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Endogenous competitive business cycles with positive outside money and positive interest elasticity of savings
eco.om|cs leff;e~ Economics Letters 52 (1996) 309-317
EISJEVIER
Endogenous competitive business cycles with positive outside money and positive interest elasticity of savings Dazhan Huang", Paul Maddenb'* "Otina Nanshan Develoinnent Inc., Shenzhen, China t'School of Econmnic Studies, Univer,~ity of Mancht:~'ter, Manchester MI3 9PL, UK Received 5 February 1996; accepted I May 1996
Abstract .An overlapping generations model is studied where young agents supply labour and old agents supply entrepreneurial ability. Endogenous competitive cycles emerge if labour demand is appropriately inelastic, with positive outside money and positive interest elasticity of savings.
Keywords: Endogenous competitive b~siness cycles JEL classification: E32
1. Introduction
Endogenous competitive business cycles have been widely studied in the last decade (Grandmont, 1985; Azariadis and Guesnerie, 1986; Farmer, 1986; Reichlin, 1986; Benhabib and Laroque, 1988, for instance). As a result, various conditions are known under which perfectly competitive equilibria with perfect foresight may be cyclical in overlapping generations models with two-period lives and stationary characteristics. To data such endogenous cycles have been found in production economies without money (Reichlin, 1986) or with negative outside money (Farmer, 1986) and in economies with a negative interest elasticity of savings (Azariadis and Guesnerie, 1986; Grandmont, 1985), the latter being associated with odd shapes of consumer offer curves violating gross substitutes. Indeed, for a family of production economies with inputs of capital and labour, Benhabib and Laroque (1988) show that endogenous cycles can occur only with either negative outside money or a negative interest elasticity of savings. The objective of this paper is to introduce a family of production economies with inputs of labour and entrepreneurship, and to show that competitive * Corresponding author. Tel.: 0161 275 4870; fax' 0161 275 4812; e-mail; [email protected] 0165-1765/96/$12.00 © 1996 Elsevier Science S.A. All rights reserved P l l S0165-1765(96)00869-5
310
D. Huang, P. Madden / Economics Letters 52 (1996) 309-317
endogenous cycles emerge with positive outside money, positive interest elasticity of savings and well-behaved offer curves satisfying gross substitutes, provided only that the demand for labour is suitably inelastic. In the model, one agent is born every period, able to work when young, and to supply one unit of entrepreneurial ability when old. Entrepreneurial ability can be modelled in two equivalent ways. On the first interpretation, entrepreneurial abiPity allows old agents to organise ~t of young labour in period t into producing an output of f(F,) of the only consumption good available in the economy. In period t, w, and p, denote the money prices on the competitive markets for labour and consumption good, respectively, and old agents in t earn a rent on their entrepreneurial ability of p,f(~)- w,~. Assuming that agents consume only when old, perfect foresight competitive equilibria reduce to a univariate difference equation, whose 'flip' bifurcations point to the role of inelastic labour demand in generating endogenous business cycles. The novelty stems from the impact of anticipated entrepreneurial rents on labour supply decisions. To see the intuition behind a two-cycle, for instance, suppose today'~ employment level on this cycle is 'low', whilst tomorrow's is 'high'. Given the inelasticity of labour demand, tomorrow's real wage must be much lower, and the real rent accruing to tomorrow's entrepreneurs must be much higher than today's. But tomorrow's entrepreneurs are today's young workers, and, with normal leisure, the expected high real entrepreneurial rent overcomes the relative price effect and creates today's low labor supply, consistent with equilibrium today. The reasoning reverses for tomorrow, suggesting the possibility of a two-cycle driven by inelastic labour demand. Alternatively, we may think of a concave, constant returns production function F(~, e,), where er is entrepreneurial ability and F(~, l ) - f(~). Now, assume that there is a competitive market in entrepreneurial ability, so that there is a full set of competitive markets as in the earlier endogenous competitive business cycle literature. The equilibrium return to entrepreneurial ability is its marginal product, which, as usual under constant returns and profit maximisation, equates to the 'rent' of the last paragraph, and the two interpretations produce equivalent dynamics. The exposition is more attractive on the first interpretation, which we follow, rather than the alternative. Section 2 describes the economy. Section 3 studies steady-state and two-periodic equilibria. Section 4 addresses cycle multiplicity and stability and also produces a striking example - with linear utility and quadratic production functions the model collapses exactly to the well-known logistic equation. Section 5 concludes. 2, The ~'om~my We study an overlapping generations models in which time runs from t = 0, 1 , 2 , . . . and in which agents live two periods. For convenience, we assume that one agent is born each period and there is a pre-existing agent at t = 0 who holds a stock M of fiat money. The young agent in period t supplies an amount of labour ~, 0 ~<~ ~
D. Huang, P. Madden / Economics Letters 52 (1996) 309-317
311
Alternatively, we may think of [ as being derived from a concave, constant returns, two-factor production function F(<,e,), where ~ is young labour supply, e , = 1 is entrepreneurship supplied, and f ( ~ ) = F(~, 1). Agents consume only in old age, and the lifetime utility of the agent born in t is V~, (L - ~) + V2(c,+!), where c,+~ is consumption of the single consumption good in period t + 1. Here, VI : [0, L]--* R is continuous, concave and strictly increasing, C 2, at least on (0, L); V2:R+--> R, is C 2, concave and strictly increasing. Note that these assumptions imply that V~(0) and ['(0) are finite. Note also that the separability assumption on utility involves little cost, since it admits, of course, the 'well-behaved' offer curves of interest. There is a fixed amount M of outside money. The money wage rate for young labour in period t is wt, and p, is the period t money price of the consumption good. There are competitive markets for the trade of labour, consumption good and money in each period t, and the old agent in t receives the profit ll,=p,f((~)-w,C, from the application of entrepreneurial knowledge; thus, (,, will be chosen to maximise 11,, given w,, p, in each period so that
['(¢'t) <~w,/p, with equality,
(1)
if (~,> 0.
On the alternative interpretation we may think of [ as being derived from a concave, constant returns production function F(~,e,), where < is young labour supply, e, is entrepreneurship, and f(e,) = F(~, 1). If s, is the money price of entrepreneurship, old agents will supply one unit and the entrepreneurship market clears if OF
OF
st=p,-~t(~, 1)=p,F(~, 1 ) - p , ~ . - ~
(~, 1)
from constant returns. Thus, in equilibrium, the return to entrepreneurship is s, = p , f ( ~ ) w , < - ~',, as before, so that the two interpretations are equivalent. We exposit the sequel in terms of the first interpletation. Assuming perfect foresight, the lifetime utility maximisation problem for the agent born in period t is max Vt(L - e,) + V2(c,+1) , ttlt'('t'Ct
÷I
S t . m t . ~ Wt~t , Pt + I Ct + I <~ m t + Ct '
0 < ~ < ~ L , c,+l ~>0, m,~>0. Here, m, is money held as savings by young agents for their old age. The following conditions characterise the solution to this utility maximisation problem:
A=B, ifO<~~B, i f ~ - O ,
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D. Huang, P. Madden I Economics Letters 52 (1996) 309-317
where
(2)
A = V~(L - (,)
and w, V" ~ w, ~ + //,+,) B-p,+1 Pl+l t P,+z (3)
wt~=m, .
A perfect foresight monetary equilibrium (an equilibrium for short) is a sequence (e,, w,. p,) with 0~;e, ~ L , w,>0, p , > 0 for t=O, 1 , 2 , . . . , which satisfies (1), (2) and (3) for all t with m , - M. A k.periodic equilibrium (steady state if k = 1) is an equilibrium in which for all t, = g+k, ~ ~ ~,+k for 0 < i < k, with similar periodic behaviour for w, and p,; such equilibria are completely described by k different numbers ( ¢ ' t , . . . , ek), which constitute the k-cycle for employment, where ~ +! follows <, i = 1 , . . . , k - 1 and e t follows ~'k, and similarly for wages and prices. The following assumption ensures that ~ < L for all t in an equilibrium. Assumption 1. V~(0) = + ~, or f'(L ) = 0.
If e, > 0 in equilibrium for a t ~ 1, then ¢,+t > 0 also, since c, =f(e,+~) in equilibrium, and if c,+ t --0, e, = 0 from utility maximisation. Also, it is clear that there cannot be an equilibrium with eo - 0; thus C, > 0 for all t in the equilibria., as follows. For all t, e, E (0, L), w, > 0, p, > O, where
f'(~)=wtlp,, V (L -
e,)=p,+--S
(4) .\p,+,
e, + p , + , /
.
w,~ = M .
(5) (6)
After some manipulation using the definition of H,+~, (4), (5) and (6) are equivalent to (4), (6) and the following (7):
~.V~(L -
~) = ~+,. f ' ( ~ + , ) . V~[f(~+,)].
(7)
Thus, equilibria are completely characterised by strictly positive sequences (£o, Cm,...) satisfying (7), with (6) and (4) describing the corresponding sequences of wages and prices. Since the left-hand side of (7) is strictly increasing in ~, we may 'invert' giving, equivalently to (7),
t, =
(8)
Reversing the direction of time gives
=
(9)
D. Huang, P. Madden I Economics Letters 52 (1996) 309-317
313
Clearly, if ( t o l , . . . , tok) is a k-periodic equilibrium of the economy, then (tok,..., to~) is a k-periodic solution of the difference equation (9), and vice-versa. So finding k-periodic equilibria reduces to finding k-periodic solutions of (9).
3. Steady-state and two-periodic equilibria We show first that there is a unique steady-state equilibrium if the following assumptions hold: Assumption 2. f'(0) > V;(C )/V~(O). This merely ensures that autarchy is inefficient. Assumption 3. At each to ~. [0, L ], either f"(to) < 0, or V'((L - to) < 0, or V['(f(to)) < O. Theorem 1. Under Assumptions 1-3, there exists a unique steady-state equilibrium ( ~, if,, ~ ) given by f ' ( ~ ) = V~(L- ~)/V~(f(~)) , ff = M / ~ ,
fi= ff,/f'(~) .
Proof. It is immediate that any steady-state equilibrium must be given by the stated equations. Let or(to)=V~(L-e)/v(,(f(e)). Then f'(0)>or(0) from Assumption 2. From Assumption 1, f ' ( 1 ) < or(l) and there must exist at least one steady-state equilibrium. From concavity, f'(~¢) is non-increasing on [0, L] and or(t°) is non-decreasing on [0, L]; from Assumption 3, at each to E[0, L] either f'(to) is strictly decreasing, or or(to) is strictly increasing, and uniqueness is guaranteed. E! To study two-cycle equilibria, notice first that from (7), ~p(0)= 0 and ,p'(e)IV;(L -- ,p(e)) - q,(e). V~(L -
~(e))l
= f ' ( e ) , v ~ ( f ( e ) ) + e . v ~ ( f ( e ) ) , f"(e) + e [ f ' ( e ) ) l ' , v','(f(e)).
(~o)
Hence, ~p'(0)=f'(0). V~(O)/V[(~e)> 1, from Assumption 2. Writing ¢2(e) for ¢(¢~,g)), ~ 2 ( 0 ) = 0 and (d~p2/dto)(0)=[q~'(0)]2>l. Also, , p ' ( ? ) = ? and d,t,*/dto=[,p'(?)] 2. If [g,,(~)]2 > 1, then there must exist e E (0, ?) where ¢2(¢,) = to, which implies the existence of a two-cycle. From (10),
~P'(~) f'V~ + ~. V~ f" + ~(f,)2. V~' =
v;-
h,';
(11) '
with derivatives of f evaluated at ~, those of V2 at f(~), and those of Vt at L - ~7. Since f ' = V~/V~ at equilibrium, it follows that ~p'(~)< 1 so that c p ' ( ~ ) < - 1 is the appropriate sufficient condition for two-cycles. To study this condition we use the following notations. (a) The elasticity of output, 71 = f'(~)~/f(~).
D. Huang, P. Madden / Economics Letters 52 (1996) 309-317
314
(b) The elasticity of labour demand, E = f ' ( ~ ) / ~ . f"(~). (c) The Arrow-Pratt relative risk-aversion coefficient of old consumers, r =3"(~).V'2'(f(~))/
V'(f(~)). (d) Consumers interest elasticity of savings"
S = -(*lr + 1)~(fir + il/'"/V~) 1 with VI derivatives evaluated at L - ~. Note. The formula in (d) derives by introducing an interest factor R, on money, so that the consumer problem becomes: choose < E [0, L] to
w, R,+lt',]
F 1I,+, +
Pt+'---T
Differentiating a first-order condition for the problem with respect to R,+ ! produces a formula for the interest elasticity of savings, (R,+~/w,e,).(d(w,~,)/dR,÷~). Use of the equilibrium conditions with R,+! = 1 produces the formula in (d), ,alter some manipulation. Notice that this interest elasticity has the sign of d~/dR,+ 1. So a positive interest elasticity of savings occurs if, and only if, the consumer's offer curve does not 'bend backwards', or equivalently if, and only if, consumer demands satisfy gross substitutes. Introducing the notation of (a)-(d) into (11) and manipulating the inequality ~,'(~) < - 1 produces the following straightforwardly.
Theorem 2. Under Assumptions 1-3, there exists a two-cycle if --tv/r~ 2 + -~ + ~ < - 2 .
(12)
Existing results (Grandmont, 1985; Azariadis and Guesnerie, 1986) suggest that a sufficiently negative S will make (12) true. Indeed, this is the case. If S < 0 , then from (d), 1 + r r / < 0 (since r < 0, 7/> 0) and (12) becomes S E
> - ( 1 + 2S)( 1 + r/r)
(13)
This is necessarily true if S < - ½ even in the linear f case, where 1/e = 0, and r/= 1, exactly as in Azariadis and Guesnerie (1986). Moreover, (13) is also true for 0 > S > - ~ provided e is sufficiently small, i.e. if labour demand is sufficiently inelastic. Moreover, inelastic labour demand can now generate endogenous cycles with a positive savings elasticity. In this case, if S > 0, then 1 + rr/> 0 and (12) becomes S (1 + v/r)(l + 2S) > - E ,
(14)
which again will be true if labour demand is sufficiently inelastic. Notice that the parameter is the only parameter in (14) that involves the second derivative f"(C). This derivative may be increased independently of the other parameters, ensuring that (14) becomes true eventually, i.e. for a sufficiently inelastic labour demand. Hence,
D. Huang, P. Madden / Economics Letters 52 (1996) 309-317
315
Corollary to Theorem 2. There exists a two-cycle with positive interest elasticity of savings if labour demand is sufficiently inelastic.
4. Uniqueness and stability of cycles Following Grandmont (1985) we study the time-reversed version of (8), the 'backward perfect foresight (bpf)' dynamics of Eq. (9) earlier. Completely analogously to Grandmont (1985) we may define a plausible learning process for expectation formation (the 'dynamics with learning'), which has the result that k-periodic equilibria are also equilibria of the dynamics with learning; moreover, these equilibria are stable in the dynamics with learning if they are stable in the bpf dynamics (see Huang, 1994, for details of this analogy). This suggests that stability in the bpf dynamics is an appropriate stability concept for the model. In the sequel we speak of convergence and stability to mean convergence and stability in the bpf, and we focus on the multiplicity and stability of cycles. A striking example emerges in the special case where utility is linear, and the production function is quadratic. Assumption 4. (a) V,(L- ~)= L - t,, (b) V2(c) = c, (c) f(t °) = A~e(1 - te/2L), where 1 < A ~<4.
Assumption 4 ensures that Assumptions 1, 2 and 3 are satisfied, Assumption 1 with f'(L) = 0. Under Assumption 4 the bpf dynamics become directly the well-known logistic equation: )=xe,
-L-
•
(15)
The unique steady-state equilibrium is ~= L(1 _ 1 ) .
(16)
The following properties of equilibrium are well-known consequences of the logistic specification. Theorem 3. Under Assumption 4: (i) For all values of h ~ (1, 4) there is at most one locally stable k-periodic equilibrium. (ii) For all h ~ (1, 3) the steady-state equilibrium ~ is the unique locally stable equilibrium. (iii) For all A E(3, 1+ V~) there is a two-periodic equilibrium that is the unique locally stable equilibrium. (iv) For each h E(1 + V~, )~*) where A* =3.57, there is a 2k(^Lperiodic equilibrium that is the unique locally stable equilibrium; as h increases in this range, k(A) increases, covering all positive integers by A* ('period doubling bifurcation').
D. Huang, P Madden / Economics Letters 52 (1996) 309-317
316
(v) For A E ( A*, 4) there are intervals of A-values for which there is a unique locally stable equilibrium that is k-periodic for some k that is not a power of 2; in particular, there is an interval in which there is a three-periodic equilibrium that is the unique locally stable equilibrium. Under Assumption 4, ~ = 1 / ( 1 - A) and so two-cycles emerge when E > - ½ (i.e. A> 3), which is condition (12) in Theorem 2 with r = 0, s = +oo. Theorem 3 shows that in this special case the emerging two-cycle is the unique locally stable equilibrium if E is 'not too much' bigger than - ~ , and that unique locally stable k-periodic equilibria emerge in a particular pattern as ¢ increases from -½ to -½. Finally, a partial generalisation of Theorem 3 is as follows, and shows that the cyclical phenomenon of this paper is robust. Suppose that an economy E is defined by the functions V~, V,, f and by the money stock M. Two economies E I and E 2 are close if g t is close to M" and'if (VI, V~, f~) is close to (V~, V 22, f :) for the C: topology (so that on any compact subset of their domains, the functions and their first two derivatives are close). Define also for an economy E the following 'slack' in inequality (12):
(
A ( E ) = - ~ + ~ r 2+~"
+~+2,
where the right-hand side is evaluated at the steady-state equilibrium of E. An economy E is then 'critical' if A(E)= 0, so that t0'(e)= - 1 at the steady-state equilibrium ~ of E; Theorem 2 says that E has a two-cycle if A(E) < 0; also the steady-state equilibrium of E is locally stable if A(E)> 0. Our differentiability assumption allows standard bifurcation results to be applied (see, for example, the exposition of Azariadis, 1993, Theorem 8.4) to give the stability result: Theorem 4. Under Assumptions 1, 2 and 3, economies that are sufficiently close to a critical economy possess a locally stable two-periodic equilibrium if A(E) < O.
$. Conclusions Endogenous competitive business cycles with outside money are known to exist if there is a negative interest elasticity of savings, associated with backward-bending offer curves and violation of gross substitutability. The relative implausibility of these conditions is often seen as a drawback in the literature. We have shown here now this drawback may be overcome. If the young supply labour and the old supply entrepreneurship, endogenous competitive cycles with positive interest elasticity of savings emerge if labour demand is sufficiently inelastic.
Acknowledgements This paper arises out of Dazhan Huang's PhD thesis (Huang, 1994). We are very grateful to Chris Birchenall and Laurence Copeland for extensive helpful comments. Thanks also to
D. Huang, P. Madden / Economics Letters 52 (1996) 309-317
317
Costas Azariadis, Roger Farmer and participants in the Manchester macro workshop for comments on an earlier draft. All shortcomings remain the authors' responsibility.
References Aziariadis, C., 1993, Intertemporal macroeconomics (Basil Blackwell, Oxford). Azariadis, C. and Guesnerie, R., 1986, Sunspots and cycles, Review of Economic Studies 53, 725-736. Benhabib, J. and Laroque, G., 1988, On competitive cycles in production economies, Journal of Economic Theory 45, 145-170. Farmer, R., 1986, Deficits and cycles, Journal of Economic Theory 40, 77-78. Orandmont, J-M., 1985, On endogenous competitive business cycles, Econometrica 53, 995-1046. Huang, D., 1994, Essays on cycles and chaos in economic dynamics, PhD thesis, University of Manchester. Reichlin, P., 1986, Equilibrium cycles in an overlapping generations model with production, Journal of Economic Theory 40, 89-102.
EISJEVIER
Endogenous competitive business cycles with positive outside money and positive interest elasticity of savings Dazhan Huang", Paul Maddenb'* "Otina Nanshan Develoinnent Inc., Shenzhen, China t'School of Econmnic Studies, Univer,~ity of Mancht:~'ter, Manchester MI3 9PL, UK Received 5 February 1996; accepted I May 1996
Abstract .An overlapping generations model is studied where young agents supply labour and old agents supply entrepreneurial ability. Endogenous competitive cycles emerge if labour demand is appropriately inelastic, with positive outside money and positive interest elasticity of savings.
Keywords: Endogenous competitive b~siness cycles JEL classification: E32
1. Introduction
Endogenous competitive business cycles have been widely studied in the last decade (Grandmont, 1985; Azariadis and Guesnerie, 1986; Farmer, 1986; Reichlin, 1986; Benhabib and Laroque, 1988, for instance). As a result, various conditions are known under which perfectly competitive equilibria with perfect foresight may be cyclical in overlapping generations models with two-period lives and stationary characteristics. To data such endogenous cycles have been found in production economies without money (Reichlin, 1986) or with negative outside money (Farmer, 1986) and in economies with a negative interest elasticity of savings (Azariadis and Guesnerie, 1986; Grandmont, 1985), the latter being associated with odd shapes of consumer offer curves violating gross substitutes. Indeed, for a family of production economies with inputs of capital and labour, Benhabib and Laroque (1988) show that endogenous cycles can occur only with either negative outside money or a negative interest elasticity of savings. The objective of this paper is to introduce a family of production economies with inputs of labour and entrepreneurship, and to show that competitive * Corresponding author. Tel.: 0161 275 4870; fax' 0161 275 4812; e-mail; [email protected] 0165-1765/96/$12.00 © 1996 Elsevier Science S.A. All rights reserved P l l S0165-1765(96)00869-5
310
D. Huang, P. Madden / Economics Letters 52 (1996) 309-317
endogenous cycles emerge with positive outside money, positive interest elasticity of savings and well-behaved offer curves satisfying gross substitutes, provided only that the demand for labour is suitably inelastic. In the model, one agent is born every period, able to work when young, and to supply one unit of entrepreneurial ability when old. Entrepreneurial ability can be modelled in two equivalent ways. On the first interpretation, entrepreneurial abiPity allows old agents to organise ~t of young labour in period t into producing an output of f(F,) of the only consumption good available in the economy. In period t, w, and p, denote the money prices on the competitive markets for labour and consumption good, respectively, and old agents in t earn a rent on their entrepreneurial ability of p,f(~)- w,~. Assuming that agents consume only when old, perfect foresight competitive equilibria reduce to a univariate difference equation, whose 'flip' bifurcations point to the role of inelastic labour demand in generating endogenous business cycles. The novelty stems from the impact of anticipated entrepreneurial rents on labour supply decisions. To see the intuition behind a two-cycle, for instance, suppose today'~ employment level on this cycle is 'low', whilst tomorrow's is 'high'. Given the inelasticity of labour demand, tomorrow's real wage must be much lower, and the real rent accruing to tomorrow's entrepreneurs must be much higher than today's. But tomorrow's entrepreneurs are today's young workers, and, with normal leisure, the expected high real entrepreneurial rent overcomes the relative price effect and creates today's low labor supply, consistent with equilibrium today. The reasoning reverses for tomorrow, suggesting the possibility of a two-cycle driven by inelastic labour demand. Alternatively, we may think of a concave, constant returns production function F(~, e,), where er is entrepreneurial ability and F(~, l ) - f(~). Now, assume that there is a competitive market in entrepreneurial ability, so that there is a full set of competitive markets as in the earlier endogenous competitive business cycle literature. The equilibrium return to entrepreneurial ability is its marginal product, which, as usual under constant returns and profit maximisation, equates to the 'rent' of the last paragraph, and the two interpretations produce equivalent dynamics. The exposition is more attractive on the first interpretation, which we follow, rather than the alternative. Section 2 describes the economy. Section 3 studies steady-state and two-periodic equilibria. Section 4 addresses cycle multiplicity and stability and also produces a striking example - with linear utility and quadratic production functions the model collapses exactly to the well-known logistic equation. Section 5 concludes. 2, The ~'om~my We study an overlapping generations models in which time runs from t = 0, 1 , 2 , . . . and in which agents live two periods. For convenience, we assume that one agent is born each period and there is a pre-existing agent at t = 0 who holds a stock M of fiat money. The young agent in period t supplies an amount of labour ~, 0 ~<~ ~
D. Huang, P. Madden / Economics Letters 52 (1996) 309-317
311
Alternatively, we may think of [ as being derived from a concave, constant returns, two-factor production function F(<,e,), where ~ is young labour supply, e , = 1 is entrepreneurship supplied, and f ( ~ ) = F(~, 1). Agents consume only in old age, and the lifetime utility of the agent born in t is V~, (L - ~) + V2(c,+!), where c,+~ is consumption of the single consumption good in period t + 1. Here, VI : [0, L]--* R is continuous, concave and strictly increasing, C 2, at least on (0, L); V2:R+--> R, is C 2, concave and strictly increasing. Note that these assumptions imply that V~(0) and ['(0) are finite. Note also that the separability assumption on utility involves little cost, since it admits, of course, the 'well-behaved' offer curves of interest. There is a fixed amount M of outside money. The money wage rate for young labour in period t is wt, and p, is the period t money price of the consumption good. There are competitive markets for the trade of labour, consumption good and money in each period t, and the old agent in t receives the profit ll,=p,f((~)-w,C, from the application of entrepreneurial knowledge; thus, (,, will be chosen to maximise 11,, given w,, p, in each period so that
['(¢'t) <~w,/p, with equality,
(1)
if (~,> 0.
On the alternative interpretation we may think of [ as being derived from a concave, constant returns production function F(~,e,), where < is young labour supply, e, is entrepreneurship, and f(e,) = F(~, 1). If s, is the money price of entrepreneurship, old agents will supply one unit and the entrepreneurship market clears if OF
OF
st=p,-~t(~, 1)=p,F(~, 1 ) - p , ~ . - ~
(~, 1)
from constant returns. Thus, in equilibrium, the return to entrepreneurship is s, = p , f ( ~ ) w , < - ~',, as before, so that the two interpretations are equivalent. We exposit the sequel in terms of the first interpletation. Assuming perfect foresight, the lifetime utility maximisation problem for the agent born in period t is max Vt(L - e,) + V2(c,+1) , ttlt'('t'Ct
÷I
S t . m t . ~ Wt~t , Pt + I Ct + I <~ m t + Ct '
0 < ~ < ~ L , c,+l ~>0, m,~>0. Here, m, is money held as savings by young agents for their old age. The following conditions characterise the solution to this utility maximisation problem:
A=B, ifO<~
312
D. Huang, P. Madden I Economics Letters 52 (1996) 309-317
where
(2)
A = V~(L - (,)
and w, V" ~ w, ~ + //,+,) B-p,+1 Pl+l t P,+z (3)
wt~=m, .
A perfect foresight monetary equilibrium (an equilibrium for short) is a sequence (e,, w,. p,) with 0~;e, ~ L , w,>0, p , > 0 for t=O, 1 , 2 , . . . , which satisfies (1), (2) and (3) for all t with m , - M. A k.periodic equilibrium (steady state if k = 1) is an equilibrium in which for all t, = g+k, ~ ~ ~,+k for 0 < i < k, with similar periodic behaviour for w, and p,; such equilibria are completely described by k different numbers ( ¢ ' t , . . . , ek), which constitute the k-cycle for employment, where ~ +! follows <, i = 1 , . . . , k - 1 and e t follows ~'k, and similarly for wages and prices. The following assumption ensures that ~ < L for all t in an equilibrium. Assumption 1. V~(0) = + ~, or f'(L ) = 0.
If e, > 0 in equilibrium for a t ~ 1, then ¢,+t > 0 also, since c, =f(e,+~) in equilibrium, and if c,+ t --0, e, = 0 from utility maximisation. Also, it is clear that there cannot be an equilibrium with eo - 0; thus C, > 0 for all t in the equilibria., as follows. For all t, e, E (0, L), w, > 0, p, > O, where
f'(~)=wtlp,, V (L -
e,)=p,+--S
(4) .\p,+,
e, + p , + , /
.
w,~ = M .
(5) (6)
After some manipulation using the definition of H,+~, (4), (5) and (6) are equivalent to (4), (6) and the following (7):
~.V~(L -
~) = ~+,. f ' ( ~ + , ) . V~[f(~+,)].
(7)
Thus, equilibria are completely characterised by strictly positive sequences (£o, Cm,...) satisfying (7), with (6) and (4) describing the corresponding sequences of wages and prices. Since the left-hand side of (7) is strictly increasing in ~, we may 'invert' giving, equivalently to (7),
t, =
(8)
Reversing the direction of time gives
=
(9)
D. Huang, P. Madden I Economics Letters 52 (1996) 309-317
313
Clearly, if ( t o l , . . . , tok) is a k-periodic equilibrium of the economy, then (tok,..., to~) is a k-periodic solution of the difference equation (9), and vice-versa. So finding k-periodic equilibria reduces to finding k-periodic solutions of (9).
3. Steady-state and two-periodic equilibria We show first that there is a unique steady-state equilibrium if the following assumptions hold: Assumption 2. f'(0) > V;(C )/V~(O). This merely ensures that autarchy is inefficient. Assumption 3. At each to ~. [0, L ], either f"(to) < 0, or V'((L - to) < 0, or V['(f(to)) < O. Theorem 1. Under Assumptions 1-3, there exists a unique steady-state equilibrium ( ~, if,, ~ ) given by f ' ( ~ ) = V~(L- ~)/V~(f(~)) , ff = M / ~ ,
fi= ff,/f'(~) .
Proof. It is immediate that any steady-state equilibrium must be given by the stated equations. Let or(to)=V~(L-e)/v(,(f(e)). Then f'(0)>or(0) from Assumption 2. From Assumption 1, f ' ( 1 ) < or(l) and there must exist at least one steady-state equilibrium. From concavity, f'(~¢) is non-increasing on [0, L] and or(t°) is non-decreasing on [0, L]; from Assumption 3, at each to E[0, L] either f'(to) is strictly decreasing, or or(to) is strictly increasing, and uniqueness is guaranteed. E! To study two-cycle equilibria, notice first that from (7), ~p(0)= 0 and ,p'(e)IV;(L -- ,p(e)) - q,(e). V~(L -
~(e))l
= f ' ( e ) , v ~ ( f ( e ) ) + e . v ~ ( f ( e ) ) , f"(e) + e [ f ' ( e ) ) l ' , v','(f(e)).
(~o)
Hence, ~p'(0)=f'(0). V~(O)/V[(~e)> 1, from Assumption 2. Writing ¢2(e) for ¢(¢~,g)), ~ 2 ( 0 ) = 0 and (d~p2/dto)(0)=[q~'(0)]2>l. Also, , p ' ( ? ) = ? and d,t,*/dto=[,p'(?)] 2. If [g,,(~)]2 > 1, then there must exist e E (0, ?) where ¢2(¢,) = to, which implies the existence of a two-cycle. From (10),
~P'(~) f'V~ + ~. V~ f" + ~(f,)2. V~' =
v;-
h,';
(11) '
with derivatives of f evaluated at ~, those of V2 at f(~), and those of Vt at L - ~7. Since f ' = V~/V~ at equilibrium, it follows that ~p'(~)< 1 so that c p ' ( ~ ) < - 1 is the appropriate sufficient condition for two-cycles. To study this condition we use the following notations. (a) The elasticity of output, 71 = f'(~)~/f(~).
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(b) The elasticity of labour demand, E = f ' ( ~ ) / ~ . f"(~). (c) The Arrow-Pratt relative risk-aversion coefficient of old consumers, r =3"(~).V'2'(f(~))/
V'(f(~)). (d) Consumers interest elasticity of savings"
S = -(*lr + 1)~(fir + il/'"/V~) 1 with VI derivatives evaluated at L - ~. Note. The formula in (d) derives by introducing an interest factor R, on money, so that the consumer problem becomes: choose < E [0, L] to
w, R,+lt',]
F 1I,+, +
Pt+'---T
Differentiating a first-order condition for the problem with respect to R,+ ! produces a formula for the interest elasticity of savings, (R,+~/w,e,).(d(w,~,)/dR,÷~). Use of the equilibrium conditions with R,+! = 1 produces the formula in (d), ,alter some manipulation. Notice that this interest elasticity has the sign of d~/dR,+ 1. So a positive interest elasticity of savings occurs if, and only if, the consumer's offer curve does not 'bend backwards', or equivalently if, and only if, consumer demands satisfy gross substitutes. Introducing the notation of (a)-(d) into (11) and manipulating the inequality ~,'(~) < - 1 produces the following straightforwardly.
Theorem 2. Under Assumptions 1-3, there exists a two-cycle if --tv/r~ 2 + -~ + ~ < - 2 .
(12)
Existing results (Grandmont, 1985; Azariadis and Guesnerie, 1986) suggest that a sufficiently negative S will make (12) true. Indeed, this is the case. If S < 0 , then from (d), 1 + r r / < 0 (since r < 0, 7/> 0) and (12) becomes S E
> - ( 1 + 2S)( 1 + r/r)
(13)
This is necessarily true if S < - ½ even in the linear f case, where 1/e = 0, and r/= 1, exactly as in Azariadis and Guesnerie (1986). Moreover, (13) is also true for 0 > S > - ~ provided e is sufficiently small, i.e. if labour demand is sufficiently inelastic. Moreover, inelastic labour demand can now generate endogenous cycles with a positive savings elasticity. In this case, if S > 0, then 1 + rr/> 0 and (12) becomes S (1 + v/r)(l + 2S) > - E ,
(14)
which again will be true if labour demand is sufficiently inelastic. Notice that the parameter is the only parameter in (14) that involves the second derivative f"(C). This derivative may be increased independently of the other parameters, ensuring that (14) becomes true eventually, i.e. for a sufficiently inelastic labour demand. Hence,
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Corollary to Theorem 2. There exists a two-cycle with positive interest elasticity of savings if labour demand is sufficiently inelastic.
4. Uniqueness and stability of cycles Following Grandmont (1985) we study the time-reversed version of (8), the 'backward perfect foresight (bpf)' dynamics of Eq. (9) earlier. Completely analogously to Grandmont (1985) we may define a plausible learning process for expectation formation (the 'dynamics with learning'), which has the result that k-periodic equilibria are also equilibria of the dynamics with learning; moreover, these equilibria are stable in the dynamics with learning if they are stable in the bpf dynamics (see Huang, 1994, for details of this analogy). This suggests that stability in the bpf dynamics is an appropriate stability concept for the model. In the sequel we speak of convergence and stability to mean convergence and stability in the bpf, and we focus on the multiplicity and stability of cycles. A striking example emerges in the special case where utility is linear, and the production function is quadratic. Assumption 4. (a) V,(L- ~)= L - t,, (b) V2(c) = c, (c) f(t °) = A~e(1 - te/2L), where 1 < A ~<4.
Assumption 4 ensures that Assumptions 1, 2 and 3 are satisfied, Assumption 1 with f'(L) = 0. Under Assumption 4 the bpf dynamics become directly the well-known logistic equation: )=xe,
-L-
•
(15)
The unique steady-state equilibrium is ~= L(1 _ 1 ) .
(16)
The following properties of equilibrium are well-known consequences of the logistic specification. Theorem 3. Under Assumption 4: (i) For all values of h ~ (1, 4) there is at most one locally stable k-periodic equilibrium. (ii) For all h ~ (1, 3) the steady-state equilibrium ~ is the unique locally stable equilibrium. (iii) For all A E(3, 1+ V~) there is a two-periodic equilibrium that is the unique locally stable equilibrium. (iv) For each h E(1 + V~, )~*) where A* =3.57, there is a 2k(^Lperiodic equilibrium that is the unique locally stable equilibrium; as h increases in this range, k(A) increases, covering all positive integers by A* ('period doubling bifurcation').
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(v) For A E ( A*, 4) there are intervals of A-values for which there is a unique locally stable equilibrium that is k-periodic for some k that is not a power of 2; in particular, there is an interval in which there is a three-periodic equilibrium that is the unique locally stable equilibrium. Under Assumption 4, ~ = 1 / ( 1 - A) and so two-cycles emerge when E > - ½ (i.e. A> 3), which is condition (12) in Theorem 2 with r = 0, s = +oo. Theorem 3 shows that in this special case the emerging two-cycle is the unique locally stable equilibrium if E is 'not too much' bigger than - ~ , and that unique locally stable k-periodic equilibria emerge in a particular pattern as ¢ increases from -½ to -½. Finally, a partial generalisation of Theorem 3 is as follows, and shows that the cyclical phenomenon of this paper is robust. Suppose that an economy E is defined by the functions V~, V,, f and by the money stock M. Two economies E I and E 2 are close if g t is close to M" and'if (VI, V~, f~) is close to (V~, V 22, f :) for the C: topology (so that on any compact subset of their domains, the functions and their first two derivatives are close). Define also for an economy E the following 'slack' in inequality (12):
(
A ( E ) = - ~ + ~ r 2+~"
+~+2,
where the right-hand side is evaluated at the steady-state equilibrium of E. An economy E is then 'critical' if A(E)= 0, so that t0'(e)= - 1 at the steady-state equilibrium ~ of E; Theorem 2 says that E has a two-cycle if A(E) < 0; also the steady-state equilibrium of E is locally stable if A(E)> 0. Our differentiability assumption allows standard bifurcation results to be applied (see, for example, the exposition of Azariadis, 1993, Theorem 8.4) to give the stability result: Theorem 4. Under Assumptions 1, 2 and 3, economies that are sufficiently close to a critical economy possess a locally stable two-periodic equilibrium if A(E) < O.
$. Conclusions Endogenous competitive business cycles with outside money are known to exist if there is a negative interest elasticity of savings, associated with backward-bending offer curves and violation of gross substitutability. The relative implausibility of these conditions is often seen as a drawback in the literature. We have shown here now this drawback may be overcome. If the young supply labour and the old supply entrepreneurship, endogenous competitive cycles with positive interest elasticity of savings emerge if labour demand is sufficiently inelastic.
Acknowledgements This paper arises out of Dazhan Huang's PhD thesis (Huang, 1994). We are very grateful to Chris Birchenall and Laurence Copeland for extensive helpful comments. Thanks also to
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Costas Azariadis, Roger Farmer and participants in the Manchester macro workshop for comments on an earlier draft. All shortcomings remain the authors' responsibility.
References Aziariadis, C., 1993, Intertemporal macroeconomics (Basil Blackwell, Oxford). Azariadis, C. and Guesnerie, R., 1986, Sunspots and cycles, Review of Economic Studies 53, 725-736. Benhabib, J. and Laroque, G., 1988, On competitive cycles in production economies, Journal of Economic Theory 45, 145-170. Farmer, R., 1986, Deficits and cycles, Journal of Economic Theory 40, 77-78. Orandmont, J-M., 1985, On endogenous competitive business cycles, Econometrica 53, 995-1046. Huang, D., 1994, Essays on cycles and chaos in economic dynamics, PhD thesis, University of Manchester. Reichlin, P., 1986, Equilibrium cycles in an overlapping generations model with production, Journal of Economic Theory 40, 89-102.