- Email: [email protected]
Delay equivalence in capital accumulation models
Journal of Mathematical Economics 46 (2010) 1243–1246
Contents lists available at ScienceDirect
Journal of Mathematical Economics journal homepage: www.elsevier.com/locate/jmateco
Short communication
Delay equivalence in capital accumulation models夽 Jonathan P. Caulkins a , Richard F. Hartl b,∗ , Peter M. Kort c,d a
Carnegie Mellon University, Qatar Campus, Heinz College’s School of Public Policy & Management, School of Information Systems & Management, 5000 Forbes Ave, Pittsburgh, PA 15213, USA b University of Vienna, School of Business, Economics, and Statistics, Bruennerstrasse 72, A-1210 Vienna, Austria c Tilburg University, Department of Econometrics & Operations Research and CentER, P.O. Box 90153, NL-5000 LE Tilburg, The Netherlands d Department of Economics, University of Antwerp, Prinsstraat 13, 2000 Antwerp 1, Belgium
a r t i c l e
i n f o
Article history: Received 25 March 2010 Received in revised form 8 July 2010 Accepted 30 August 2010 Available online 16 September 2010 JEL classification: C61 D92
a b s t r a c t We study delays in capital accumulation models. Our contribution is threefold. First, we identify a class of models that can be transformed into standard optimal control models without delay. Second, in the single state versions of these models the state trajectory is monotonic in the optimal solution. This is noteworthy given the common belief that adding delays facilitates oscillatory behavior of capital, output and investment. Third, we identify an equivalence result between time-to-install/deliver problems and time-to-build problems. © 2010 Elsevier B.V. All rights reserved.
Keywords: Capital accumulation Delayed response Time-to-build Time-to-install/deliver Optimal control
This paper studies capital accumulation models with delays. Capital accumulation models have been investigated extensively, but until recently usually without delays. That is striking inasmuch as there usually is a delay between the decision to launch a capital project and when that investment first bears fruit, whether the investment is in physical assets, such as building a factory, knowledge assets, such as inventing a new technology or product, or human capital, such as raising the educational level of one’s workforce. Here we in some sense defend the traditional emphasis on models without delays by showing that an important class of models with delays can be transformed into equivalent optimal control problems without delays. This result both extends the relevance of past work to some problems with delay and provides a strategy for analyzing certain types of models with delay. The equivalence holds regardless of the dimension of the state space. In the special case of one-dimensional models, the existence of an equivalent problem without delays implies that the optimal solution to the model with delays cannot involve oscillation. This is in agreement with, e.g., Benhabib and Rustichini (1991), who arrive at non-oscillatory behavior under exponential depreciation, which is also assumed by us. However, Benhabib and Rustichini (1991) also show that, as soon as we depart from the usually assumed exponential depreciation, then the optimal solution becomes oscillatory under
夽 The authors like to thank an associate editor, two anonymous reviewers, Mauro Bambi, Raouf Boucekkine, Gustav Feichtinger, and Omar Licandro for their helpful comments. ∗ Corresponding author. Tel.: +43 1 4277 38091; fax: +43 1 4277 38094. E-mail address: [email protected] (R.F. Hartl). 0304-4068/$ – see front matter © 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.jmateco.2010.08.021
1244
J.P. Caulkins et al. / Journal of Mathematical Economics 46 (2010) 1243–1246
some quite standard assumptions on the utility and production function. Differences with Benhabib and Rustichini (1991) are that, where Benhabib and Rustichini (1991) focus on the nation-state (general equilibrium), we consider behavior at the level of the firm (partial-equilibrium). Second, where they concentrate on different depreciation profiles, we consider specific delay problems as “time-to-deliver” and “time-to-build”. The two just mentioned delay problems are considered in the next two sections. We show that both can be transformed into a standard optimal control problem, so in the single state variant optimal behavior is monotonic. Then, we show that, provided the length of the delays are equal and initial conditions match, the firm’s optimal investment behavior in both delay situations are equivalent. Finally, we generalize our findings beyond the framework of capital accumulation. 1. Time-to-install/deliver Suppose there is a fixed delay before capital goods paid for today can be delivered or installed. This problem is modeled as a capital accumulation model with control delay, in which Ic denotes investment. (The subscript c refers to a model with control delay, as opposed to a model with a state delay below that has a subscript s.) If Kc is the capital stock, R(Kc ) the revenue from producing with capital stock Kc , and C(Ic ) the investment cost:
∞
Jc =
e−rt (R(Kc (t)) − C(Ic (t)))dt → max,
(1)
0
K˙ c (t) = Ic (t − ) − ıKc (t),
(2)
with r being the discount rate and ı being the depreciation rate. Throughout the paper R(Ki ) is continuously differentiable in Ki , and C(Ii ) is continuous in Ii (cf. Hartl (1987)), where i ∈ c, s. The initial conditions are Kc (0) = Kc0 , Ic (t) = I¯c (t)
(3) for − ≤ t ≤ 0.
(4)
Employing the state-time transformation: x(t) = Kc (t + ),
(5)
the problem can be rewritten as
Jc
∞
=
e−rt (R(x(t − )) − C(Ic (t)))dt
0
∞
=
−
˙ x(t)
∞
e−r(t+) R(x(t))dt −
e−rt C(Ic (t))dt → max,
0
= Ic (t) − ıx(t),
with initial condition x(−) = Kc0 . On the interval [ − , 0] the control problem is trivial, because the control is given exogenously by Ic (t) = I¯c (t), and the state ˙ follows from x(t) = I¯c (t) − ıx(t). Hence, given the initial conditions, x(t) with t ∈(− , 0] is determined exogenously. The (exogenously given) profit in this first interval is
0
e−r(t+) R(x(t))dt =
c0 = −
e−rt R(x(t − ))dt.
(6)
0
Now the final form of the optimization problem becomes
Jc − c0 =
∞
e−rt (e−r R(x(t)) − C(Ic (t)))dt → max,
(7)
0
˙ x(t) = Ic (t) − ıx(t),
(8)
x(0) = x0 .
(9)
Summing up, the problem with delay in the control can be transformed into a standard (non-delayed) optimal control problem. The special feature is that the revenue R(.) is multiplied by the constant e−r implying that the effect of the delay on the firm performance is negative. The transformation can be carried out in the case of several capital goods, i.e. where Kc and Ic are vectors. However, when the problem has a single state, we have the following result:
J.P. Caulkins et al. / Journal of Mathematical Economics 46 (2010) 1243–1246
1245
Proposition 1. Consider optimal control problem (1)–(4) with a one-dimensional the state space. If the optimal solution is unique, its state trajectory is monotonic. Proof.
See Hartl (1987).
2. Time-to-build A related problem is known as time-to-build (see, e.g., Bambi et al. (2009)). In it, capital equipment is both paid for and acquired at time t, but it does not become productive until time t + , even though it starts to depreciate as soon as it is acquired, so
∞
Js =
e−rt (R(Ks (t − )) − C(Is (t)))dt → max,
(10)
0
K˙ s (t) = Is (t) − ıKs (t),
(11)
Ks (t) = K¯ s (t)
(12)
for
t ∈ [−, 0],
The difference is whether depreciation starts when the capital is delivered (as in the time-to-install/deliver formulation) or from the moment it is ordered (as in the time-to-build formulation). The former makes sense when depreciation is caused by use, as with most physical production assets. The latter makes sense when depreciation is driven by technological obsolescence, as with some software products. Since Ks (t) cannot be influenced for t < 0, the term
s0 =
0
e−rt R(Ks (t − ))dt =
e−r(t+) R(K¯ s (t))dt
(13)
−
0
is a given constant. Hence the objective (10) can be reformulated as
Js
∞
=
e−r(t+) R(Ks (t))dt −
−
= s0 +
∞
∞
e−rt C(Is (t))dt
0
e−rt (e−r R(Ks (t)) − C(Is (t)))dt,
0
yielding a standard optimal control problem without delay:
∞
Js − s0 =
e−rt (e−r R(Ks (t)) − C(Is (t)))dt → max,
(14)
0
K˙ s (t) = Is (t) − ıKs (t),
(15)
Ks (0) = K¯ s (0),
(16)
to which the monotonicity property of Proposition 1 holds. A crucial feature is that the state equation does not contain a delayed term Ks (t − ). This implies that the model studied in, e.g., Bambi et al. (2009), is not covered by this formulation, which is why its solution admits oscillatory behavior. Provided that the initial conditions are compatible, the two general problems from this section and the previous one, are equivalent. Theorem 2. Consider the control delay problem (1)–(4) with optimal solution (Ic , Kc ) and the state delay problem (10)–(12) with optimal solution (Is , Ks ). Furthermore let the initial conditions be such that Kc () = Ks (0). The two problems are equivalent in the sense that the optimal solutions (if they are unique) satisfy: Kc (t + ) = Ks (t) for t ≥ 0, Ic (t) = Is (t) for t ≥ 0. The optimal objective function values Jc and Js satisfy Jc − c0 = Js − s0 , where the exogenous initial profits c0 and s0 are given by (6) and (13), respectively Proof. Follows directly, since, from time t = 0 onwards, the problems after transformation are identical and just the revenues and costs c0 and s0 incurred on an initial interval of length can be different.
1246
J.P. Caulkins et al. / Journal of Mathematical Economics 46 (2010) 1243–1246
Remark 3. It can be shown that the transformations and the delay equivalence still hold in a finite time framework without salvage value. However, adding a salvage value complicates matters considerably and the delay equivalence property is lost; see Hartl and Kort (2010). 3. Beyond capital accumulation We abstract from the capital accumulation problem to describe what is needed in the formulation of the delay problem, such that it can be transformed into a standard optimal control problem without delay. To do so we introduce a more general notation: x(t) stands for the vector of state variables, while u(t) is the vector of the control variables. In general, a delay problem that one is able to transform into a standard problem without delays, must have the following features: • The objective function may contain any number of additively separable terms of the form fi (x(t − i ), u(t − i )) as long as all state and control variables in a given term are lagged by the same amount. • The state and control can appear in any arbitrary nonlinear combination in the objective function as long as their relative lag matches that in the state equation. That is, one can have a nonlinear interaction in the objective function of x(t − ı) and ˙ u(t − ) so long as also the state dynamics are x(t) = f (x(t), u(t − ı − )). • The state equation cannot contain a lag in the state variable. References Bambi, M., Fabbri, G., Gozzi, F., 2009. Optimal policy and consumption smoothing effects in the time-to-build AK model. Working Paper. Benhabib, J., Rustichini, A., 1991. Vintage capital, investment, and growth. Journal of Economic Theory 55, 323–339. Hartl, R.F., 1987. A simple proof of the monotonicity of the state trajectories in autonomous control problems. Journal of Economic Theory 41, 211–215. Hartl, R.F., Kort, P.M., 2010. Delay in Finite Time Capital Accumulation. Central European Journal of Operations Research, forthcoming; DOI:10.1007/s10100010-0170-7.
Contents lists available at ScienceDirect
Journal of Mathematical Economics journal homepage: www.elsevier.com/locate/jmateco
Short communication
Delay equivalence in capital accumulation models夽 Jonathan P. Caulkins a , Richard F. Hartl b,∗ , Peter M. Kort c,d a
Carnegie Mellon University, Qatar Campus, Heinz College’s School of Public Policy & Management, School of Information Systems & Management, 5000 Forbes Ave, Pittsburgh, PA 15213, USA b University of Vienna, School of Business, Economics, and Statistics, Bruennerstrasse 72, A-1210 Vienna, Austria c Tilburg University, Department of Econometrics & Operations Research and CentER, P.O. Box 90153, NL-5000 LE Tilburg, The Netherlands d Department of Economics, University of Antwerp, Prinsstraat 13, 2000 Antwerp 1, Belgium
a r t i c l e
i n f o
Article history: Received 25 March 2010 Received in revised form 8 July 2010 Accepted 30 August 2010 Available online 16 September 2010 JEL classification: C61 D92
a b s t r a c t We study delays in capital accumulation models. Our contribution is threefold. First, we identify a class of models that can be transformed into standard optimal control models without delay. Second, in the single state versions of these models the state trajectory is monotonic in the optimal solution. This is noteworthy given the common belief that adding delays facilitates oscillatory behavior of capital, output and investment. Third, we identify an equivalence result between time-to-install/deliver problems and time-to-build problems. © 2010 Elsevier B.V. All rights reserved.
Keywords: Capital accumulation Delayed response Time-to-build Time-to-install/deliver Optimal control
This paper studies capital accumulation models with delays. Capital accumulation models have been investigated extensively, but until recently usually without delays. That is striking inasmuch as there usually is a delay between the decision to launch a capital project and when that investment first bears fruit, whether the investment is in physical assets, such as building a factory, knowledge assets, such as inventing a new technology or product, or human capital, such as raising the educational level of one’s workforce. Here we in some sense defend the traditional emphasis on models without delays by showing that an important class of models with delays can be transformed into equivalent optimal control problems without delays. This result both extends the relevance of past work to some problems with delay and provides a strategy for analyzing certain types of models with delay. The equivalence holds regardless of the dimension of the state space. In the special case of one-dimensional models, the existence of an equivalent problem without delays implies that the optimal solution to the model with delays cannot involve oscillation. This is in agreement with, e.g., Benhabib and Rustichini (1991), who arrive at non-oscillatory behavior under exponential depreciation, which is also assumed by us. However, Benhabib and Rustichini (1991) also show that, as soon as we depart from the usually assumed exponential depreciation, then the optimal solution becomes oscillatory under
夽 The authors like to thank an associate editor, two anonymous reviewers, Mauro Bambi, Raouf Boucekkine, Gustav Feichtinger, and Omar Licandro for their helpful comments. ∗ Corresponding author. Tel.: +43 1 4277 38091; fax: +43 1 4277 38094. E-mail address: [email protected] (R.F. Hartl). 0304-4068/$ – see front matter © 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.jmateco.2010.08.021
1244
J.P. Caulkins et al. / Journal of Mathematical Economics 46 (2010) 1243–1246
some quite standard assumptions on the utility and production function. Differences with Benhabib and Rustichini (1991) are that, where Benhabib and Rustichini (1991) focus on the nation-state (general equilibrium), we consider behavior at the level of the firm (partial-equilibrium). Second, where they concentrate on different depreciation profiles, we consider specific delay problems as “time-to-deliver” and “time-to-build”. The two just mentioned delay problems are considered in the next two sections. We show that both can be transformed into a standard optimal control problem, so in the single state variant optimal behavior is monotonic. Then, we show that, provided the length of the delays are equal and initial conditions match, the firm’s optimal investment behavior in both delay situations are equivalent. Finally, we generalize our findings beyond the framework of capital accumulation. 1. Time-to-install/deliver Suppose there is a fixed delay before capital goods paid for today can be delivered or installed. This problem is modeled as a capital accumulation model with control delay, in which Ic denotes investment. (The subscript c refers to a model with control delay, as opposed to a model with a state delay below that has a subscript s.) If Kc is the capital stock, R(Kc ) the revenue from producing with capital stock Kc , and C(Ic ) the investment cost:
∞
Jc =
e−rt (R(Kc (t)) − C(Ic (t)))dt → max,
(1)
0
K˙ c (t) = Ic (t − ) − ıKc (t),
(2)
with r being the discount rate and ı being the depreciation rate. Throughout the paper R(Ki ) is continuously differentiable in Ki , and C(Ii ) is continuous in Ii (cf. Hartl (1987)), where i ∈ c, s. The initial conditions are Kc (0) = Kc0 , Ic (t) = I¯c (t)
(3) for − ≤ t ≤ 0.
(4)
Employing the state-time transformation: x(t) = Kc (t + ),
(5)
the problem can be rewritten as
Jc
∞
=
e−rt (R(x(t − )) − C(Ic (t)))dt
0
∞
=
−
˙ x(t)
∞
e−r(t+) R(x(t))dt −
e−rt C(Ic (t))dt → max,
0
= Ic (t) − ıx(t),
with initial condition x(−) = Kc0 . On the interval [ − , 0] the control problem is trivial, because the control is given exogenously by Ic (t) = I¯c (t), and the state ˙ follows from x(t) = I¯c (t) − ıx(t). Hence, given the initial conditions, x(t) with t ∈(− , 0] is determined exogenously. The (exogenously given) profit in this first interval is
0
e−r(t+) R(x(t))dt =
c0 = −
e−rt R(x(t − ))dt.
(6)
0
Now the final form of the optimization problem becomes
Jc − c0 =
∞
e−rt (e−r R(x(t)) − C(Ic (t)))dt → max,
(7)
0
˙ x(t) = Ic (t) − ıx(t),
(8)
x(0) = x0 .
(9)
Summing up, the problem with delay in the control can be transformed into a standard (non-delayed) optimal control problem. The special feature is that the revenue R(.) is multiplied by the constant e−r implying that the effect of the delay on the firm performance is negative. The transformation can be carried out in the case of several capital goods, i.e. where Kc and Ic are vectors. However, when the problem has a single state, we have the following result:
J.P. Caulkins et al. / Journal of Mathematical Economics 46 (2010) 1243–1246
1245
Proposition 1. Consider optimal control problem (1)–(4) with a one-dimensional the state space. If the optimal solution is unique, its state trajectory is monotonic. Proof.
See Hartl (1987).
2. Time-to-build A related problem is known as time-to-build (see, e.g., Bambi et al. (2009)). In it, capital equipment is both paid for and acquired at time t, but it does not become productive until time t + , even though it starts to depreciate as soon as it is acquired, so
∞
Js =
e−rt (R(Ks (t − )) − C(Is (t)))dt → max,
(10)
0
K˙ s (t) = Is (t) − ıKs (t),
(11)
Ks (t) = K¯ s (t)
(12)
for
t ∈ [−, 0],
The difference is whether depreciation starts when the capital is delivered (as in the time-to-install/deliver formulation) or from the moment it is ordered (as in the time-to-build formulation). The former makes sense when depreciation is caused by use, as with most physical production assets. The latter makes sense when depreciation is driven by technological obsolescence, as with some software products. Since Ks (t) cannot be influenced for t < 0, the term
s0 =
0
e−rt R(Ks (t − ))dt =
e−r(t+) R(K¯ s (t))dt
(13)
−
0
is a given constant. Hence the objective (10) can be reformulated as
Js
∞
=
e−r(t+) R(Ks (t))dt −
−
= s0 +
∞
∞
e−rt C(Is (t))dt
0
e−rt (e−r R(Ks (t)) − C(Is (t)))dt,
0
yielding a standard optimal control problem without delay:
∞
Js − s0 =
e−rt (e−r R(Ks (t)) − C(Is (t)))dt → max,
(14)
0
K˙ s (t) = Is (t) − ıKs (t),
(15)
Ks (0) = K¯ s (0),
(16)
to which the monotonicity property of Proposition 1 holds. A crucial feature is that the state equation does not contain a delayed term Ks (t − ). This implies that the model studied in, e.g., Bambi et al. (2009), is not covered by this formulation, which is why its solution admits oscillatory behavior. Provided that the initial conditions are compatible, the two general problems from this section and the previous one, are equivalent. Theorem 2. Consider the control delay problem (1)–(4) with optimal solution (Ic , Kc ) and the state delay problem (10)–(12) with optimal solution (Is , Ks ). Furthermore let the initial conditions be such that Kc () = Ks (0). The two problems are equivalent in the sense that the optimal solutions (if they are unique) satisfy: Kc (t + ) = Ks (t) for t ≥ 0, Ic (t) = Is (t) for t ≥ 0. The optimal objective function values Jc and Js satisfy Jc − c0 = Js − s0 , where the exogenous initial profits c0 and s0 are given by (6) and (13), respectively Proof. Follows directly, since, from time t = 0 onwards, the problems after transformation are identical and just the revenues and costs c0 and s0 incurred on an initial interval of length can be different.
1246
J.P. Caulkins et al. / Journal of Mathematical Economics 46 (2010) 1243–1246
Remark 3. It can be shown that the transformations and the delay equivalence still hold in a finite time framework without salvage value. However, adding a salvage value complicates matters considerably and the delay equivalence property is lost; see Hartl and Kort (2010). 3. Beyond capital accumulation We abstract from the capital accumulation problem to describe what is needed in the formulation of the delay problem, such that it can be transformed into a standard optimal control problem without delay. To do so we introduce a more general notation: x(t) stands for the vector of state variables, while u(t) is the vector of the control variables. In general, a delay problem that one is able to transform into a standard problem without delays, must have the following features: • The objective function may contain any number of additively separable terms of the form fi (x(t − i ), u(t − i )) as long as all state and control variables in a given term are lagged by the same amount. • The state and control can appear in any arbitrary nonlinear combination in the objective function as long as their relative lag matches that in the state equation. That is, one can have a nonlinear interaction in the objective function of x(t − ı) and ˙ u(t − ) so long as also the state dynamics are x(t) = f (x(t), u(t − ı − )). • The state equation cannot contain a lag in the state variable. References Bambi, M., Fabbri, G., Gozzi, F., 2009. Optimal policy and consumption smoothing effects in the time-to-build AK model. Working Paper. Benhabib, J., Rustichini, A., 1991. Vintage capital, investment, and growth. Journal of Economic Theory 55, 323–339. Hartl, R.F., 1987. A simple proof of the monotonicity of the state trajectories in autonomous control problems. Journal of Economic Theory 41, 211–215. Hartl, R.F., Kort, P.M., 2010. Delay in Finite Time Capital Accumulation. Central European Journal of Operations Research, forthcoming; DOI:10.1007/s10100010-0170-7.