Competitive Equilibrium in an Overlapping Generations Model with Production Loans

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1 Front. Econ. China 2017, 12(2): DOI /s RESEARCH ARTICLE Dihai Wang, Gaowang Wang, Heng-fu Zou Competitive Equilibrium in an Overlapping Generations Model with Production Loans Abstract The paper shows that there do exist two kinds of steady states equilibria in the overlapping generations models with consumption and production loans, similar to the pure exchange economies examined by Gale (1973). Furthermore, the local stability properties of these two (kinds of) steady states are also investigated: In the classical case, the golden-rule steady state is stable and the balanced steady state is saddle-point stable; however, in the Samuelson case, the golden-rule steady state is saddle-point stable and the balanced steady state is stable. Keywords multiple equilibria, overlapping generations model, production loans JEL Classification E21, O41 1 Introduction In his seminal paper, Samuelson (1958) constructs the overlapping generation model Received August 15, 2016 Dihai Wang School of Economics, Fudan University, Shanghai , China wangdihai@fudan.edu.cn Gaowang Wang ( ) Center for Economic Research, Shandong University, Jinan , China gaowang.wang@sdu.edu.cn Heng-fu Zou China Economics and Management Academy, Central University of Finance and Economics, Beijing , China hzoucema@gmail.com

2 Competitive Equilibrium in an Overlapping Generations Model 269 with consumption loans, examines the determination of interest rates and suggests for further research the OLG model with capital. 1 By introducing production employing a durable capital good into the Samuelson (1958) model, Diamond (1965) examines the long-run competitive equilibrium in a growth model with both consumption and production loans and explores the effects on this equilibrium of government debt. Galor and Ryder (1989) and Konishi and Perera-Tallo (1997) establish existence theorems of non-trivial (positive capital stock) steady-state equilibrium in Diamond s (1965) overlapping-generations model with production. Gale (1973) finds two kinds of steady state paths and examines their global stability properties in a Samuelson-type pure exchange economy. And this paper wants to answer the question whether or not there exist multiple kinds of steady state paths in the Diamond (1965) model with production loans. By utilizing an overlapping generations model with both consumption and production loans, this paper shows that there exist two kinds of steady state paths. Furthermore, the local stability of those multiple (kinds of) steady states is examined. Specifically, in the classical case, the golden-rule steady state is stable and the balanced steady state is saddle-point stable; however, in the Samuelson case, the golden-rule steady state is saddle-point stable and the balanced steady state is stable. The paper is organized as follows. Section 2 describes the model. Section 3 examines competitive quilibrium and multiple steady state paths. By executing the phase diagram analysis, section 4 investigates the local stability of the steady states. The conclusion is in section 5. The proof appears in the Appendix. 2 Model The economy being considered is assumed to have an infinite future. Each individual lives for two periods, working in the first period (young) while being retired in the second (old). The birth rate of the population is a given constant, n. And the total number of people in period t is the sum of the young L t = L 0 (1 + n) t and the old L t 1 = L 0 (1 + n) t 1, namely, P t L t + L t 1 = L 0 (2 + n)(1 + n) t 1, which shows that the population growth rate is also n. It is assumed that all consumers are identical as to preferences, and income streams, differing only as regards their dates of birth. A consumer who is born in period t works with wage w t, consumes c y,t and 1 It is well known that Samuelson s model provides an analytical framework for the existence of money.

3 270 Dihai Wang, Gaowang Wang, Heng-fu Zou saves s t (= w t c y,t ) in period t; and in period t +1he consumes the gross interest income of the savings of period t, i.e., c o,t+1 =(1+r t+1 )s t and dies. His utility function is U(c y,t,c o,t+1 )=u(c y,t )+βu(c o,t+1 ), (1) where β is the time preference rate, and u(c) is a strictly positive, strictly increasing, and strictly concave function satisfying the Inada conditions. Then, given the prices w t and r t+1, the maximization problem of the consumer born in period t is to maximize (1) subject to the budget constraints The optimality condition is c y,t + s t = w t,c o,t+1 =(1+r t+1 )s t. (2) u (w t s t )=β(1 + r t+1 )u ((1 + r t+1 )s t ), (3) which determines implicity the savings function of the consumer who was born in period t as a function of those prices by the strict concavity of u, s t = s(w t,r t+1 ). (4) And then he consumes c y,t = w t s(w t,r t+1 ) in period t and consumes c o,t+1 = (1 + r t+1 )s(w t,r t+1 ) in period t +1. Following Arrow and Kurz (1970) s seminal work, we assume that there are two types of capital (private and public) and output depends on both of them. The intuition of public capital entering private production is that government investment is primarily invesment (which forms public capital) in pubic or collective goods, such as highways, airports, roads, transit systems, and railways etc., which are very helpful for private production. 2 Specifically, we assume that the total productive capital K t includes private capital K s,t and public capital K g,t, i.e., K t = K s,t + K g,t,andthat public capital has an exogenously given accumulation process with the same rate of return as private capital, namely, K g,t+1 =(1+r t )K g,t. (5) Equipped with the neoclasscial production technology Y t = F (K t,l t ) and the depreciation rate of physical capital δ (> 0), a representative firm produces the unique 2 Lynde and Richmond (1992) show empirically that public capital has positive marginal product and that private and public capital are complements in production.

4 Competitive Equilibrium in an Overlapping Generations Model 271 commodity in the economy by employing both capital K t with the rental rate r t and labor L t with the wage cost w t. The labor force in period t must be the young people in period t, L t. The profit maximization problem of the representative firm is as follows, Π t = F (K t,l t ) w t L t (r t + δ)k t. With the definitions of the per-labor capital k t K t /L t and the per-labor production f(k t ) F (K t,l t )/L t, optimality leads to r t = f (k t ) δ, w t = f(k t ) k t f (k t ). (6) 3 Competitive Equilibrium and Steady States In order to examine the competitive equilibrium, we should attach the market clearing condition of the commodity market: c o,t L t 1 + c y,t L t + K t+1 (1 δ)k t = F (K t,l t ), (7) whichsaysthatinperiodt the sum of the total consumption c o,t L t 1 + c y,t L t and total investment K t+1 (1 δ)k t is equal to total production F (K t,l t ).InSamuelson (1958) and Gale (1973) s pure exchange economies, only consumption loans are considered. And the market-clearing condition is that the total savings in the economy must cancel out to zero in every period. However, the model considered here is a Diamond-type OLG model with both consumption loans and production loans. In every period, the total saving F (K t,l t ) (c o,t L t 1 + c y,t L t ) is equal to the total investment (K t+1 (1 δ)k t ) in each period. And the amount of total saving or investment is exactly the amout of production loans. 3 By substituting the consumption functions, the savings function (4), and the pricing functions (6) into equation (7), we obtain the dynamic accumulation equation of capital per labor: [f (k t )+(1 δ)] [s (w(k t 1 ),r(k t )) (1 + n)k t ] =(1+n)[s (w(k t ),r(k t+1 )) (1 + n)k t+1 ]. (8) In order to obtain the steady state equilibria, setting k t+1 = k t = k, s t+1 = s t = s in equation (8) gives rise to [f (k ) (δ + n)] [s (1 + n)k ]=0, (9) 3 Certainly, if the production function is just a linear funtion about labor and the total investment is zero in each period, the model here will degenerate to the Samuelson (1958) and Gale (1973) models without production loans..

5 272 Dihai Wang, Gaowang Wang, Heng-fu Zou from which we can derive the following two kinds of steady states f (k 1 ) δ = n, s (w(k 2 ),r(k 2 )) = (1 + n)k 2. (10) Before presenting the propositions, we examine the two kinds of steady states at first. The steady state capital stock of the first type satisfies the modified golden rule which implies that the net return of physical capital equals the population growth rate. 4 Furthermore, equations (6) and (9) tell us that r1 (= f (k1 ) δ) =n, which shows that the equilibrium interest rate depends only on the structure of the population (i.e., the population growth rate) and has nothing to do with technology and preferences in the economy. The equilibrium interest rate determined by the structure of the population is called the biological rate of interest defined by Samuelson (1958) and followed by Gale (1973). To comprehend the steady state of the second type, we multiply both sides of equation (9) by L t 1, s (w(k 2 ),r(k 2 )) L t 1 = k 2 L t(= K ). (11) The left-hand side of equation (11) is the total private savings in period t, and the righthand side stands for the aggregate capital in period t. Their equality tells us that the public capital (or debt) is zero in the economy, since the capital stock (K t ) at any time t is the sum of the private capital (K s,t s t 1 L t 1 ) and the public capital (K g,t ). This second kind of steady state is called balanced ones in the sence of zero public capital. 5 Obviously, there must be an equilibrium interest rate r2 (= f (k2) δ) associated with the balanced steady state. Proposition 1. In the overlapping generations model with production loans, there exist two kinds of steady states: one is the golden-rule steady state with the biological rate of interest, and the other is the balanced steady state without any role of public capital (or debt). 6 The existence of two kinds of steady states is very similar to Gale (1973) s pure exchange economies. However, this extension to include production loans displays two important merits different from the Gale (1973) pure exchange model. On one 4 This is the standard result on the nature of the golden rule path, see, e.g., Phelps (1961) and Diamond (1965). 5 Gale (1973) calls the no-trade (neither borrow nor lend, i.e., aggregate assets are zero) stationary equilibrium balanced in pure exchange economies. And this balance is in the sense of balanced accounts, not balanced growth. In this paper, the balanced steady states are those without public assets or debts. 6 We exclude the case that the two kinds of equilibria coincide (i.e., r2 = n).

6 Competitive Equilibrium in an Overlapping Generations Model 273 hand, the equilibrium interest rate associated with the golden-rule level capital stock is endogenously determined by the model economy in general equilibrium. On the other hand, there is no trade between generations in the second steady state equilibrium of the Gale (1973) model. However, in our model with production loans, due to the existence of production loans, the production can be well organized at any period by combining capital (savings) and labor force provided by the old and young men respectively, which shows that there exist self-interested exchanges among goods, capital and labor markets. Theorem 2. The golden-rule steady state with r1 = n is Pareto optimal, however, the balanced steady state with r2 > (or <) n is not optimal.7 Definition 3. A model will be called classical (Samuelson) if k1 >k 2 (k 1 <k 2 ). The definition is a generalization (to the economy with production) of Gale s (1973) definition which depends on the exogenous parameters in pure exchange economies. And they are essentially the same. The intuition will be revealed in the next section: in the classical case people consume more and save less in both kinds of stationary equilibria, and in the Samuelson case people consume less and save more in these equilibria. Theorem 4. A model is classical (Samuelson) if and only if r2 >n(r2 <n). Proof. It can be easily proved by using (6) and (10). And we omit it. 4 Local Stability of the Equilibria and Phase Diagram Analysis In this section, we will examine the local stability of these two (kinds) of steady states and their local dynamics. For the second kind of steady state, there may exist multiple equilibria with different assumptions on the production and utility functions, to which we will not attach importance. 8 Hence, for simplicity, we assume log utility, Cob- Douglas technology, and complete depreciation: u(c) =lnc, Y = AK α L 1 α,and δ =1. Substituting the assumptions into the optimality conditions in the above section 7 The proof has the same procedure as Stein (1969) (section (1)). And we omit it. 8 In his seminal paper, Samuelson (1958) has pointed out the existence of multiple equilibria. And Bliss (2008) and Hiraguchi (2012) give some particular examples of multiple equilibria in the Diamond capital model numerically and analytically.

7 274 Dihai Wang, Gaowang Wang, Heng-fu Zou leads to the savings function the consumption functions s t = β 1 β w t, (12) w t c y,t = 1 β, c o,t+1 = β(1 + r t+1) w t, (13) 1 β the pricing functions and the two (kinds of) steady states r t = Aαk α 1 t 1,w t = A(1 α)k α t, (14) ( ) 1 ( ) 1 Aα k1 = 1 α,k A 1 α 1+n 2 =, (15) (1 + n)() where k1 is the golden rule level of the steady state and k 2 is the balanced steady state. α Proposition 5. A model is classical (Samuelson) if and only if 1 α > β ( α 1 α < β ). Proof. The conclusion can be drawn from the relations k1 > (<) k 2 r 2 > (<) n α 1 α > (<) β, which are derived from definition 2, theorem 3, and (15). The term α is the relative income share of capital to labor, which is an increasing function of α. As an increasing function of the time preference rate β, the 1 α term β also measures the degree of patience of the consumer. α 1 α > β implies that compared to the relative share of these two factors (capital and labor), the consumers with low time preference rates are relatively impatient and they prefer larger consumption in their earlier periods of their lives and less later on, which corresponds to the classical case examined by Fisher (1961). Conversely, α 1 α < β shows that with a relatively high time preference rate, consumers are relatively patient and hence they prefer future consumption, which corresponds to the Samuelson case investigated by Samuelson (1958). In order to examine the stability of the steady state and the dynamics of the system,

8 Competitive Equilibrium in an Overlapping Generations Model 275 we are going to execute the phase diagram analysis. Define k g,t K g,t and k s,t L t K s,t as the public capital per labor and private capital per labor. Hence, L t K t = K s,t + K g,t k t = k s,t + k g,t. (16) On one hand, equations (5), (14) and (16) give rise to k g,t+1 = Aα (k g,t + k s,t ) α 1 k g,t ; (17) 1+n on the other hand, by the assumption of complete depreciation, we know that the private capital stock of any time t is the total savings of the old people at time t, namely, K s,t+1 = L t s t. Putting (12), (14), and (16) into the above equation and dividing the derived equation by L t+1 turn out to k s,t+1 = A(1 α)β (k g,t + k s,t ) α. (18) (1 + n)(1 + β) Equations (17) and (18) can determine the whole dynamics of the model. With the definition of the steady state (ks,k g ) which subjects k s = k s,t+1 = k s,t and kg = k g,t+1 = k g,t, it is straightforward to derive the two (kinds of) steady states from equations (17) and (18): ( ( ) 1 ( ) 1 ) (ks1,k g1) A 1 α α = α 1 α Aα 1 α, k 1+n 1+n s1, (19) (k s2,k g2) = ( ( ) 1 ) A 1 α, 0, (20) (1 + n)() with the associated steady state levels of interest rates r1 =n, (1+n) α (()) r 2 =. And it is easy to check that k1 = k s1 + k g1,andk 2 = k s2 + k g2. The dynamics of the system composed by equations (17) and (18) can be described in the phase space (k s,k g ). From equation (17), we know that public capital keeps constant on the straight line AB i.e.,k g + k s = and the ( ( ) 1 ) Aα 1 α 1+n k s -axis (i.e., k g 0). Furthermore, if k g > 0 (i.e., public capital), then public capital decreases above the line AB while it increases below it; if k g < 0 (i.e., public debt), then public debt increases above the line AB while it decreases below it. Meanwhile, from (18), we know that private capital keeps constant on the smooth

9 276 Dihai Wang, Gaowang Wang, Heng-fu Zou ( curve OCD i.e.,k g = ( ) 1 (1 + n)() α 1 k α s k s ), which is strictly concave A and passes through the origin. It is not hard to know that private capital increases above the curve and decreases below it. If putting these two curves into the same phase space, it is easy to find that the line AB and the curve OCD intersect only once in the space, but with two possibilities: the classical and Samuelson case. ( α (1) The classical case 1 α > β ). In the classical case (figure 1), (ks1,k g1 ) (E 1 in figure 1) is the golden-rule steady state with the biological interest rate r1 (= n) and (k s2,k g2 ) (E 2 in figure 1) is the balanced steady state with the balanced interest rate r2 (> n). They satisfy ks1 >ks2, kg1 >k g2 (= 0), and hence k 1 >k g2. The direction field of the system in figure 1 shows that the golden rule steady state is locally stable and the balanced steady state is locally saddle-point stable. The rigorous proof of the stability results can be found in the mathematical appendix. Figure 1 The Classical Case ( α (2) The Samuelson case 1 α < β ). Similarly, in the Samuelson case (Figure 2), (ks1,k g1 ) (E 1 in Figure 2) is the golden-rule steady state with the biological interest rate r1 (= n) and (ks2,k g2)

10 Competitive Equilibrium in an Overlapping Generations Model 277 (E 2 in Figure 2) is the balanced steady state with the balanced interest rate r2 (< n). Andtheysatisfyks1 <k s2, k g1 <k g2 (= 0), and hence k 1 <k g2. The direction field of the system in Figure 2 shows that the golden rule steady state is locally saddle-point stable and the balanced steady state is locally stable. The proof can also be found in the mathematical appendix. Figure 2 The Samuelson Case Proposition 6. In the overlapping generations model with production loans, in the classical case the golden-rule steady state is locally stable and the balanced steady state is locally saddle-point stable, however, in the Samuelson case the golden-rule steady state is locally saddle-point stable and the balanced steady state is locally stable. 5 Conclusion By utilizing an overlapping generations model with both consumption and production loans, the paper shows that there do exist two kinds of steady state paths, similar to the pure exchange economies examined by Gale (1973). Furthermore, the local stability of these multiple (kinds of) steady state is investigated. Specifically, in the classical case, the golden-rule steady state is stable and the balanced steady state is saddle-point stable; however, in the Samuelson case, the golden-rule steady state is saddle-point stable and the balanced steady state is stable. Acknowledgements The authors thank the reviewer for valuable comments and suggestions. Wang Gaowang acknowledges the support of Humanities and Social Sciences of Ministry of Education (No. 16YJC790095).

11 278 Dihai Wang, Gaowang Wang, Heng-fu Zou References Arrow K, Kurz M (1970). Public Investment. The Rate of Return and Optimal Fiscal Policy. Baltimore, MA: The Johns Hopkins University Press Bliss C (2008). Multiple equilibrium in the Diamond Capital Model. Economics Letters, 100: Diamond P (1965). National debt in a neoclassical growth model. The American Economic Review, 55 (5): Fisher I (1961). The Theory of Interest. New York, NY: Augusus M. Kelley Publishers Gale D (1973). Pure exchange equilibrium of dynamic economic model. Journal of Economic Theory, 6: Galor O, Ryder H (1989). Existence, uniqueness, and stability of equilibrium in an overlappinggenerations model with production capital. Journal of Economic Theory, 49: Hiraguchi R (2012). Multiple equilibrium in the overlapping generations model revisited. Economics Letters, 115: Konishi H, Perera-Tallo F (1997). Existence of steady-state equilibrium in an overlappinggenerations model with production. Economic Theory, 9: Lynde C, Richmand J (1992). The role of public capital in production. The Review of Economics and Statistics, 74: Phelps E (1961). The golden rule of accumulation: A fable for growthmen. American Economic Review, 51: Samuelson P (1958). An exact consumption-loan model of interest with or without the social contrivance of money. Journal of Political Economy, 66: Stein J (1969). A minimal role of government in achieving optimal growth. Economica, 36: Appendix In this appendix, we examine the local stability of the two (kinds of) steady states. Define k i,t = k i,t ki, i = g, s. The dynamic system given by (17) and (18) can be linearized around the steady state as k g,t+1 k gg kg,t + k gs kg,t, ks,t+1 k sg kg,t + k ss ks,t, ( ) ( (α 1)kg (kg +k s) +1, k gs = where k gg = Aα(k g +k s) α 1 (1+n) Aα(1 α)β(k g +k s) α 1 (1+n)(), k ss = Aα(1 α)β(k g +k s) α 1 (1+n)(). Aα(α 1)(k g +k s) α 2 k g (1+n) ), k sg =

12 Competitive Equilibrium in an Overlapping Generations Model 279 Since the procedure is similar, we consider only the classical case. In the goldenrule steady state E 1, the matrix form of the linearized system becomes: ( ) ( ) [ ] 1 (α 1) 1 (α 1) 1 kg,t+1 α (1 + β) α (1 + β) k s,t+1 [ ] kg,t. k s,t Define the Jacobian matrix of the linearized system as M 1 and its two eigenvalues as λ 1, λ 2. Hence, we have λ 1 +λ 2 = trace (M 1 )=α+ α (1 + β),λ 1λ 2 =det(m 1 )= α (1 + β).their solutions are: λ 1 = α (1 + β) (0, 1), λ 2 = α (0, 1), which show that the golden-rule steady state is locally stable. Similarly, in the balanced steady state E 2, the matrix form of the linearized system is [ kg,t+1 k s,t+1 ] α (1 + β) α 0 α [ ] kg,t. k s,t Define the Jacobian matrix of the linearized system as M 2 and its two eigenvalues α (1 + β) as μ 1, μ 2. Hence, μ 1 + μ 2 = trace (M 2 )=α +,μ 1μ 2 =det(m 2 )= α 2 (1 + β). The corresponding solutions are: μ α (1 + β) 1 = (1, ), μ 2 = α (0, 1), which tell that the balanced steady state is locally saddly-point stable.