**1.6 The multilingual challenge: a complex issue?**

**1.6.4 The trade-off between mobility and inclusion**

In the introduction it was mentioned that one needs to pay particular at-tention to causal links when implementing a policy addressing a complex issue. The reason for this is that there might exist links between apparently unrelated variables. Therefore, actively affecting one variable might acciden-tally affect other variables. In this section I discuss, both analytically and in practice, how policies addressing seemingly unrelated issues could work in opposite directions, practically cancelling their respective effects. In par-ticular, I draw from Grin (2018a) and consider the example of mobility and inclusion in multilingual Europe. By "mobility" I refer to the fact that people are nowadays more and more likely to move throughout their lifetime for various purposes. By "inclusion" I refer to people’s feeling of belonging in a given context or within a given community.

Mobility and inclusion might pull in non-converging (potentially oppo-site) directions. Investing resources to increase one dimension could affect the other in a negative way. An individual or society can at best strike a bal-ance between mobility and inclusion, so that utility (i.e. the level of satisfac-tion or, in this case, the aggregate value accrued to society) is maximized un-der a specific constraint. The combination of mobility and inclusion defines a certain level of utility within society. Reasonably, an efficient or cost-effective use of resources is preferred to any other, as it guarantees the highest possible result given a certain resource endowment. This trade-off can be graphically represented as in Figure1.13, drawn from Grin et al. (2014, p.17).

It is easy to see that this is the best possible result because any move-ment along the constraint would provide people with less desirable condi-tions. Any other pair along the constraint is indeed sub-optimal, as it could be matched by other pairs providing the same level of satisfaction and con-suming less resources.

This whole idea of the trade-off can be rewritten in analytical terms. By creating a model, one usually means to use simple mathematical represen-tations to fit the relationship between mobility and inclusion just discussed.

One can think of mobility and inclusion as the elements of a two-good econ-omy and some form of social value as the utility resulting from a combination of them. For simplicity, one can think of the relation between these three vari-ables (utility, mobility, and inclusion) as a two-input Cobb-Douglas function (more specifically, I discuss utility as a function of mobility and inclusion):

U(M,I) =kM* ^{α}*I

*(1.25)*

^{β}whereU, M, and I represent, respectively, utility, mobility, and inclusion, k
is a positive constant, and *α* and *β*are constants whose values are included
between 0 and 1.^{53} Basic calculus tells us that the behaviour of this function
with respect to its variables is described by the following first-order partial
derivatives:

UM =kαM^{α}^{−}^{1}I* ^{β}* >0 (1.26)
UI =kβM

*I*

^{α}

^{β}^{−}

^{1}>0 (1.27) The fact that both first-order derivatives are positive tells us that, if either mobility or inclusion increase while the other dimension is kept constant, the level of utility increases (though we have not specified in what proportion).

In other words, the marginal contribution of either dimension is positive.

The second-order partial derivatives are:

U_{M,M} =kα(* _{α}*−

_{1})M

^{α}^{−}

^{2}I

*<*

^{β}_{0}

_{(1.28)}UI,I =

_{kβ}(

*−*

_{β}_{1})

_{M}

^{α}_{I}

^{β}^{−}

^{2}<

_{0}

_{(1.29)}U

_{M,}

_{I}=U

_{I,M}=kαβM

^{α}^{−}

^{1}I

^{β}^{−}

^{1}>

_{0}

_{(1.30)}

53Note that in many cases, especially those concerning production functions, the sum of
the exponents*α*and*β*is also defined, in that exponents summing up to a value lower, equal,
or higher than one imply, respectively, diminishing, constant, and increasing return to scale
However, I am not interested here in cardinality properties, which are key in production
functions. What matters here is that our function preserves ordinality of preferences (utility).

What do the signs of the partial derivatives tell us? The unmixed
second-order derivatives are both negative, suggesting that the contribution of one
dimension, though being always positive, diminishes when the other is held
constant. However, the sign of the mixed second-order derivatives (which
are equal, by Young’s theorem) is positive. This tells us that mobility and
in-clusion are complementary goods and that more of one increases the marginal
utility of the other.^{54}

Is it reasonable to think of the relationship between inclusion and mobility in this sense? The indifference curves resulting from this conceptualization are downward-sloping and convex, as depicted in Figure1.13. The negative slope represents the willingness to make a trade-off, i.e. to exchange mobility for inclusion and vice versa. The convexity of these curves tells us that there exists a preference for variety between inclusion and mobility. Besides, the shape of the curves makes sure that corner solutions (where one dimension is completely abandoned in favor of the other) are not possible. The underlying assumption is that policies "going all-in" on either one or the other dimension are in no way satisfying. A completely mobile society, with no inclusion, and a sclerotic though extremely inclusive society are equally unfavorable con-ditions. Besides, if the indifference curves were concave, society would be ready to give up more and more of one dimension for less and less of the other. In other words, for example, people would be ready to give away an increasing amount of mobility to make society only slightly more inclusive.

However, again, I do not contemplate extreme positions favoring one dimen-sion over the other. As said, these curves are characterized by diminishing marginal rates of substitution. It means that people’s preferences are charac-terized by a decreasing willingness to give up one dimension for the other.

Simply put, it means that if an individual (or society) has already a very little amount of, say, inclusion, a further reduction can be compensated only by a great increase in mobility. In a symmetrical way, when inclusion is abun-dant, it will not be a problem to give up a greater portion of inclusion to have a little more mobility. In analytical terms, the marginal rate of substitution is equal to the ratio of the two first derivatives:

MRSM,I = ^{U}^{M}

54However, it should be noted that, by this reasoning, mobility and inclusion cannot be consideredperfectcomplements, i.e. goods thatmustbe consumed together in a fixed ratio.

In that case, more of one good does not imply that the marginal utility of the other increases.

As said, the marginal rate of substitution (how much inclusion we need to
give up to increase mobility while keeping the level of satisfaction intact)
de-creases in the variable M, meaning that the more mobility we get, the less
inclusion we will be ready to give up. Needless to say, the symmetrical trend
is also true, both analytically and intuitively. Finally, it can be easily shown
that the elasticity of substitution of these indifference curves is constant and
equal to one. This metric describes the willingness of people to substitute
one dimension for the other. As a general rule, if elasticity tends to
infin-ity, the goods are considered substitutes, while if it tends to zero, they are
complements and substitution is less and less possible (or welcome). A
uni-tary elasticity of substitution means that people are willing to obtain utility
by switching between the two dimensions based on their relative costs, but
there is a preference for a diverse bundle rather than extreme solutions.^{55}

We could theoretically find an infinite number of combinations of inclu-sion and mobility and, in virtue of the non-satiation principle, the more, the better. However, the indifference curves are just a way of showing that, given a certain level of utility, we have an infinite number of equally satisfying combinations, and that all of these combinations lie on the geometric locus described by the curve. Shifting to another curve means moving to a whole different set of combinations which are more or less satisfying than the pre-vious one (which is why indifference curves can never intersect). To sum up, moving along the curve means choosing among different combinations that guarantee the same level of utility accruing to society, while jumping from one curve to the other means not only changing the combination (although the relative proportions might well stay the same), but also switching to a different (lower or higher) level of social utility.

Indifference curves alone say nothing about the limitations to utility pro-vided by constraints of different nature. Indeed, these curves only inform us about the "behaviour" of individual (on a micro-level) and social (on an aggregate macro-level) preferences with respect to the goods that consumers are provided with (inclusion and mobility, in this case), given a pre-specified amount of utility. Therefore, the next step in the formulation of our mobility-inclusion model is giving a formal definition to the constraint. This step is fundamental to define the pre-specified amount of utility just mentioned and

55Of course, this is a strong assumption, as one may as well argue that certain societies prefer mobility over inclusion (or the other way round). Note, however, that this assumption can be easily relaxed by switching from a Cobb-Douglas to a common constant-elasticity-of-substitution (CES) function.

find an optimal combination of inclusion and mobility. Describing a con-straint means defining the relative cost of either dimension in terms of the other at any moment, i.e. how much, say, inclusion must be given up to in-crease mobility by one unit (assuming that we have a unit of measures for either dimension). It should be noted that, although this argument might sound similar to the one just made about the indifference curves, it is not the same thing. In the case of indifference curves, I was referring to people’s preferences, what makes them satisfied and how. On the contrary, I am now referring to the trade-off caused by the fact that measures supporting one dimension often clash with measures supporting the other. Here are some examples drawn from Grin et al. (2014):

• Encouraging the acquisition of language skills to sustain mobility risks amplifying social differences, causing eventually the exclusion of a part of the population. This is especially the case when opportunities to ac-quire language skills (as many other professional skills) are associated with better socio-economic conditions (i.e. better-off people have more chances to acquire these skills), such as when languages are mostly ac-quired through private means. If a policy maker fails to see this connec-tion, implementing a mobility-enhancing policy that exclusively relies on the acquisition of language skills risks exacerbating the social divide, causing further social fragmentation (that is, less inclusion).

• Strengthening the association of a territory to one (or more) specific lan-guages (the so-called "territoriality principle") may work in favor of a cohesive society. Measures such as teaching the local language are gen-erally recognized as very effective tools to foster inclusion, especially of immigrants. However, relying extensively on such policies to include people reinforces the relation between one place and one language, and therefore it may actually hinder mobility, not to mention the fact that it might be considered anachronistic in a highly mobile world.

The reason why it is important to define a constraint is that it helps us under-stand how consumers can maximize their utility, given a certain availability of resources. We can think of the constraint as a line or a curve that lies on the same plane as our indifference curves, crossing some of them and being tangent to only one. Individuating the point of tangency means maximizing the level of utility in terms of the two dimensions. In analytical form:

s.t.B(maxM,I,p_{M},p_{I})U(M,I) (1.32)

or, in explicit terms:

s.t.B(maxM,I,p_{M},p_{I})kM* ^{α}*I

*(1.33) In the expression s.t. stands for "subject to" and B(M,I,pM,pI) is a func-tion that describes the constraint in terms of mobility, inclusion, and their re-spective prices, all being positive values.*

^{β}^{56}What does the constraint mean?

As mentioned, the constraint is placed in the same plane as the indifference curves. However, it does not show all the equivalent pairs of goods, but individuates all affordable pairs, which are those in the area between the constraint line (the purple line in Figure 1.13) and the axes. Intuitively, the closer the constraint to the origin, the fewer the affordable pairs. Any in-crease in the budget will shift the constraint line further from the origin (the dotted line in Figure 1.13), increasing the range of choice (this can also be the consequence of a fall of the cost of one or both dimensions). Maximiz-ing our utility function simply means findMaximiz-ing the pair lyMaximiz-ing at the same time on the constraint and on the highest possible indifference curve. In terms of our analysis, (monetary and non-monetary) resources to push mobility and inclusion are limited and need to be allocated (ideally, in an optimal way) between the two. In theory, corner pairs (resources being completely allo-cated on one of the two) are available in the case of the constraint, but the possibility of these extreme solutions is anyway ruled out by the shape of the indifference curves.

Up until now, we have been thinking of the constraint as a linear function,
mostly for the sake of simplicity. However, the price to pay for simplicity is
often a loss in terms of realism. As a matter of fact, the constraint we are
dealing with is rather complex and is definitely best described by non-linear
functions. It is easily understood that price of either mobility or inclusion
is a function of their relative quantities. The shape of the constraint is
piv-otal for our maximization problem: a line or a convex (upward) curve would
guarantee the existence of a single pair that maximizes utility, while a
non-convex curve would allow for several possibilities, such as multiple local
and global optima.^{57} The latter case is particularly interesting: as the
maxi-mization problem is solved and policy makers come up with equally optimal

56Note that I keep the function describing the constraint in implicit terms and avoid using a simplistic additive form such asB = pMM+pII. For a discussion on non-linear budget constraints, see Quigley (1982).

57It should be noted that the word "convex upward" is used here to refer to a specific property of certain real-valued functions, for which the following inequality holds true for allxandyin their domain:

f((1−*α*)x+*αy*)≥(1−*α*)f(x) +*α*f(y), ∀*α*∈[0, 1] (1.34)

solutions, choice could be driven not just by the political climate but also by the preferences of the government currently in charge. As a matter of fact, right-wing and left-wing parties have different positions concerning mobil-ity and inclusion. Thinking of the constraint of policy making processes in linear terms would be inevitably reductive, given the extremely changing en-vironment in which they take place and the high level of interaction between policies at all levels. Policy makers need to tackle ever more complex and sometimes not well understood problems, in terms of both causes and con-sequences. It is important to note that the constraint is further complicated by the geographical scope of the policy. If the policy is to target a very wide and socio-economically diverse area, the number of elements making up the constraint increases and things that can be taken for granted in certain areas may be completely different in other areas. We could say that these differ-ences can be attributed to different social archetypes. By this I mean that dif-ferent societies, as those making up complex organizations like the European Union, possess different collectively-inherited ideas that shape different so-cial patterns of thought and behaviour with respect to specific issues. In other words, different social entities might react differently to similar situations on the basis of underlying hypotheses internal to their group, non-universally shared and which have existed for a long time. Besides, current conditions in a different jurisdiction translate into different needs and constraints. As mentioned, different policies are in constant interaction. Therefore, a com-plete harmonization of a specific policy across a wide area is seldom possible, unless the different authorities agree on a larger view. This, however, would be somewhat unrealistic at the moment. For this reason, a complex language policy should not aim at finding a one-fits-all solution. It should rather offer an algorithm or a model that could elaborate on the information provided and come up with an optimal strategy on a case-by-case basis.

If the equality is never satisfied, we can speak ofstrictupward convexity. This is equivalent to saying that, if we traced a segment connecting two points of the convex upward curve, the segment would always sit below the graph of the function between the two points. Note that "convex upward" is synonymous with "concave".