An Application of the Levels of Organization in Biology to Process Formation in an Industrial Cluster: The Economies of Sequence

Abstract

Kuchiki and Mizobe focus on the processes of the formation of an industrial cluster in the field of manufacturing. The methodology seeks to apply the concept of the levels of organization to the formation processes of an industrial cluster. A cluster is efficiently formed when the sequence of segment formation is in the optimal order. The economies of sequence are defined as a concept by which the costs of forming an industrial cluster are most efficiently controlled by optimally sequencing the order of segment formation in the cluster. The automobile industry cluster in the Eastern Seaboard region of Thailand and the electronics industry cluster in northern Vietnam are discussed.

Appendices

Appendix A: The Definition of the Economies of Sequence

This section illustrates and defines the concept of the economies of sequence as corresponding to the concepts of the economies of scale and the economies of scope as defined in economics. The economies of sequence can be defined as the concept by which the costs of forming the segments of an industrial cluster are reduced when the sequence of segment formation is optimized, while the costs of forming an industrial cluster are uneconomical if the sequence is incorrectly ordered. The sequence of segment formation is crucial to the normal birth of an organism. An organism is formed from the head to the heart and legs in a specific order, with the sequence of the Hox genes determining the sequence of segment formation. We can illustrate the importance of the economies of sequence as follows.

Case 1:

Model construction: A plastic model of an airplane needs to be assembled according to the sequence provided in the instructions.

Case 2:

Painting: The role of the undercoat is to protect against rust, the role of the middle coat to improve durability and the role of the final coat to enhance the outside appearance. The correct sequence of undercoat, middle coat and final coat must be followed for an effective paint job.

Case 3:

Construction of a house: Carpenters build the framework of a house, plasterers put up its walls, painters fix the outside appearance and the electrical work is the final stage.

The above three examples intuitively demonstrate the existence of the economies of sequence.

These can be strictly defined as follows. Suppose that there are three periods: the first, second and third. Let us examine two examples of segment formation sequencing in an industrial cluster. Policy measures form an organ, and there are two candidate sequences of policy measures: A and B. Suppose a cluster consists of three segments {s ₁, s ₂, s ₃}. The difference between A and B lies in the ordering of s ₂ and s ₃. In A, the sequence of segment formation is supposed to be {s ₁➔s ₂➔s ₃}, and we can assume that the sequence of policy measures to form a production function in A is {s ₁➔s ₂➔s ₃}. We can therefore describe A and B as follows:

$$ A=\left\{{s}_1,{s}_2,{s}_3\right\}, $$

$$ B=\left\{{s}_1,{s}_3,{s}_2\right\}. $$

The production functions for sequences A and B can be given as Y _A and Y _B. The production function of sequence A changes from CRS to IRS in sequence A(Y _A ) by implementing the sequence of policy measures. The production function after implementing the successful sequence of policy measures of {s ₁, s ₂, s ₃} is Y _A = f ({s ₁, s ₂, s ₃}). Suppose the production function in sequence A and B in the third period is Y _A and Y _B , respectively, or,

$$ {Y}_A=f(A)=f\left(\left\{{s}_1,{s}_2,{s}_3\right\}\right), $$

$$ {Y}_B=f(B)=f\left(\left\{{s}_1,{s}_3,{s}_2\right\}\right). $$

We can assume without loss of generality that the interest rate in all periods is the same at r. The production costs of Y _B and Y _A are

$$ {C}_B = {\left(1+r\right)}^2{C}_1\left({s}_1\right)+\left(1+r\right){C}_2\left({s}_2\right)+{C}_3\left({s}_3\right), $$

$$ {C}_A = {\left(1+r\right)}^2{C}_1\left({s}_1\right)+\left(1+r\right){C}_2\left({s}_3\right)+{C}_3\left({s}_2\right), $$

where C _i (s _k ) (k = 1, 2, 3; i = 1, 2, 3) is the total costs in implementing the policy measure to form the segment s _k in the period i.

Now we can compare the productivity of A and B in the third period. Suppose there exist economies of sequence between s₂ and s₃. That is, Y _B is very small while Y _A is large. The sequence of segment formation from s ₃ to s ₂ is inefficient in comparison to that from s ₂ to s ₃.

Definition: there exist economies of sequence between s ₂ and s ₃, such as

$$ {Y}_B/{C}_B < {Y}_A/{C}_A. $$

Sequence A is more efficient than sequence B owing to the IRS for sequence A.

Appendix B: The Existence of Economies of Sequence

This appendix considers the case in which the local government imposes a tax for cultural production on individuals. We assume that the production function of culture enhances the level of culture by increasing the tax revenue of the local government. The level of culture gives utility to the individuals. Our model demonstrates the existence of economies of sequence.

Let m, l, x and u denote goods, leisure time, culture elements and the utility function of the representative individual, respectively:

$$ u=u\left(m,l,x\right). $$

(8.1)

The maximization problem is solved under the constraint of τ = 24 hours:

$$ l+a=\tau, $$

(8.2)

where a denotes labor time.

Here, if w, t and T _i denote wage rate, tax rate for an individual i and the total tax for an individual i, then the budget constraint is

$$ wa\ge m+{T}_i. $$

(8.3)

If T _i is tax for the provision of cultural public goods, then

$$ {T}_i\le twa $$

(8.4)

as the total tax is twa. The total tax of a region (T) can therefore be given as

$$ T=n{T}_i, $$

where n is the population.

The social production function of culture is

$$ x=H(T), $$

(8.5)

Where

$$ T=f\left(\left\{{H}_1,{H}_2,{H}_3\right\}\right), $$

$$ 0=f\left(\left\{{H}_1,{H}_3,{H}_2\right\}\right), $$

and

$$ 0=f\left(\left\{{H}_2,{H}_1,{H}_3\right\}\right). $$

Here we suppose that a cluster consists of three segments: H ₁ , H ₂ and H ₃. The segments are formed one by one along the time axis. This means that the production level will be zero if the sequence of organ formation is incorrect.

Proposition: There exist economies of sequence under certain conditions.

Proof:

The Lagrangian function, taking considerations from (1) to (5), is

$$ L= nu\left(l,m,x\right)-\alpha \left\{n\left(1-t\right)wa- nm\right\}-\beta \left(l+a-\tau \right)-\gamma \left({T}_i-twa\right)-\delta \left(x-H\left(n{T}_i\right)\right). $$

The solution is

$$ \frac{\partial u}{\partial l}=w\cdot \frac{\partial u}{\partial m}+\left(n\cdot \frac{\partial \mathrm{H}}{\partial \mathrm{T}}-\frac{\partial u}{\partial m}\right)tw. $$

In the case of t = 0 (i.e. where no tax is imposed on the population),

$$ \frac{\partial u}{\partial l}=w\cdot \frac{\partial u}{\partial m}, $$

(8.6)

where the left side, which is positive, is the marginal utility of leisure and the right side is the marginal utility of goods and services gained by the wages paid for the supply of labor.

The case of t > 0 can be given as

$$ \left(n\cdot \frac{\partial \mathrm{H}}{\partial \mathrm{T}}-\frac{\partial u}{\partial m}\right)tw. $$

The difference between the two cases is

$$ \left(n\cdot \frac{\partial \mathrm{H}}{\partial \mathrm{T}}-\frac{\partial u}{\partial m}\right)tw. $$

Here we assume that $\left(n\cdot \frac{\partial \mathrm{H}}{\partial \mathrm{T}}-\frac{\partial u}{\partial m}\right)>0$.

So

$\left(-\frac{\partial u}{\partial m}\right)tw$, in the case of the incorrect sequence for the formation of organs, or

$$ \left(n\cdot \frac{\partial \mathrm{H}}{\partial \mathrm{T}}=0\right). $$

The sign is minus to explain the negative effect of the economies of sequence in the case where the sequence of organ formation is incorrect.

The left side is the marginal utility of leisure time and the right side is the marginal utility of the goods and services earned by the wages. Both sides are equal and positive.

In the case of the existence of external economic effects, equation (8.6) becomes

$$ \frac{\partial u}{\partial l}=w\cdot \left(1-t\right)\frac{\partial u}{\partial m}+twn\left(\left(\partial u/\partial x\right)\frac{\partial \mathrm{H}}{\partial \mathrm{T}}\right). $$

The left side is the marginal utility of leisure time and the right side is the sum of the first and second terms. The first term is the marginal utility of the goods and services gained by the wages after the reduction in tax. With regard to the second term, the sum of the multiplication of the wage (w) and the tax rate (t) is the tax revenue for the local government from the representative individual, and the sum of the multiplication of wt and the population, or wtn, is the total tax revenue for the local government from the society. The middle parenthesis of the second term$\left(\left(\partial u/\partial x\right)\frac{\partial \mathrm{H}}{\partial \mathrm{T}}\right)$ is the multiplication of the marginal utility of the individual by the external effect of culture and $n\left(\left(\partial u/\partial x\right)\frac{\partial \mathrm{H}}{\partial \mathrm{T}}\right)$is the marginal product of the external effect by the total tax revenue. In sum, the second term is the multiplication of the total tax revenue and the parenthesis—that is, the external effect for the society. This equation becomes

$$ \frac{\partial u}{\partial l}=w\cdot \frac{\partial u}{\partial m}+\left(n\cdot \left(\partial u/\partial Y\right)\frac{\partial \mathrm{H}}{\partial \mathrm{T}}-\frac{\partial u}{\partial m}\right)tw. $$

(8.7)

In the case that the second term of equation (8.7) is positive, we can enhance the level of utilities of n persons by imposing tax on the n persons and providing them with the external effects. Comparing equation (8.7) with equation (8.6), we find the following difference in the second term:

$$ \left(n\cdot \left(\partial u/\partial Y\right)\frac{\partial \mathrm{H}}{\partial \mathrm{T}}-\frac{\partial u}{\partial m}\right)tw>0. $$

We assume that the wage is positive (w > 0) and the tax rate is positive (t > 0). Transforming the above equation gives

$$ \left(twn\cdot \left(\partial u/\partial Y\right)\frac{\partial \mathrm{H}}{\partial \mathrm{T}}>tw\frac{\partial u}{\partial m}\right). $$

(8.8)

This inequality means that the community obtains a higher level of utility in the case of the intervention of the local government than in the case of no intervention if the total external effects are greater than the marginal utility forgone by the tax payment of ntw. The inequality in (8.8) becomes

$$ \left(twn\cdot \left(\partial u/\partial Y\right)\frac{\partial \mathrm{H}}{\partial \mathrm{T}}-tw\frac{\partial u}{\partial m}\right)>0. $$

The inequality shows the deviation of the net total external effect from the market solution without intervention.

Now we will explain the case of H (C _A ) = 0 in equation (8.5). In that case, the economies of sequence do not hold and the sequence is incorrect from the point of efficiency. The marginal utility of the representative individual from the external effect of culture for the population is zero. On the external effect of culture$\left(\left(\partial u/\partial Y\right)\frac{\partial \mathrm{H}}{\partial \mathrm{T}}\right)$, the marginal production of the total tax is zero, $\left(\frac{\partial \mathrm{H}}{\partial \mathrm{T}}=0\right)$. We have a negative effect in this case. This means that the marginal utility $\left(\frac{\partial u}{\partial m}\right)$of the tax on the wages of the representative individual (tw) is negative:

$$ \left(-tw\frac{\partial u}{\partial m}\right)<0. $$

Consequently, we show that the marginal utility is negative if the economies of sequence do not hold and the sequence is incorrect. In summary, the external effect on culture by the local government is positive if the local government imposes a tax on culture, the economies of sequence hold and condition (8.8) holds. However, we can show that the external effect of culture becomes a disexternal effect if the economies of sequence do not hold. This appendix therefore explains the above economies of sequence from the point of view of welfare economics.