Ad hoc bounds in linear models to constrain overspecialization

2.2.2. Ad hoc bounds in linear models to constrain overspecialization

In the applied linear programming models designed for policy analysis this possibility of overspecialization was a bothering fact (it challenged the relevance of the models) and therefore the modellers tried to avoid by introducing upper and/or lower bounds on some key variables. We will illustrate this technique and its consequences in our model by introducing upper and lower bounds on the volume of exports

z^l≤ z ≤ z^u, that is, z_i^l≤ z_i≤ z_i^u, and on the ratios of import/domestic supply:

r^ml<x^h> ≤ m ≤ r^mu<x^h>, that is, r_i^mlx_i^h ≤ mi ≤ riml

x_i^h, where x_i^h = x_i− z_i as before, and r_i^m = m_i/ x_i^h.

As a result we get the following, slightly modified version of the previous LP problem.

(LP-2.2-3a) (P) (D)

What becomes apparent at the first glance that there is a price for keeping all output, export and import levels all positive, not falling too far from their observable levels (for the sake of simplicity we assumed that all the sectoral commodities were traded in the base). The dual problem became much less transparent and it is less obvious how one can interpret its optimal solution. Since all the primal variables are positive, the dual constraints will be fulfilled in the form of equations. The first two sets

pjh

are basically the same as before, except for the notation. The first primal constraint belongs to the domestic output, the second to domestic (composite) supply. This is why we assigned to them the dual variables pjh

and pihm

, in line with notation introduced in the previous section.

The meaning of the

equations can be deciphered on the basis of the following economic reasoning.

Observe that the formulation of the problem implicitly assumes, that domestic outputs and imports are perfect substitutes, therefore, their prices should be equal in perfect equilibrium.

And in fact, both should be sold at prices pihm

. However, the purchasing price of the imports is v·piwm

, whereas the producers’ price of domestic output is pih

, and they will be different, as a rule. The shadow prices assigned to the individual constraints, confining their ratios into the given range, generate such taxes/subsidies that equalize them. If the lower limit is binding,

which indicates that the import is more expensive than the domestic production, than τiml

will be positive and it will lower the domestic sales price of the import and increase that of the domestic output in order to equalize them.

It can be shown that in effect p_i^hm will be equal to the weighted average of the component prices,

p_i^hm = p_i^h⋅s_i^h + v·p_i^wm⋅s_i^m,

just the way as we defined the price of the domestic/import composite it in the previous section. Observe, however, that here the prices are equalized at the same time, and the taxes/subsidies serve for this purpose. In the previous section we did not assume that, we just calculated the average price. Thus, according to the logic of the model these taxes/subsidies just redistribute income among the users of the commodity, so it will not affect the total available net income, which is given by the objective function of the dual problem as before.

The equality of the optimal values of the two objective functions ensures the fulfilment of Walras’s law in this case too:

y = y⋅p⁰s^y = w⁰·L₀ + q⁰·K₀ + v⁰·d_e + τieu

·z_i^u−τiel

·z_i^l.

As we can see, these taxes/subsidies do not appear in the net income, unlike those related to the regulations of the export prices, which follows a different logic. Since the prices of the domestic products is determined by their cost, to make the producers sell on both the domestic and the foreign market, the export price have to be made equal to the former. This is exactly the meaning of the dual equation

v·p_i^we – τieu

+ τiel

= p_i^h.

The producers sell their products at prices p_i^h, whereas the foreign buyers pay v·p_i^we. The price differences (taxes/subsidies) can be interpreted here as income transfers between the domestic tax authority and the foreign buyers, which modify the level of domestic income.

We can illustrate the logic followed in the case of import constraint on Figure 2.3.

Figure 2.3

The logic and working of the import constraints in the LP problem 2.2-3

x^h x^h0

the level of use value the graph of the demand implied x^hm

demand ’curve’

r^m

r^mu r^m0

r^ml m

m⁰

The above graph should be all familiar from the microeconomics textbooks, except for the use of piece-wise linear rather than smooth indifference and demand curves. The algebraic representation of the problem illustrated on the graphs is the following

min p_i^h·x_i^h + p_i^m·m_i, s.t. x_i^h + m_i = x_i^hm, r_i^ma·x_i^h ≤ mi ≤ rimf

·x_i^h,

and x^h0, m⁰ and r⁰ on the graph represent the observed (base, not the optimal!) values of the variables. Note also, that p_i^h, p_i^m and x_i^hm are considered here to be parameters, unlike in the model itself.

The original indifference curves were linear, in line with the implicit assumption that the domestically produced and the imported goods are perfect substitutes. What the individual bounds do is they turn this relationship into less then perfect substitutability. In a rather rigid manner: they are perfect substitutes between the given bounds, and perfect complements beyond them.

Consider also the case, when domestic outputs and imports are assumed to be perfect complements, that is, their ratio is fixed: r_i^m0 = m_i/ x_i^h. In the above linear model this would mean that r_i^ma = r_i^mf = r_i^m0. Modifying the problem accordingly, we would only have one equality condition,

r_i^m0x_i^h = m_i

instead of the pair of inequality constraints, r_i^mlx_i^h ≤ mi, and m_i ≤ rimu

x_i^h,

and the sign of the dual variable assigned to it (τim

) would be undetermined, and the corresponding dual constraints would take the following forms:

p_i^hm ± τim

·r_i^m0 ≤ p_i^h (x_i^h)

p_i^hm ≤ v·piwm ± τim

(m_i) The optimality conditions would, thus, only slightly change. What we wanted to illustrate, as a matter of fact, was that letting the ratio of import/domestic supply (r_i^m) move within some bounds could also be interpreted as relaxing a former assumption of perfect complementarity.