Tradable permits and production-inventory strategies

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Dec 23, 2006 - Seierstad, A., Sydsaeter, K., 1987. Optimal Control Theory with. Economic Applications. North-Holland, Amsterdam. Wagner, H.M., Whitin, T.M., ...
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Int. J. Production Economics 108 (2007) 329–333 www.elsevier.com/locate/ijpe

Tradable permits and production-inventory strategies of the firm Imre Dobos Institute of Business Economics, Corvinus University Budapest, H-1053 Budapest, Veres Pa´lne´ u. 36, Hungary Available online 23 December 2006

Abstract The paper deals with the effect of introduction of tradable permits on the production-inventory strategy of a firm. The basic model of Arrow and Karlin [1958. Production over time with increasing marginal costs. In: Arrow, K.J., Karlin, S., Scarf, H. (Eds.), Studies in the Mathematical Theory of Inventory and Production. Stanford University Press, Stanford, pp. 61–69] assumes that the firm minimizes the sum of linear holding and convex production costs. These costs depend explicitly on time. After introducing emission trading, the cost function will contain linear emission procurement/selling cost. We compare two optimal production-inventory strategies of a firm to examine changes of optimal strategies. The first strategy is the optimal solution of the Arrow–Karlin model. Then we analyze this model extended with costs of tradable permits. r 2007 Elsevier B.V. All rights reserved. Keywords: Emission Trading; Tradable Permits; Environmental Licenses; Optimal Control; Production; Inventory; Environmental Management

1. Introduction The aim of the paper is to analyze the effects of tradable permits (emission trading or environmental license) on production and inventory. Tradable permits are a combined instrument of environmental regulation. (Field, 1997) The government can reduce the emission of firms with the help of pollution (emission) taxes and/or environmental standards. These two instruments of environmental regulation are not market conform in the sense that it is the government that influences the functioning of the firm and not the market. In the case of tradable permits the firm can purchase or sell its pollution level on the market of pollution rights. The price of emission (pollution) is determined on the emission market, so it is market conform. On E-mail address: [email protected].

the one hand, the purchased/sold pollution level acts for the firm first as an environmental standard (upper bound) (Dobos, 1998), on other the hand as a pollution tax (extra costs or revenue) (Dobos, 1999). We will investigate the effect of the introduction of emission trading for a production firm. There are two basic models to investigate the production-inventory decision processes of a firm. Both models include a stock-flow identity and a cost function. The difference lies in the type of the cost functions. The well-known lot sizing model of Wagner and Whitin (1958) examines a concave cost function. The second basic model has a convex cost function. The basis of this investigation is the wellknown Arrow–Karlin-type dynamic productioninventory model. (Arrow and Karlin, 1958) This model belongs to the second category: the inventory holding cost is linear, while the production cost is a non-decreasing and convex function of the

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I. Dobos / Int. J. Production Economics 108 (2007) 329–333

production level. Recent empirical analysis shows that the convexity of the cost model can be experienced in empirical cases. (Ghali, 2003) This is the cause why we have decided to examine the convex cost model of Arrow and Karlin (1958). The firm is able to purchase additional rights or to sell from its rights after the introduction of emission trading. This means that the basic Arrow–Karlin model will be expanded with a cost tag of the purchased/sold emission rights. It is assumed that the emission permits are completely divisible and the costs of the purchased/sold licenses are linear, but the market price of the emission rights is dependent on time in the planning horizon. The instantaneous pollution is a non-decreasing and convex function of the production level. The goal of the decision maker is to minimize the relevant costs. The following question is asked: how much pollution rights to procure or to sell and what are the effects of these rights on the production-inventory strategy. In this paper we continue the examination started in a former paper. (Dobos, 2005) In this paper, we generalize the above-mentioned paper in two ways. First we assume that the price of unit tradable permit is time dependent. After introduction of new products or services the prices can change very fast. A second extension is that we calculate the instantaneous costs of procured environmental licenses. It is a consequence of changing prices. The paper is organized as follows. In the second section the variables and parameters will be defined and the model will be shown. The optimal solution is presented in Section 3. In the next section we illustrate the results of the paper with a numerical example. In the last section we will summarize the results of the paper. 2. Model with tradable permits The following notations will be used: Decision variables: I(t) P(t) u(t)

inventory status in point t, non-negative, production rate at time t, non-negative, the change of the production level.

The control variable of the models is the rate u(t) and the state variables are the inventory level I(t) and the production rate P(t). The material and cost flow of the model are depicted in Fig. 1.

I(t) P(t)

h I(t)

Market S(t)

Production F(P(t))

g(P(t))

Environment

d(t)⋅ g(P(t))u(t) Fig. 1. Material and cost flow in the model.

Parameters: T I0

length of the planning horizon, positive, the initial inventory level at the beginning of the planning horizon, non-negative, S(t) the known deterministic demand rate at time t, non-negative, h the linear inventory holding costs, positive, F(P(t)) the costs of production at time t, nondecreasing, strictly convex function, nonnegative, d(t) the costs of one unit tradable permit, nondecreasing in time, positive, g(P(t)) the rate of emission at time t, non-decreasing, strictly convex function, non-negative.

The Arrow–Karlin model extended with the costs of environmental licenses has the following form:

ðPÞ

IðtÞX0; PðtÞX0; uðtÞ 2 > > > > < ð1 þ 0:05tÞð1 þ sin ðtÞÞ; 0 P ðtÞ ¼ 0:125t þ 0:689; > > > ð1 þ 0:05tÞð1 þ sin ðtÞÞ; > > > : 0:125t þ 0:049;

2.5 2

0pto2:5462406; 2:5462406pto3:56993; 3:56993pto9:0007043;

1.5

9:0007043pto10:7429; 10:7429ptp12;

0.5

I1(t)

1 I0(t)

0

and

-0.5

8  R t > 1:0 þ 0 P0 ðtÞ  SðtÞ dt; > > > > > > < R0;  0  t I 0 ðtÞ ¼ 3:56993 P ðtÞ  SðtÞ dt; > > > 0; > >  0  R > > : t 10:7429 P ðtÞ  SðtÞ dt;

0

0pto2:5462406; 2:5462406pto3:56993; 3:56993pto9:0007043; 9:0007043pto10:7429; 10:7429ptp12:

The optimal production rate and the optimal inventory level are shown in Figs. 2 and 3. The minimal production-inventory costs are 93.649 units. Now let us examine the model with emission trading. The pollution function of the firm is g(P(t)) ¼ 2  P2(t) and the market price of permits is d(t) ¼ sin (t)+1. The modified model has the following form: IðtÞX0; PðtÞX0; uðtÞ 2 > 42 cos ðtÞ ; > > > > ð1 þ 0:05tÞð1 þ sin ðtÞÞ; > < 2t7:347 1 P ðtÞ ¼ 42 cos ðtÞ þ 0:583; > > > ð1 þ 0:05tÞð1 þ sin ðtÞÞ; > > > > 2t21:699 : 42 cos ðtÞ þ 0:016;

0pto3:1436456; 3:1436456pto3:673404; 3:673404pto9:3453493; 9:3453493pto10:849585; 10:849585ptp12;

and 8  R t > 1:0 þ 0 P1 ðtÞ  SðtÞ dt; > > > > > > < R0;  1  t 1 I ðtÞ ¼ 3:673404 P ðtÞ  SðtÞ dt; > > > 0; > >  1  R > > : t P ðtÞ  SðtÞ dt; 10:849585

0pto3:1436456; 3:1436456pto3:673404; 3:673404pto9:3453493; 9:3453493pto10:849585; 10:849585ptp12:

The optimal total (production-inventory and permits) costs are 83.347 units after introduction of emission trading. The production-inventory costs are 99.297 units, which means an increase of 6.03% of production and inventory costs. Fig. 2 shows the smoothing properties of the emission trading on production rate. Fig. 3 demonstrates the increase in inventory level.

3 S(t) 2

P1(t)

2

5. Conclusion

P0(t)

1

0 0

2

4

6

8

10

12

t Fig. 2. The optimal production rates and the demand rate.

In this paper we have investigated the effect of the introduction of an emission trading program on the production-inventory strategy of a firm in an Arrow–Karlin model. We have shown that the firm will sell a part of its pollution rights to minimize the total costs, if the emission permit is divisible. After the sale the firm uses all of the available rights. The

ARTICLE IN PRESS I. Dobos / Int. J. Production Economics 108 (2007) 329–333

production-inventory costs will be higher after the introduction of the process. In this model we have assumed that the firm does not change its technology with the emission trading program, i.e., the costs of production are unchanged. A further research direction would be needed to examine the situation when the firm introduces a new technology with the tradable permits. References Arrow, K.J., Karlin, S., 1958. Production over time with increasing marginal costs. In: Arrow, K.J., Karlin, S., Scarf, H. (Eds.), Studies in the Mathematical Theory of Inventory and Production. Stanford University Press, Stanford, pp. 61–69. Dobos, I., 1991. The Modigliani-Hohn model with capacity and warehousing constraints. International Journal of Production Economics 24, 49–54.

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Dobos, I., 1998. Production-inventory control under environmental constraints. International Journal of Production Economics 56–57, 123–131. Dobos, I., 1999. Production strategies under environmental constraints in an Arrow–Karlin model. International Journal of Production Economics 59, 337–340. Dobos, I., 2005. The effects of emission trading on production and inventories in the Arrow–Karlin model. International Journal of Production Economics 93–94, 301–308. Feichtinger, G., Hartl, R.F., 1986. Optimale Kontrolle o¨konomischer Prozesse: Anwendungen des Maximumprinzips in den Wirtschaftswissenschaften, de Gruyter, Berlin. Field, B.C., 1997. Environmental Economics: An Introduction, second ed. McGraw-Hill, Inc., New York. Ghali, M.A., 2003. Production-planning horizon, production smoothing, and convexity of the cost function. International Journal of Production Economics 81–82, 67–74. Seierstad, A., Sydsaeter, K., 1987. Optimal Control Theory with Economic Applications. North-Holland, Amsterdam. Wagner, H.M., Whitin, T.M., 1958. Dynamic version of the economic lot size model. Management Science 5, 89–96.