Optimal and politically opportune language policies for the vitality of minority languages

Language competition and language policy

Early models generally considered the number of speakers as the endogenous driving force behind language change whereas other variables, like status – or prestige –, are exogenously given. The models of Abrams and Strogatz (2003) without bilinguals and Minett and Wang (2008) with bilinguals inevitably lead to the extinction of one language, which one depends on the initial distribution of speakers on the two languages. By introducing a process of family formation depending on the size of the three groups of speakers (monolinguals in each of the two languages and bilinguals) and preferences for choosing a partner with a similar linguistic profile, Wickström (2005) showed that all kinds of steady-state equilibria are possible (only monolinguals in either language, monolinguals in one or both languages as well as bilinguals, and only bilinguals). Again, at which equilibrium the system finally ends up depends on the initial situation and the exogenous relative status of the two languages. ²

Minett and Wang (2008) incorporate policy in the model in a very straight-forward (albeit non-realistic) manner: by changing the relative status of the two languages, the policy maker can affect the size of the threshold of speakers necessary for the language to survive. Each time a language is on its way to extinction, the policy maker increases its status thereby reversing the process. Fernando et al. (2010) show how a language policy which introduces education in the minority language and status planning, which increases its visibility in the linguistic landscape, can guarantee its survival. These contributions demonstrate that a given policy can preserve a minority language. The next step is to characterize an optimal policy. Given that public budgets are limited, the policy maker has to set priorities.

The cost-effectiveness of policy measures will depend on the characteristics of the group of beneficiaries, especially on the number of beneficiaries and their spatial habitation patterns. For example, mounting street signs in a minority language causes costs that increase with the territory but are independent of the numerical size of the beneficiaries; on the contrary, the costs of answering emails in the language of the clients depend on the number of clients, but not on where they live. The costs of other measures, like publishing law texts in a minority language, are independent of both habitation patterns and numerical size, and providing home health care causes costs dependent on both the size and the domicilation of the clients. The optimal choice of measures in a simple static setting with measures being perfect substitutes as far as their effect is concerned, is explored in Wickström (2021).

The main contributions of the present essay are: (1) the extension of the analysis to policy measures that show different degrees of complimentarity/substitutability in a fully dynamic setting; (2) the optimal relative strength of an instantaneous and a sustained, protracted measure under a given budget; ³ and (3) the implications of a finite time horizon.

Investing in the status or prestige of a minority language is an example of a protracted measure. The higher the status of the language, the more likely its speakers are to pass it on to the next generation, thereby contributing to its vitality. Status-building measures are typically symbolic and directed to the collective of language users, like the naming of public institutions, bilingual names of cities and towns, the use of the language in prestigious publications, or, at important events, by prominent personalities. Such measures take effect peu à peu, but the effect typically lasts for a long time. Other measures, like the requirement that public servants be forced to use the language when interacting with its speakers, would increase the visibility and actual use of a language directly and faster. Teaching the language as a compulsory subject at schools is also a measure with a rather rapid effect. These two more or less instantaneous measures are also examples of measures with a different degree of substitution possibilities with the status-building measures. Compulsory teaching increases the number of people knowing the language, but does not necessarily increase its use. There exists considerable anecdotal evidence that a language must also be attractive to the speakers, that is, have a high status. We would claim that there is a high degree of complementarity between the acquisition planning in the form of compulsory learning and status planning. This is probably not so in the case of civil servants being forced to use the language in communicating with its speakers; that has a certain status effect of its own and the measure is a substitute to pure status planning.

The structure of the essay

The point of departure of this essay is that the vitality of a minority language (defined as its use by many individuals in many domains) is affected by language policy. The more resources that are spent on the policy measures, the higher the vitality of the language. ⁴ A language policy is here defined as a collection of policy measures that are implemented in order to create incentives for the speakers of a certain language to increase its use and to give it on to the next generation. Given that the implementation of a measure is benefiting the speakers of the language, and the aggregate benefits of all speakers are compared to the implementation costs in order to decide if the measure be implemented or not, we talk about a welfare – or cost-benefit – approach to language policy; see Wickström (2017, 2019). Since the estimation and quantification of benefits in a cost–benefit analysis can be very problematic – due both to practical difficulties of measurement and to theoretical inconsistencies and path dependencies – an alternative approach is cost-effectiveness analysis. Here the language planner aims at the fulfillment of objectively observable goals such as the vitality of a language in a given area or in given domains. ⁵ The degree of goal fulfillment then depends on the employment of different policy measures. ⁶ However, the choice of policy measures is constrained by the budget of the policy maker. The expenditure side of the budget is, of course, mainly determined by the implementation costs of the various enacted measures. The planner’s task is then to attempt to reach a high goal fulfillment within the constraints of the budget.

What both cost-benefit and cost-effectiveness analyses have in common is that the cost structures of different measures are very important; see Wickström (2017, 2020); Wickström et al. (2018). The costs of different language-planning measures can be classified over (at least) two dimensions, degree of rivalry and degree of spatiality of the resulting language-related goods. See the discussion above on language competition. Different types of language-policy measures cause costs of different magnitudes per capita of the beneficiaries in dependency on their number and habitation patterns. In other words, some planning measures, like the use of a language in public documents, are almost perfectly collective goods, being both non-rival and non-spatial. Only fixed costs – independent of the number of beneficiaries and their geographical distribution – are to be covered. To the other extreme, some planning measures, like providing social services – for instance home nursing – in a certain language, are almost purely individual goods. Here the costs are more or less proportional to the number of beneficiaries and to the size of the geographical region where the beneficiaries live.

The cost structure has important implications for the choice of policy in a cost-effectiveness analysis. Assume that the planner wants to preserve or increase the vitality of a minority language, and both status planning, like using the language in official documents or in the public space, and acquisition planning, like teaching the language in public schools, serve this purpose. Given that the costs of status planning are basically independent of the number of beneficiaries or their geographical distribution – the status resulting from the planning measure is a non-rival good – whereas the costs of acquisition planning are more or less proportional to both – teaching material and teaching as such are to a large extent rival goods, at least between schools – an implication might be that the effectiveness of status planning in comparison to acquisition planning increases with the size of the community of beneficiaries and with the spread of their living areas. To receive a certain effect in a small, concentrated community by a given budget, it can be sensible to put more effort into acquisition planning in relation to status planning than it is in a larger community spread over a large area.

In this essay, this observation is applied in a dynamic analysis. If the size of the target community increases, this is without consequences for the costs of provision of a non-rival good like status with only fixed costs, whereas, in the case of proportional costs, the costs increase as the number of beneficiaries increases. Given that the influence on an average individual of a given planning measure is independent of the size of the community, the effect of status planning at given costs is independent of the size of the community, whereas, for acquisition planning, the effect decreases with the increasing size of the community since available resources are divided among more individuals. This makes status planning progressively more attractive as vitality increases. However, since both budget and effect are influenced by changes in the measures, it is not obvious at the first glance how budget shares should change, and we discuss both the change in the optimal employment of different policy measures and in the corresponding budget shares as the vitality of a supported language increases. In the analysis, we limit ourselves to two types of planning measures, a perfectly non-rival and a perfectly rival one.

As we mentioned above, with the introduction of time, the possible types of goods are further differentiated with a dimension describing how long the good and its effects last. We can talk of instantaneous goods whose benefits last a very short time, for instance a glass of wine; once it is drunk it is gone. ⁷ On the other hand, there are goods with an extended life period, like the reputation of Château Margaux, which is reinforced every time we enjoy a few drops of the product. We then speak of a protracted good. Protracted goods like status or goodwill act over an extended period. In choosing between instantaneous and protracted goods to produce incentives for increased language use, policy makers might make different choices depending on whether they are myopic or hyperopic. In Templin et al. (2016); Wickström (2013, 2016); Templin (2020), the rôle of protracted status in a dynamic setting is analyzed. Here, we extend it to discussing effects due to both types of policy measures.

In the next section “Claims”, we intuitively discuss some expected implications for an optimal language policy of rival versus non-rival and immediate versus protracted language-related goods. Part of the discussion is made precise in the section “The dynamics of language use” with the help of a simple model presented in Sections “The model” and “The steady-state static optimization problem” and further illustrated in some simulations in Section “Examples”. Most of the formal analysis is found in online appendices.

Family behavior and language choice

We describe the behavior of families with functions α ( F , p , β ) . These functions can be derived from optimization of family utility. This is shown in some detail in Wickström (2005). Here, we give an outline of the argument. We assume that there are three types of individuals with respect to language use: monolinguals in the high-status language, H, monolinguals in the low-status language, M, and bilinguals in both languages, L. That is, we have added the category M to those discussed above. The frequencies of speakers are written as ( p H , p M , p L ) , p H + p M + p L = 1 . A typical family i of type F ϵ { H H , M M , L L , H M , H L , M L } considers costs and benefits when choosing the linguistic repertoire of the children. The benefits are the direct communication possibilities B _C and intrinsic values B _I due to the language as carrier of identity and culture. The latter depends on the social status of the language S which is influenced by government policy. The costs C are learning costs carried by the family which are also determined by government language policy.

The communication benefits depend on the number of individuals with whom the children can communicate. If the child is socialized as an H-individual, it can communicate with the H- and L-individuals, that is a fraction 1 − p M of the individuals in society; a child socialized as an M - individual with a fraction 1 − p H of the population; and an L-socialized child with the total population. Normalizing, we can write 1 = B C ( L ) ≧ B C ( H ) ≧ B C ( M ) ≧ 0 . The intrinsic benefits can be assumed to be, on the average, the larger, the stronger is the language by the parents. The learning costs can also be assumed to depend on the language use in the family as well as government policy.

Each family will rank the three linguistic profiles (H, M, and L) and choose the highest ranked one for their children. The communication benefits are the same for all families; the intrinsic benefits of each of the profiles increase with a stronger presence of the language(s) of the profile in the family, ¹³ and the costs will move in the opposite direction. We can then expect the average ranking of a certain profile to be higher, the stronger are its languages in the family. An increase in the status of the minority language through language policy will increase its intrinsic value in families where it is represented and make the choice of a profile involving the minority language (L or M) more likely; similarly, acquisition planning in favor of the minority language will decrease its learning costs and, hence, make the choice of profile L or M in the family more likely. The smaller is p _M the higher the difference in communication costs between profile M and the other two profiles, making the choice of profile M less likely. We make the assumption that profile M will not be chosen due to its small communication benefits. ¹⁴ This makes the communication benefits of the two other profiles identical and equal to one, and the choice between H and L dependent on the intrinsic benefits that are dependent on the language profiles of the parents and influenced by the status planning, as well as on the learning costs that are influenced by the acquisition planning.

The function α ( F , p , β ) is the central behavioral element of the model and is defined to describe the average fraction of bilingual children of type L emerging from a family of type F if the distribution of individuals is given by p and language policy results in language-related goods β = ( β 1 , β 2 , … ) . Following the discussion above, we can make some assumptions:

Assumption 1

α ( F , p , β ) does not depend on p.

The motivation of parents of providing their children with a certain linguistic repertoire is, as mentioned, at least twofold. On the one hand, the children should be able to communicate with other members of society; ¹⁵ on the other hand, there is an emotional motivation for passing the language on to the children since the language is an important carrier of the parents’ identity. In our setting, we assume that there are no monolinguals in the minority language, and the first motivation is irrelevant for the language decision of the parents. ¹⁶ As a consequence, we will drop the p dependence of α.

Assumption 2

1 > α ( L L , β ) ≥ α ( H L , β ) > α ( H H , β ) = 0 .

If both parents are speakers of the minority language it follows from the discussion above that more children who are bilingual in the minority language emerge from such a family than from a family with only one parent speaking the minority language. 1 − α ( L L , β ) is the rate of attrition of the minority community if there is no influx from outside. α ( H L , β ) correspondingly reflects the rate of influx due to the mixed marriages. In a family where both parents are speakers of the majority language, it is assumed that no children are raised as bilinguals, α ( H H , β ) = 0 . ¹⁷

In addition, to simplify specific calculations, we will at times appeal to three technical assumptions:

Assumption 3

α ( H L , β ) = δ α ( L L , β ) , 1 > δ > 0 .

This simplifies the analysis considerably. We can describe the functional form of each α by the same function which is further specified:

Assumption 4

Let q ( β ) be a concave function with the property q 0 = q 0 with 0 the vector of 0’s. Then α L L , β = A L L q β Q + q β ⇒ A L L ≥ α L L , β ≥ A L L q 0 Q + q 0 = : A L L . Q is here a constant.

α ( L L , β ) ranges between A^LL and A_LL and α ( H L , β ) between δ A L L and δ A L L .

For analytic purposes, we can – as in standard micro-economic theory – describe the function q using equivalence classes, the combination of β s giving the same value of the function. Below we will reduce the set of policy variables to two, β 1 and β 2 . In the ( β 1 − β 2 ) -space we can then call the equivalence classes iso-effect curves. The absolute value of the (negative) slope of a iso-effect curve, h ( β 1 , β 2 ) , can be found from

Due to the concavity of q, h ( β 1 , β 2 ) decreases in β 1 and increases in β 2 . One can make further assumptions on q making the analysis more tractable. One such assumption is:

Assumption 5

The function q is homothetic. That is, the function h is homogeneous of degree zero: h β 1 , β 2 = h 1 , β 2 β 1 ; we will write this as h 0 ( β 2 β 1 ) , a function increasing in its argument.

Steady-State solution

If in a given period the distribution of the two types is (p _H , p _L ) and is designated by ( p H + , p L + ) in the next period, the new fraction of speakers of L can be written as

In the online appendix, the following results are derived:

Proposition 1

If the intermarriage rate between the language communities and the fraction of bilingual children emerging from mixed marriages on the average are sufficiently high, specifically if 2 λ α H L , β > 1 - 1 - λ α L L , β , the size of the minority community displays a positive and stable steady state.

If λ = 1 , the stability condition becomes α ( H L , β ) > 1 / 2 . That is, a stable steady state exists if more than half the children emerging from mixed marriages become bilingual.

The steady-state static optimization problem

The planner wants to maximize the vitality of the minority language, as reflected in the value of p, and is constrained by the budget. Maximization of the steady-state vitality can then formally be written as

It is readily seen that the budget set is not convex. We, therefore, first have to investigate if the problem has a unique solution. In the online appendix, it is shown that this is the case. In addition, the following results are derived in the appendix:

Proposition 2

The maximum steady-state vitality of a minority language, defined as the highest possible proportion of bilingual speakers of the language,

1. increases with the size of the budget for language stimulating policies, both in the fixed part of the budget and in the part proportional to the number of speakers.

The results to a large degree depend on the fact that status expenditures have a non-rival cost structure.

The dynamics of language use

The dynamics of p, given by equation (3), can be transformed into continuous time with a suitable normalization:

That is

The general dynamic problem of the policy maker can be written as

The first expression, 14a, is the maximand. If we divide it by the time horizon T it expresses the discounted average size of p, the vitality. This is what is to be maximized. Since T is an exogenously given constant, maximizing the expression 14a is the same as maximizing the discounted average size of the vitality. The discount rate r can be set equal to zero, as long as the time horizon is finite. The next two expressions, 14b and 14c, describe the dynamics of the vitality p and the status S, respectively. The speeds of adjustment, ω and η , are important for the following discussion, and the fact that 14c is linear in S is crucial for Proposition 5. The budget constraint, expression 14d, must hold at each instant of time. The dependence of the budget on the vitality of the minority language, quantified by the parameter b, plays a certain rôle for the optimal values of the policy measures as the vitality changes, see below. The next two expressions fix the initial values of the dynamic variables, and the last two expressions make sure that the expenditures on the policy measures are non-negative.

This system is the most general formulation of the planner’s problem. Before attacking the general formulation, we consider two special cases of the status variable. We first ignore its protracted character in order to discuss the influence of the type of language-related goods on the optimal planning path. This is equivalent to letting the change in p, described by the parameter ω , be slow in comparison to the change in S, described by the parameter η and letting the time horizon increase in proportion to the rate of change in p. We then consider the case of the measures S and a being perfect complements before using the general formulation to discuss the optimal policy if the status variable is protracted.

Instantaneous status adjustment

In order to see how the two control variables behave before the time horizon makes its influence felt, we let the status adjust very fast, that is we let η → ∞ and we concentrate on the complementarity/substitutability of the two measures.

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Figure 1.

Budget constraints when S adjusts fast, expression 16b. (a) Fixed budget, (b) Proportional budget, (c) General budget.

The optimization problem 14 of the planner now takes the form

max a , e S , p , S ∫ 0 T p e - r t d t

(15a)

s u b j e c t t o p ˙ = p ω k ( p , S , a , e S ; λ )

(15b)

e S = ρ S

(15c)

B 0 + N p b = e S + N p a

(15d)

p ( 0 ) = p 0

(15e)

a ≥ 0

(15f)

e s ≥ 0

(15g)

The equation describing the dynamics of S, 14c, in the general formulation has been replaced by a non-dynamic constraint that has to be satisfied at each instant of time.

In the online appendix, we find the relevant conditions

p N ρ = ∂ q ∂ a ∂ q ∂ S

(16a)

0 = B 0 + N p b - ρ S - N p a

(16b)

Equations (16a) and (16b) are most conveniently analyzed in diagrams. The budget restriction 16b is readily presented. Two polar cases can be distinguished. With a fixed budget (b = 0), the budget set in a ( S − a ) diagram decreases with an increasing value of p. The reason, of course, is that the cost structure of S is non-rival, whereas the cost structure of a is rival. That is, the costs of maintaining a certain level of S are independent of the size of the minority, but the costs of a are proportional to the size of the minority, and with an increasing minority the amount of resources per capita decreases. The reverse holds for a proportional budget ( B 0 = 0 ) , that is, the budget increases with the size of the supported minority. This is illustrated in diagrams 1a and 1b. The general case with both B₀ and b being positive is shown in diagram 1c.

Equation (16a) can also be pictured in the same diagram Under the assumption of homotheticity, it is very straight-forward, see Figure 2. In general, we only know that the implied constraint has a positive slope and that the curve moves upwards as p increases. In the case of a growing p and a homothetic function q it is clear that the ratio S/a will increase, since equation (16a) only depends on that ratio

∂ q ∂ a ∂ q ∂ S = h 0 ( S a )

(17)

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Figure 2.

Constraint imposed by optimality condition 16a. (a) Homothetic q. (b) General q.

For a fixed budget, a will decrease, and for a proportional budget S will increase. See Figure 3. In general, however, we cannot conclude that the ratio S/a increases as p increases, see Figure 3(c). In Figure 3(c), the impact of the status variable strongly diminishes after reaching a certain level, whereas the impact of a continues to be strong.

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Figure 3.

S and a in dependence of p due to the constraints imposed by 16a and 16b. (a) Fixed budget and homothetic q. (b) Proportional budget and homothetic q. (c) Proportional budget and general q.

Limiting ourselves to the homothetic case, and writing the ratio of the budget shares as

Λ : = ρ S p N a

(18)

allows us to write equation (16a) as

h 0 ( S a ) = h 0 ( p N ρ Λ ) = p N ρ

(19)

Using the definition of the elasticity of substitution between S and a

σ ≔ h 0 ( y ) y ∂ h 0 ( y ) ∂ y

(20)

where y is defined as y := S/a, and writing Λ as a function of p, equation (16a) becomes an identity that can be differentiated with respect to p

N ρ = ∂ h 0 ∂ y N Λ ρ + ∂ Λ ∂ p p N ρ 1 p = ∂ h 0 ∂ y h 0 y 1 p + ∂ Λ ∂ p 1 Λ 1 p = 1 σ 1 p + ∂ Λ ∂ p 1 Λ 1 p σ − 1 = ∂ Λ ∂ p 1 Λ

(21)

The sign of ∂Λ/∂p depends on the elasticity of substitution between S and a. If σ = 1 the budget shares stay constant as p changes. If the elasticity is less than one, the variables are complementary, the budget share of S decreases, although S/a increases. For the case that S and a are substitutes, both S/a and the budget share of S increase. This discussion, of course, only applies if we are dealing with inner solutions. For corner solutions, S = 0 or a = 0, the budget shares are constant and equal to zero or one, respectively. We have a new

Proposition 3

The optimal vitalization policy for a minority language under the assumption of a homothetic influence of policy measures on linguistic behavior implies that policy measures leading to rival language-related goods should be partially or fully replaced by measures leading to non-rival language-related goods as the vitality of the language increases. The budget shares for the two types of measures, however, should move in the opposite direction if the language-related goods are complements in their impact on the vitality, that is, if the elasticity of substitution is less than one. If the two types of measures are substitutes, that is, the elasticity of substitution is greater than one, the budget share of the measure leading to a non-rival good should increase with an increase in the vitality. In the case of corner solutions, the budget shares and the ratio of the measures do not change as long as one measure is not used at all.

We gave some intuition for which measures are complements and which measures are substitutes in the discussion above in the introductory section. We argued that compulsory teaching and general status increasing measures probably are complementary. Without making the language attractive to the young people they might not like to use it, or even pay attention in the class room; and without the teaching, a high general status might not lead to much use of the language. Hence, Proposition 3 suggests that in case the language policy consists of general status-building measures and acquisition planning, the budget share for teaching should increase as the policy shows success and that of general status-increasing measures, such as government publications in the language should decrease. These results are independent of the type of budget. However, the analysis in Figures 3(a) and (b) suggests that in case of a fixed budget restriction the costs per minority student should decrease, whereas the total budget for the status-building measures should stay more or less the same. On the other hand, if the budget is proportional to the size of the minority, the total budget for the status-building measures should increase and the budget per minority student stay more or less constant.

If the relevant measures are instead general status building and the right to correspond with the government in one’s language of choice, we would argue that these measures are basically substitutes. That means that as vitality increases, the budget share of general status-building measures should increase and the share allocated to the rival good should decrease, again independently of the type of budget. The influence of the type of budget on the absolute expenditures would be the same as in the example above.

The general formulation

In the case of the two variables not being perfect complements, the general dynamic problem of the policy maker is given by equation (14).

Claim 2.2 in Section “Claims” is readily proven, since the equation of motion of S, equation 14c, is linear and additive; that is, each addition to S from the status-building measure can be treated separately from each other addition and then added to all the others in order to find the total value of S. Take any (small) interval ∆ t beginning at time t^∗ and let the expenditure for the status-building measure in that interval be equal to e _S (t^∗). Then the contribution to S in this interval, is given by ∆ S ( t ∗ ) = η e S ( t ∗ ) ∆ t . This ∆ S(t^∗) will decay exponentially at a rate of ηρ, but always be positive and contribute to the vitality of the language until the time horizon. If e _S (t^∗) is part of an optimal path, then e _S (t^∗) ∆ t will be equal to the value of the (marginal) contribution of the protracted ∆ S(t^∗) to the vitality at all times from t^∗ until the time horizon. If the time horizon is shortened, the contribution of ∆ S(t^∗) after the new time horizon is lost and the costs of the policy on the interval ∆ t are bigger than the benefits. The same argument holds for all intervals up to the new time horizon. That is, the formerly optimal path of e _S is at every time interval up to the new horizon too big. Applying Le Chatelier’s principle we know that on the new optimal path the value of e _S should be reduced on each time interval. This, of course, only applies if the optimal status S and, hence, e _S are positive.

This gives us:

Proposition 5

If the status of a minority language contributes to its vitality, the optimal size of the protracted status-building measure decreases (or remains equal to zero) at each point of time if the time horizon is shortened. In comparison to a situation with an infinite time horizon, a policy consisting of a sequence of optimal planning decisions, each with a finite time horizon, is sub-optimal.

In the online appendix, it is demonstrated that the status-building variable goes to zero as the time horizon approaches:

Proposition 6

If the status of a minority language contributes to the vitality of the language, and the status is a protracted variable (state variable), a planner planning for a high vitality of the language, will let the budget share allocated to the measure increasing the status of the language become zero, as the time horizon approaches.

In other words, at the end of the planning period, the planner implements the rule « après nous le déluge ».

This corresponds to the intuition expressed in the introductory claims.

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Table 1.

Values of the parameters of the model in the examples.

	B 0	b	ε	ζ	q 0	Q	ALL	ρ	ω	λ	δ	r	N	p 0	S 0
Fast change in S	100	0	20	15	200	20	0.75	1	1	1	0.75	0	100	0.05	—
Slow change in S	100	0	20	15	200	20	0.75	1	1	1	0.75	0	100	0.05	10

Optimal proportions of policy measures

We simulate the vitality of the minority language for different values of the elasticity of substitution: σ = 0 . 1 , σ = 0 . 5 , σ = 1 . 2 , and σ = 1 . 5 . The time horizon is T = 50. The results are given in Figures 4 and 5. The values of the budget shares are given on the left and that of the vitality of the minority language on the right. In Figure 4, the two measures are complements; that is, they reinforce one another. As expected, the budget share of S is going down and that of a is increasing. The corresponding results for a high substitution elasticities are found in Figure 5. The general discussion is confirmed here, too. The budget share of a is decreasing and that of S is increasing as p grows. We also note that the higher is the elasticity of substitution, the higher the budget share of S. In Subfigure 5b, for σ = 1 . 5 , we are getting close to a corner solution with over 80% of the budget going to the status-building measure.

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Figure 4.

Policy measures complements, status adjustment instantaneous ( η → ∞ ). (a) σ = 0 . 1 . (b) σ = 0 . 5 .

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Figure 5.

Policy measures substitutes, status adjustment instantaneous ( η → ∞ ). (a) σ = 1 . 2 , (b) σ = 1 . 5 .

The efficiency gain due to the flexible policy compared to a policy with fixed budget shares given by the optimum values in period one varies strongly with the substitution elasticity, Measured as the change in W (T), it is, in the case of σ = 0 . 1 , 1.72%; for σ = 0 . 5 , 0.78% and for σ = 1 . 5 , only 0.44%. The more flexible the vitality-promoting “technology”, the smaller the costs of not adjusting.

Effects of slow adjustment of the status variable

We conjectured in our claims and proved in the previous section that the budget share of the status-building measure goes to zero at the time horizon. That way the optimal budget share first increases and then decreases when the two measures are substitutes. In Figure 6 this is illustrated in the case of a still fairly rapid adjustment of the status and an elasticity of substitution equal to 1.5.

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Figure 6.

Policy measures substitutes ( σ = 1 . 5 ), status adjustment fast ( η = 0 . 4 ).

If the status adjustment is even slower, the increasing phase of the budget share might be dominated by the decreasing time-horizon effect. This is the case where the elasticity of substitution is the same as in Figure 6 but the status accumulation considerably slower by a factor of eight. In Figures 7 and 8, we also compare the optimal policy when the time horizon is T = 50 with the optimal policy when it is T = 25 and the optimization is repeated for a second planning regime of 25 periods. As expected, there is a trade-off between a gain in vitality after 25 periods and a loss after 50 periods in the case of two planning periods. For the myopic planner the vitality at the horizon at T = 25 outweighs the horizon at T = 50. The vitality can, in the short run, be increased by increasing the budget for the rival good at the costs of not maintaining the status. Since the decline in status reacts slowly to the budget decrease and the effect of a budget increase on the rival good is fast and direct, there is an increase in vitality at the end of the planning periods. During the last periods before the time horizon, vitality increases compared to a policy of maintaining the status. The costs of this increase comes in the next planning regime, since the status at the beginning of the next periods is lower than it would have been if the planner had continued keeping e _S on a high level. This has to be built up again at the beginning of the next period. We have a short-term gain and a long-term loss. There are discontinuities at the end of every planning period with the control variables making jumps. The vitality p and the status S, of course, will be continuous but the path of especially the status will not be smooth.

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Figure 7.

Policy measures complements ( σ = 0 . 5 ), status adjustment slow ( η = 0 . 05 ). (a) Long time horizon (T = 50) (b) Short time horizon (T = 25).

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Figure 8.

Policy measures substitutes ( σ = 1 . 5 ), status adjustment slow ( η = 0 . 05 ). (a) Long time horizon (T = 50), (b) Short time horizon (T = 25).

In the example, the value of the objective W (T) will be higher at T = 25 in the case of two consecutive planning regimes of 25 periods each than in the case of one planning regime of length 50 periods, but it will be lower at T = 50 with short consecutive planning regimes than with one longer one. This effect is the more pronounced the higher the elasticity of substitution between the planning measures. That is, it is higher for substitutes than for complements. In our example, the short-run gain in the case of complements is 1.01% and the long-run loss is 0.88%. In the case of substitutes, the short-run gain is 2.11% and the long-run loss 3.82%. Because of the flexibility in the planning “technology” the planner can move further away from the long-run optimum with the higher elasticity of substitution.

Also the overall lower budget for the status-building measure with a short time horizon in comparison to the case with a long one is clearly visible in the figures. Fewer resources are allocated to the protracted status variable S. This effect is also stronger in the case of the policy measures being substitutes.