in period, it is announced that productivity φ will increase by percent for one period in period 6. All variables are in percentage deviation from their steady state level. The parameters configuration is the benchmark one. σ is the elasticity of substitution between and labor in the production function We might want to say more on this 7
Figure : News Shock in the Social Planner Problem, One-Sector ILG Model.5.5 3 σ= σ= σ= ψ 4 Agg. Consumption.6 5.4. Investment..5..5 Agg. Labor..5..5 st Period Labor nd Period Labor This figure displays the response of socially optimal allocations when the Social Planner maximizes a geometrically discounted sum of different generations utility, with a constant discount factor δ S. The shock is the following news shock: in period, it is announced that productivity will increase by percent for one period in period 6. All variables except utility are in percentage deviation from their steady state level. The parameters configuration is the benchmark one. σ is the elasticity of substitution between and labor in the production function 8
An OLG Model with Two Sectors It is a two-sectors model with two-periods overlapping generations. The particular assumption that is made is that (second period of life) agents work in the investment good sector while the (first period of life) ones work in the consumption good sector.. The Setup.. Technology The consumption good is produced according to a constant return technology, using labor and capital services as input: C t = F (Kt C, L Y t ; ψ t ) (6) L Y can be only supplied by agents. ψ is a technological shift. The capital good is also produced according to a constant return technology, using labor and capital services as input. In some specific examples, we will assume that only labor enters as input. In that case, the technology will be decreasing returns to scale. I t = G(K K t, L O t ; φ t ) (7) L O can be only supplied by agents. φ is a technological shift. Aggregate capital accumulated with a one period time to build, according to K t+ = K C t+ + K K t+ = ( δ)k t + I t (8) The whole sequences of the shift parameters φ and ψ are assumed to be known, and we therefore study only perfect foresight dynamics of the model... Preferences In every period, one new agent is born, that represents the current generation (we assume no population growth). Each generation lived for two periods, works and consumes in both periods. Preferences of the representative agent of the generation born in period t are given by U(C Y t, C O t+, L Y t, L t+) = ( (C Y t ) ρ C O t+) χ χ θ η (L Y t ) η θ η (L O t+) η (9) 9
..3 Markets Organization There exist one market for the consumption good, that is assumed to be the num raire, and one for the investment good, whose price will be denoted p. Labor markets exist separately for the capital sector specific labor (supplied by the agents) and for the consumption good sector (supplied by the agents), with respective wages w O and w Y. The rental rate of capital is denoted z. Firms in both sectors live for one period, and are owned by agents, who receive profits Π if any. We assume that all markets are competitive.. A First Illustrative Example We first consider an (admittedly) particular case where some analytical results can be obtained... Restrictions on Technology and Preferences As for technology, we assume that capital fully depreciates in on period, so that equation (8) becomes K t = I t (3) The consumption good technology is assumed to be Cobb-Douglas, so that (6) writes C t = ψ t (I t ) α (L Y t ) α (3) with < α <. while we assume that only labor input enters the capital good production function, which is not constant returns to scale: I t = φ t (L O t ) β (3) with < β <. As for preferences, we assume that each generation consumes only in its second period of life (ρ = with our notations), that consumption intertemporal elasticity of substitution is infinite (χ = ) and that disutility of labor is linear (η = η = ), so that utility of generation t is given by U(Ct+, O L Y t, L t+) = Ct+ O θ L Y t θ L O t+ (33)
.. Competitive Allocations A typical generation t agent solves the following problem: max C O t+,l Y t,lo t+,it Ct+ O θ L Y t θ L O t+ s.t. p t I t wt Y L Y t (λ Y t ) Ct+ O wt+l O O t+ + z t+ I t + Π t (λ O t+) where the λs are Lagrange multipliers. One can write the problem first order conditions as λ O t+ = θ = λ Y t wt Y θ = λ O t+wt+ O λ Y t p t = λ O t+z t+ Firms optimal behavior implies the equalization of marginal productivity of factor with their prices, so that w Y t = ( α)c t /L Y t z t = αc t /I t w O t = ( β)p t φ t (L O t ) β Profits are null in the consumption good sector, positive in the capital good one, and equal to Π t = βp t I t Optimality conditions, budget constraints and market equilibrium conditions characterize competitive equilibrium allocation. The system of equations can be collapsed in two equations in L Y and L O : { ψt It (L α Y t ) α = θ β LO t + ψαit α (L Y t ) α I t (A) ( α)( β) α θ θ = ψ t ψ t+ It (L α Y t ) α (L O t ) β+(α )( β) (L Y t ) α (B) Proposition 3 The example two-sector OLG model displays Expectation Driven Business Cycles (EDBC) Proof : Simple manipulation of equation (A) gives us L O t = ( β)( α) θ ψ t I α t (L Y t ) α (C)
Following any shock to expectations, the economy moves along equation (C), which implies a a joint movement of L O and L Y, meaning a joint movement of employment in the consumption and the investment sector. As capital is predetermined, investment and consumption quantities will move together. If the news leads to an increase in investment, it will also lead to an increase in employment and consumption. Q.E.D. One can go further in this model, as the economy competitive path can be analytically fully characterized. Plugging (C) into (B), and using the fact that I t = φ t (L O t ) β, one obtain where K is a constant. L O t+(l O t ) /( α) (L O t ) αβ/( α) ψ t φ α t = K (34) Given the log-linearity of equation (34), one can analytically solve the model (see appendix for more details). Denoting l O t written as l O t = l O λ t + = log L O t, the model s solution (dropping constants) can be c ψ i log ψ t+i + i= c φ i log φ t+i (35) i= where λ ], ] is the smallest root of ( α)λ λ + αβ =, and where c ψ =, c ψ ( α)λ i = ( α)cψ i ( α)λ c φ = α ( α)λ, c φ i = ( α)cφ i ( α)λ From this solution, we can prove the following proposition Proposition 4 In the example two-sector OLG model, a future increase in consumption or investment good productivity generates an aggregate boom (in consumption, investment and employment) during the interim periods (the periods between the announcement of the shock and its implementation) Proof : Note that all the c φ i and c ψ i are positive, so that the news of an increase of phi or ψ at any future period causes a increase in L O, and therefore an aggregate boom (for labor, consumption and investment) for all the interim periods. Q.E.D...3 A Social Planner Problem Again, to contrast this model with a representative one, let us consider the case of a Social Planner that maximizes the geometrically discounted sum of utilities of the different generations, with discount rate δ ], [:
W = C O θ L O + δ ( ) t Ct+ O θ L Y t θ L O t+ t= Note that such an objective would correspond to the intertemporal discounted utility of an pseudo representative agent whose discount factor will be δ and instantaneous utility C O t δθ L Y t θ L O t and for I given.. The Social Planner maximizes W subject to the two technological restrictions resource constraints C t = ψ t (I t ) α (L Y t ) α and I t = φ t (L O t ) β. The problem can be written as a problem in L Y and L O : max L Y t,l O t ψ (I ) α (L Y ) α θ L O + { } t= δt ψ t+ (I t ) α (L Y t+) α θ L Y t θ L O t+ and the first order conditions are given by { δψt+ (L O t ) β+(α )( β) α( β) α (L Y t+) α = θ (A ) ψ t ( α)( β) α (L O ) α( β) (L Y t ) α = δθ (B ) Proposition 5 In the example two-sector OLG model, allocations that maximize a geometrically discounted sum of generations utility do not display EDBC. (36) Proof : Equation (B ) shows that a change in ψ t+ (or any future variable) will not affect L Y t, that is solely determined by past and current variables. Therefore, consumption Ct O will not increase following a news. Q.E.D. In sharp contrast with the competitive allocations, we do not get an aggregate boom...4 Discussion In the next paragraph, we write down a more general version of this model, with incomplete depreciation, capital input in both sectors and consumption in both periods. The qualitative properties of the simple model are preserved in the more general setting..3 A More General OLG Model.3. Technology and Preferences Capital and labor enter now in the production of both capital and consumption. Production functions are now given by C t = ψ t (Kt C ) α (L Y t ) α (37) I t = φ t (Kt K ) β (L O t ) β (38) 3
where K C and K K represent capital services hired in the consumption and capital good sector. Accumulation is done according to (8), and capital can be freely allocated between sectors at the beginning of each period, with the constraint Kt C + Kt K K t (39) Finally, preferences are assumed to take the general formulation (9), meaning that we allow for consumption in both periods of life, and for non constant marginal disutility of labor..3. Competitive Allocations A typical generation t agent solves the following problem: max C Y t,c O t+,ly t,lo t+,it ((C Y t )ρ C O t+) χ χ θ η (L Y t ) η θ η (L O t+) η s.t. Ct Y + p t K t+ wt Y L Y t (λ Y t ) Ct+ O wt+l O O t+ + (z t+ + p t+ ( δ)k t+ + Π t (λ O t+) where the λs are Lagrange multipliers. One can write the problem first order conditions as θ (L Y t ) η θ (L O t+) η = λ Y t w Y t = λ O t+w O t+ ρ(c O t+) χ (C Y t ) ρ( χ) = λ Y t (C O t+) χ (C Y t ) ρ( χ) = λ O t+ λ Y t p t = λ O t+(z t+ + p t+ ( δ)) Firms optimal behavior implies the equalization of marginal productivity of factor with their prices, so that wt Y = ( α)c t /L Y t z t = αc t /Kt C wt O = ( β)p t I t /L O t z t = βp t I t /Kt K At a competitive equilibrium, all firms profits are null, so that Π t =. Optimality conditions, budget constraints and market equilibrium conditions characterize competitve equilibrium allocation. We cannot derive simple analytical results in this setting, and therefore solve numerically for the non-linear perfect foresight dynamics of the model. 4
.3.3 A Social Planner Problem Again, we will contrast the competitive equilibrium allocations properties with the ones of allocations derived from a social optimum, where the Social Planner maximizes the geometrically discounted sum of utilities of the different generations, with discount rate δ S ], [: W = ( (C Y ) ρ C O χ ) χ θ η (L O ) η + {( (C Y (δ S ) t t ) ρ Ct+ O χ t= ) χ θ η (L Y t ) η θ η (L O t+) η } (4) The maximization is done subject to the following resource constraints at every periods: Ct Y + Ct O ψ t (Kt C ) α (L Y t ) α I t φ t (Kt K ) β (L O t ) β Kt C + Kt K K t K t+ ( δ)k t + I t The Social Planner problem can be rewritten as max C O t,c Y t,ly t,lo t,it,kc t W ( s.t. Ct Y + Ct O ψ t (Kt C ) α (L Y t ) α ( (δ S ) t λ C t K t+ ( δ)k t + φ t (K t Kt C ) β (L O t ) β (δ S ) t λ K t ) ) where the λs are discounted Lagrange multipliers. conditions as One can write the problem first order ρ(c O t+) χ (C Y t ) ρ( χ) = λ C t (C O t+) χ (C Y t ) ρ( χ) = δ S λ K t θ (L O t ) η θ (L Y t ) η = δ S λ K t ( β)i t /L O t = λ C t ( α)c t /L Y t λ C t αc t /Kt C = λ K t βi t /Kt K ( ) λ K t = δ S λ C t+ ( δ + βit+ /Kt+ K Again, we make use of numerical simulations to study the property of those socially optimal allocations. 5
.3.4 Numerical results As emphasized before, we provide here a counter example to the impossibility of EDBC for realistic values of the parameters. In the benchmark case we choose the following parameters values: Table : Parameters Values in the Benchmark Case Elasticity of consumption production to capital input α.33 Elasticity of investment production to capital input β.33 Disutility of labour () θ and η.5 and Disutility of labour () θ and η.5 and Utility of consumption when parameter ρ. Consumption intertemporal substitution parameter χ. Capital depreciation δ. Social Planner discount factor δ S.99 Consumption TFP level ψ Investment TFP level φ We implement the following sequence of events in both the competitive and Social Planner economies. Before period, the economy is at its steady state. Exogenous variables φ and ψ are constant. In period, it is unexpectedly announced that a shock will occur 5 periods ahead (in period 6). From now on, no more unexpected shock occurs. In period 5, TFP levels φ and ψ increase for, and go back to their pre-shock values. What we are interested in is the periods that go for to 5 i.e. the periods in which the economy is solely responding to news about the future, what we have called news and that is a shock to expectations. Figures 3 and 4 display the economy responses (see figures 9 and in the appendix for the response over a longer horizon). Note that we observe an aggregate boom in the competitive equilibrium, while the Social Planner problem, that is isomorphic to an intertemporal representative agent model does not (as expected from what we have proven elsewhere). Note that the existence of Expectation Driven Business Cycles is not warranted by the OLG structure. If we assume enough curvature of the utility for consumption function (χ = ), agents of the last period before the shock choose optimally to consume less but work less, as marginal utility of consumption is strongly decreasing. In that case (figure 5), investment and consumption are negatively correlated from period to 5 (see for the response after the 6
Figure 3: News Shock in the Competitive Equilibrium, Benchmark Case.5 φ ψ.8.6.5.8.6.4. labor.4.3.4. consumption.4.3.... Investment.7.795.79 Agg. Consumption.785 Welfare of the Generation born in t This figure displays the response of the competitive economy to the following news shock: in period, it is announced that productivity in the consumption and investment good sectors (φ and ψ) will increase by percent for one period in period 6. All variables except utility are in percentage deviation from their steady state level. The parameters configuration is the benchmark one. Figure 4: News Shock in the Social Planner Problem, Benchmark Case.5 φ ψ.3..5. 3 consumption..3.4.5. labor.5..5. Investment.894.89.89.6 Agg. Consumption.888 Welfare of the Generation born in t This figure displays the response of socially optimal allocations when the Social Planner maximizes a geometrically discounted sum of different generations utility, with a constant discount factor δ S. The shock is the following news shock: in period, it is announced that productivity in the consumption and investment good sectors (φ and ψ) will increase by percent for one period in period 6. All variables except utility are in percentage deviation from their steady state level. The parameters configuration is the benchmark one. 7
shock). Figure 5: News Shock in the Competitive Equilibrium, χ = Case.5 φ ψ.5.5.4...5 labor.3...4 consumption.4... Investment.356.357.358.4 Agg. Consumption.359 Welfare of the Generation born in t This figure displays the response of the competitive economy to the following news shock: in period, it is announced that productivity in the consumption and investment good sectors (φ and ψ) will increase by percent for one period in period 6. All variables except utility are in percentage deviation from their steady state level. In this experiment, a larger than in the benchmark case curvature of the utility for consumption function is assumed (χ = ). Note finally that the economy can display a very different dynamics in the interim period (before the announce and then actual implementation of the shock), compared to its dynamics after the realization of the shock. An example is given in the case of a large curvature if the utility for consumption function (χ = 4). Figure 6 shows that the competitive allocations display large oscillations before the shock, while it can be seen of figure 7 that the economy response is very smooth once the shock is implemented. Also note that those oscillations are not observed in the Social Planner responses (figure 8). 8
Figure 6: News Shock in the Competitive Equilibrium, χ = 4 Case, Interim Period.5 φ ψ 4.5.5.5 consumption 4 labor Investment.66.5.5.68.7 Agg. Consumption.7 Welfare of the Generation born in t This figure displays the response of the competitive economy to the following news shock: in period, it is announced that productivity in the consumption and investment good sectors (φ and ψ) will increase by percent for one period in period 6. All variables except utility are in percentage deviation from their steady state level. In this experiment, a larger than in the benchmark case curvature of the utility for consumption function is assumed (χ = 4). Figure 7: News Shock in the Competitive Equilibrium, χ = 4 Case, Period after the Shock Implementation.5.5 φ ψ.5 6 7 8 9.5 6 7 8 9 labor.5.5 6 7 8 9 consumption.5 6 7 8 9 Investment.66.5.68.7 6 7 8 9 Agg. Consumption.7 6 7 8 9 Welfare of the Generation born in t This figure displays the response of the competitive economy to the following news shock: in period, it is announced that productivity in the consumption and investment good sectors (φ and ψ) will increase by percent for one period in period 6. All variables except utility are in percentage deviation from their steady state level. In this experiment, a larger than in the benchmark case curvature of the utility for consumption function is assumed (χ = 4). 9
Figure 8: News Shock in the Social Planner Problem, Benchmark Case.5.5 φ ψ.5..5.5. labor..5.4.6.8 consumption...4. Investment.4.44.46.6 Agg. Consumption.48 Welfare of the Generation born in t This figure displays the response of socially optimal allocations when the Social Planner maximizes a geometrically discounted sum of different generations utility, with a constant discount factor δ S. The shock is the following news shock: in period, it is announced that productivity in the consumption and investment good sectors (φ and ψ) will increase by percent for one period in period 6. All variables except utility are in percentage deviation from their steady state level. In this experiment, a larger than in the benchmark case curvature of the utility for consumption function is assumed (χ = 4).
3 A Model With Some Infinite Horizon Agents We now consider a different setup that also has the property of displaying Expectation Driven Business Cycles. The economy will be composed of one intertemporal agent and a continuum of non overlapping ones that live each one period. Again, labor input will be agent specific, static agents working in the investment sector while the intertemporal one working in the investment good one. 3. The Setup 3.. Technology The consumption good is produced according to a constant return technology, using labor and capital services as input: C t = F (Kt C, L I t ; ψ t ) (4) L I can be only supplied by the intertemporal agent. ψ is a technological shift. The capital good is also produced according to a constant return technology, using labor and capital services as input. In some specific examples, we will assume that only labor enters as input. In that case, the technology will be decreasing returns to scale. I t = G(Kt K, L S t ; φ t ) (4) L S can be only supplied by static agents. φ is a technological shift. Aggregate capital accumulated with a one period time to build, according to K t+ = K C t+ + K K t+ = ( δ)k t + I t (43) The whole sequences of the shift parameters φ and ψ are assumed to be known, and we therefore study only perfect foresight dynamics of the model. 3.. Preferences In every period, two agents coexist. The first one is an intertemporal one, that consumes, works and accumulate capital. Preferences of the intertemporal agent of the generation born are given by U I = t= {( ) C I χ } γ t t χ θ (L I t ) η η (44)
In every period, one static agent is born and lives for the current period. Each generation works and consumes. Preferences for consumption and leisure are given by ( U S = Ct S θ ) χs (L S t ) η (45) χ S η Note that the value of χ S is irrelevant for the competitive equilibrium, but will play a role in the socail planner problem we will consider later. 3..3 Markets Organization There exist one market for the consumption good, that is assumed to be the num raire, and one for the investment good, whose price will be denoted p. Labor markets exist separately for the capital sector specific labor (supplied by the static agents) and for the consumption good sector (supplied by the intertemporal agents), with respective wages w S and w I. The rental rate of capital is denoted z. Firms in both sectors are owned by the intertemporal agents, who receive profits Π if any. We assume that all markets are competitive. 3. A First Illustrative Example We first consider a particular case in which some analytical results can be obtained. 3.. Restrictions on Technology As for technology, we assume that capital fully depreciates in one period, so that (43) writes K t = I t (46) The consumption good technology is assumed to be Cobb-Douglas, so that (4) becomes C t = ψ t (I t ) α (L I t ) α (47) with < α <. while we assume that only labor input enters the capital good production function, which is not constant returns to scale: I t = φ t (L S t ) β (48) with < β <.
3.. Competitive Allocations The intertemporal agent solves the following problem: { (C I t= γt t ) χ max C I t,l I t,it χ θ η (L I t ) η s.t. C I t + p t I t w I t L I t + z t I t + Π t (λ t ) where λs is Lagrange multiplier associated to the budget constraint of period t. One can write the problem first order conditions as } λ t = (C I t ) χ (49) θ (L I t ) η = λ t w I t (5) plus the budget constraint and a transversality condition. The static agent of period t solves the following problem: ( max C S t,l S t χ S Ct S θ η (L S t ) η λ t p t = γλ t+ z t+ (5) ) χs s.t. Ct S wt S L S t One can write the problem first order conditions as ( w L S S t = t θ ) /(η ) (5) C S t = θ /η (w S t ) +/η (53) Firms optimal behavior implies the equalization of marginal productivity of factor with their prices, so that w I t = ( α)c t /L I t (54) z t = αc t /I t (55) w S t = ( β)p t φ t (L S t ) β (56) Profits are null in the consumption good sector, positive in the capital good one, and equal to Π t = βp t I t Optimality conditions, budget constraints and market equilibrium conditions characterize competitive equilibrium allocation. We then have the following proposition. Proposition 6 The example model with some infinite horizon agents displays Expectation Driven Business Cycles (EDBC) 3
Proof : Let us derive a static relation between L I and L S at the competitive equilibrium. From (5) and (56), one obtain p t = θ β (φ t) (L S t ) η +β (57) Using (57) and the budget constraint of the static agent, that can be written C S t = ( β)p t I t, one obtains C S t = θ (L S t ) η (58) Equation (49), (5), (54) and (47) gives (C I t ) χ ( α)ψ t I α t (L I t ) α = θ (L I t ) η+α (59) Plugging the production function (47) and the expressions of C S t obtained in (58) and C I t obtained in (59) into the equation C t = C S t + C I t, we obtain ( α) ( ψ t I α t (L I t ) α θ (L S t ) η ) χ ψt I α t = θ (L I t ) η +α ) (6) Fully (log) differentiating (6) around any competitive allocation, and denoting X the log deviation of X, and assuming that ψ t does not change, we obtain χ CS C (η I ) L S t = (η + α + χ CC ) ( α) L I I t (6) As η and η are greater than one, (6) implies that any news shock that increases investment and therefore L S t will also increase L I t and therefore consumption, creating an aggregate boom in consumption, investment and employment. Q.E.D. 3..3 A Social Planner Problem Again, to contrast this model with a representative one, let us consider the case of a Social Planner that maximizes the sum of the intertemporal utility of the intertemporal agent and the geometrically discounted sum of utilities of the different static agents, with discount rate γ ], [: W = t= ( γ t Ct S θ ) χs (L S t ) η + χ S η t= {( ) C I χ } γ t t χ θ (L I t ) η η (6) 4
The Social Planner chooses the sequence of L C, L S, C I, C S and I in order to maximize W subject to an initial level of capital I and to the technological restrictions resource constraints C t = ψ t (I t ) α (L I t ) α and I t = φ t (L S t ) β that we can collapse into a single resource constraint to which we associate the multiplier λ The problem first order conditions are ( C S t θ (L S t ) η η ( θ (L S t ) η Ct S θ (L S t ) η η C I t + C S t = ψ t ( φt (L S t ) β) α (L I t ) α (63) (C I t ) χ = λ t (64) θ ) χs = λ t ( α)φ α t ψ t (L S t ) α( β) ( β) α (L I t ) α (65) = λ t (66) ) χs = γλ t+( β) α αφ α t ψ t+ (L S t ) α( β) (L I t+) α (67) plus the resource constraint and a transversality condition. Using those equation, we can prove the following proposition. Proposition 7 In the example model with some infinite horizon agents, allocations that maximize the utility of the intertemporal agent plus a geometrically discounted sum of static agents utilities do not display EDBC. Proof : From equations (5) to (8), one obtains Y t = αδ S I t. Therefore, the socially optimal allocations imply no current reaction of the economy to news, and does not exhibit EDBC. Q.E.D. 3..4 Discussion In the next paragraph (TO BE WRITTEN), we write down a more general version of this model, with incomplete depreciation, capital input in both sectors and consumption in both periods. setting. The qualitative properties of the simple model are preserved in the more general 5
References Beaudry, P., and F. Portier (7): When Can Changes in Expectations Cause Business Cycle Fluctuations in Neo-Classical Settings?, Journal of Economic Theory, 35(), 458 477. 6
A The Two-Sector Analytical Model We provide here some intermediate algebra in the computation of equation (35). We start from equation (34): L O t+(l O t ) /( α) (L O t ) αβ/( α) ψ t φ α t = K (34) Dropping constants and taking logs, denoting l O t = log L O t, (34) can written as ( α)l O t+ l O t + αβl t = αφ t ψ t (68) The characteristic polynomial of this equation is P (λ) = ( α)λ λ + αβ. The associated discriminant is 4( α)αβ, which is always positive as α and β are strictly between and. This polynomial has therefore two real roots. The sum of the roots is equal to /( α) >, while the product αβ/( α) is also positive. Therefore, the two roots are positive. Furthermore, P () = α( β) <, so that one of the two roots of P (noted λ) is inside the positive unit interval, while the other one is strictly greater than one. The model therefore exhibits saddle-path stability, and its solution will be of the type l O t = λl O t + + i= c ψ i log ψ t i + + i= c φ i log φ t i (69) We use (68) and (69) to identify the coefficients c φ i and cψ i. We restrict ourselves to sequences that satisfy c φ i = for i < and that c ψ i terms, we obtain = for i <. Plugging (69) into (68) and rearranging (( α)λ λ + αβ) l o t + i= (( α)λ ) cψ i log ψ t+i + i= ( α)cψ i log ψ t+i+ + i= (( α)λ ) cφ i log ψ t+i + i= ( α)cφ i log ψ t+i+ = αφ t ψ t As this equation must h for any l O t and any sequences of ψ and ψ, on must impose ( α)λ λ + αβ = (( α)λ )c ψ = c ψ i = ( α)cψ i i > ( α)λ (( α)λ )c φ = α c φ i = ( α)cφ i i > ( α)λ The first equation of this system is satisfied as λ is the root between and of the polynomial P (λ). Then the other equations can be solved recursively to compute the sequences of c ψ i c φ i. 7 and
B Figures Figure 9: News Shock in the Competitive Equilibrium, Benchmark Case.5 φ ψ.5 4 6 8.5 4 6 8 labor 4.5 4 6 8 consumption.5.5 4 6 8 Investment.73.7 4 6 8 Agg. Consumption 4 6 8 Welfare of the Generation born in t This figure displays the response of the competitive economy to the following news shock: in period, it is announced that productivity in the consumption and investment good sectors (φ and ψ) will increase by percent for one period in period 6. All variables except utility are in percentage deviation from their steady state level. The parameters configuration is the benchmark one. 8
Figure : News Shock in the Social Planner Problem, Benchmark Case.5 φ ψ.5.5 4 6 8 3 4 6 8 consumption.5 4 6 8 labor.5.5.5 4 6 8 Investment.95.9 4 6 8 Agg. Consumption.85 4 6 8 Welfare of the Generation born in t This figure displays the response of socially optimal allocations when the Social Planner maximizes a geometrically discounted sum of different generations utility, with a constant discount factor δ S. The shock is the following news shock: in period, it is announced that productivity in the consumption and investment good sectors (φ and ψ) will increase by percent for one period in period 6. All variables except utility are in percentage deviation from their steady state level. The parameters configuration is the benchmark one. Figure : News Shock in the Competitive Equilibrium, χ = Case.5.5 φ ψ.5 4 6 8.5.5 4 6 8 labor.5 4 6 8 consumption 4 6 8 Investment.34.5.35.36.5 4 6 8 Agg. Consumption.37 4 6 8 Welfare of the Generation born in t This figure displays the response of the competitive economy to the following news shock: in period, it is announced that productivity in the consumption and investment good sectors (φ and ψ) will increase by percent for one period in period 6. All variables except utility are in percentage deviation from their steady state level. In this experiment, a larger than in the benchmark case curvature of the utility for consumption function is assumed (χ = ). 9