Flexible versus rigid individual bounds: nonlinear approach

One may rightly ask, would it be a better solution to introduce imperfect substitutability by a smooth relationship, as it is usually assumed in microeconomics. The corresponding graph, the equivalent of Figure 2.3 is illustrated on Figure 2.4.

The introduction of smooth substitution possibility would have the same effect as the individual bounds, and in a flexible way. The larger is the difference between their shadow prices, the further the ratio of the two components may depart from their observed value (r⁰).

Unlike in the case of the rigid bounds, where it jumps to the lower or to the upper bound, whenever they are different. So it makes sense to experiment with such smooth curves.

All we have to do is to replace the constraints

Σ

j aij·xj + y·siy ≤ xih

+ mi

r_i^ml·x_i^h ≤ mi

And we will soon see, exactly this is what we do in the applied models of general equilibrium as well.

Figure 2.4

The logic and working of the flexible bounds

The shadow demand curve expresses the ratio of the two components (r^m) as a function (together with the cost shares) the own and cross price elasticity of demand. In applied models imperfect substitutability is most often represented by CES (constant elasticity of substitution) functions, the parameters of which are easy to estimate, once we set the elasticity of substitution. The size of the elasticity is usually chosen on the bases of somewhat ad hoc expert judgment. But as long we want to use such functions to generate flexible bounds in applied resource allocation models, the ad hoc choice of the elasticity parameters is perhaps still superior to the ad hoc choice of rigid lower and upper bounds on certain variables.

If we carry out this replacement, the so far linear model of optimal resource allocation will become nonlinear. But this would not matter nowadays, since we have rather powerful algorithms and software that could solve a nonlinear programming model, as long as the nonlinear functions used in the model are well-behaved. So let us do that, and only in the case of the domestic/import supply, but in other parts of the model too. In the case of the ratio of

a közömbösségi ’görbe’

x^hm

the graph of the demand implied

x^h

domestic sales and exports (r_j^e = z_j/ x_j^h) we can also use flexible bounds by means of properly chosen x_j(x_j^h, z_j) transformation functions.

We could and will go on and extend further the scope of using similar smooth functions elsewhere in the model too, for example, in the case of labour and capital, the composition of the personal consumption. But before we do that, we will illustrate with help of the yet simple enough model, how the conditions of optimality will be modified as a result of changing its specification. The table bellow contains the nonlinear version of the LP-2.2-3a model, where in the place of the dual conditions we put the first order necessary conditions of maximum derived by means of the Lagrange (or Kuhn–Tucker) method. Since we assume the observed values of all variables were and remain positive in the optimal solution (including the Lagrange multipliers), we may represent the conditions as equalities (in general, we should use in equalities and complementary slackness conditions as in the case of the LP problem).

NLP-2.2-1 (P) (KTD) from the differentiation of the following Lagrangian function:

L = y −

Σ

j p_j^a·

{

^xj(x_j^h, z_j) − xj

}

−

Σ

i p_i^hm·

{ Σ

j a_ij·x_j + y·s_i^y − xihm

(x_i^h, m_i)

}

− − w·

{ Σ

j lj·xj − L0

}

− q·

{ Σ

j kj·xj − K0

}

− v·

{ Σ

i(piwm

· mi − piwe

·zi) − de

}.

We have changed the notation of the dual variable (the Lagrange multiplier) assigned to the first constraint, because here the output is also a composite good, and its shadow price will be interpreted as its equilibrium cost-price, as it will become clear later. In order to be able to decipher the meaning of the dual prices and constraints, we introduce a few auxiliary symbols (pih

, pim

and pie

) by means of the following definitions (the second and the third elements of the equations are taken from the optimality conditions):

p_i^h = p_i^hm· _h

The names given suggest already in advance their intended meaning. We will show that p_i^h, p_i^m and p_i^e can indeed be interpreted as the equilibrium price of the domestically produced and sold, imported and exported product variety of the same sectoral origin, which are assumed to be imperfect substitutes.

One can easily show that the dual conditions are the same as the ones which characterize the optimal solutions of the following constrained cost minimization and revenue maximization problems:

min p_i^h·x_i^h + p_i^m·m_i, s.t. x_i^hm = x_i^hm(x_i^h, m_i), max p_j^h·x_j^d + p_j^e·z_j s.t. x_j = x_j(x_j^d, z_j),

where the variables are x_i^h, m_i and p_i^hm Lagrange multiplier in the first problem, and x_j^d, z_j and p_j^a Lagrange multiplier in the second problem. (Notice that in the programming model both the domestic demand for (x_i^h) and the supply of home produced commodities (x_i^d) are denoted by the same variable (x_i^h), whereby we implicitly assume the fulfilment of the x_i^h = x_i^d equilibrium condition. We will switch to this notation in the following discussion as well.) Let us show it for the first case, where the Lagrangian functions takes the following form:

L(x_i^h, m_i, p_i^hm) = p_i^h·x_i^h + p_i^m·m_i – p_i^hm·{x_i^hm(x_i^h, m_i) – x_i^hm}.

Thus, as in the case of the linear programming solution, the optimal the shadow, that is the equilibrium cost-price of the domestic/import composite commodity will be the weighted averages of component prices, where sih import (mi) supply, but their nonlinear aggregate: xihm

= xihm

(xih

, mi), where xihm

measures the

joint use value of the two components. The values of s_i^h and s_i^m are the cost minimizing levels of the components making up at least one unit use value. In the jargon of competitive equilibrium, they could be interpreted as demand functions. s_i^h + s_i^m > 1, as a rule, except for the base equilibrium at unit level prices, when s_i^h + s_i^m = 1, as the share coefficients in the linear case

Following the same line of argument we can show that the first order conditions in the case of the constrained revenue maximization problem are the following:

∂L/∂xjh

are again functions of the prices, homogenous of degree zero:

sjd

which can be interpreted as supply functions of a firm in a perfectly competitive market.

Next, note that there are three unknowns in both optimization problems and same number equations representing the first order necessary conditions of the optimal solutions. If the chosen functions are well-behaved and simple, we can explicitly solve analytically the model before any calculation, and express the values of the unknown variables (e.g., xih

, mi and pihm

) as functions of the variables (e.g. pih

, pim

and xihm

) assumed to be known and the parameters of the substitution functions. Thus, if we wish, we can derive the closed analytical forms of the the first two of which are derived demand functions, the third a price index aggregator.

We can in fact arrive at the optimal values of the variables in various ways. Because of the assumed first order homogeneity of the substitution (aggregation) functions, the optimal ratios, such as sih optimality conditions with equivalent alternative forms, which may be more familiar for the user or the reader. For example, the following three sets of the dual equations,

p_i^hm· _h represented by the following six sets of equations:

p_i^m = v·p_i^wm, p_i^hm = p_i^h·s_i^h(p_i^h, p_i^m) + p_i^m·s_i^m(p_i^h, p_i^m), m_i = r_i^hm(p_i^h, p_i^m)·x_i^h, from solving parametrically the following cost minimization problem:

min pih solving the following revenue maximization problem:

max pjh

Such a formulation of the dual condition of the optimal solution would be more familiar for a former student of economics than the original ones. They are also closer to the forms input-output models.

Let us now make some steps forward and make use of the possibility provided by the use of nonlinear functional forms in our resource allocation model. Let us first of all separate, as we did in the case of the applied input-output models, the main components of final use,

yi = yicv

+gi·yg +

Σ

j bij·yjb

+ ci0

, where ci0

denotes the fixed (‘committed’) part, and the sectoral levels of gross investments (yjb

) degree of substitutability by introducing the following,

ycv = yv(y1cv

linear homogenous utility (welfare) function to determine variable consumption yicv

, where the optimality conditions take the following forms:

max yv(y1cv

where p_cv is the Lagrange multiplier. We expect and will see these latter conditions to appear among the dual constraints of the optimality conditions.

Production technology will be represented by Johansen type production functions, which were introduced in the previous chapter. We will thus allow for substitution between labour and capital, by using smooth, well-behaved production functions given in the forms of

x_j = f_j(L_j, K_j).

At factor prices w_j and q_j the optimality conditions of cost minimum are as follows:

cj ·

They will also appear among the necessary conditions of the optimal macroeconomic resource allocation.

As long as functions f_j are homogeneous of degree one and well-behaved, as we usually assume, one can derive the identities

cj·

What concerns the flexible bounds one can use in the case of exports, we have in fact two possibilities. One of them is the x_j(x_j^d, z_j) transformation functions, used in the previous model, which put limits on the movement of the export volume from the supply side. Instead of or together with it, we can also use smooth export demand functions, which can also effectively constrain their levels. We will introduce them in the form of indirect demand functions, p_i^we(z_i), which define their external market price in the foreign trade balance. As a result we will end up with the following model and its optimality conditions. We will discuss later the potential side effects of this solution.

As a result of introducing the suggested changes into our optimal resource allocation model, we will arrive at the following nonlinear programming model and optimality conditions. Note, that y_g and y_j^bn are treated as exogenous variables in this version of the model. In the column on the left we find the resource allocation constraints (4n + 4 numbers of equations in terms of 8n + 1 number of variables) and the assigned Lagrange multipliers. On the right, we have listed the Lagrange (or Kuhn–Tucker) first order necessary conditions of the optimal solution (8n + 1 number of equations in terms of 4n + 4 numbers of variables). We have attached to each of them the primal variable according to which we differentiated the Lagrangian function in order to derive them. The total number of unknowns and equations in the Lagrange (Kuhn–Tucker) conditions of the optimal solution are thus 12n + 5, arranged into 17 blocks.

NLP-2.2-2 (P) (KTD)

As before, it will be useful to introduce some auxiliary variables, which make it easier to interpret the solution. Thus, we may introduce again the symbols pih

, pim

and pie

with slightly different definitions as above (the second and the third members of the equations are again implied by the second, fourth and fifth optimality conditions):

pih

Their meaning is suggested by the chosen notation. They are in fact the shadow prices of the capital goods in the different sectors. They differ from each other only to the extent the amortization rates (rja

) and the prices of the composite sectoral capital goods (pjb

) are different.

The above definition of the cost of capital is slightly different from Walras’s, which would be qj = (rja

+ πj)·pjb

.

The reason behind this difference is that in the capital constraint we treat capital as a homogeneous factor, which takes up a uniform net shadow rate of return, ρ. Unlike its

counterpart in Walras’s definition, the uniform rate of return is defined here in relation to the physical volume of capital (K_j) used and not on its value (p_j^b·K_j). In Walras’s definition this latter is assumed to be uniform in competitive equilibrium, whereas here they are different, as a rule, since πj = ρ^/pjb

.

What explains this strange logic is the somewhat contradictory treatment of capital goods in the above model. In the capital constrain capital is considered to be freely mobile across sectors, which would imply uniform composition (b_i), price (p_b =

Σ

j p_i^hm·b_i) and a uniform rate of return (ρ on the physical volume and π ⁼ ρ^/pb on the value of capital used).

We could dissolve this contradiction in two ways. The first possibility is to enforce fully the assumption that capital is a homogeneous good and revise the definition accordingly, that is, replace the yjb

variables with a single y^v scalar, and the sectorally different bij investment coefficients with bi’s, as suggested above. The other possibility is to treat capital as sector specific goods in all its appearance, thus, replace the single capital constrain with sector specific constraints: Kj = Kj0. As a result of this solution the net return both on the physical volume (ρj) and the value of the capital (πj) would be different in the various sectors in general.

This latter differentiation would, however, violate the requirement of competitive equilibrium.

Neither treatment provides, thus, a fully satisfactory solution. The root of this dilemma lies basically in the problem of macro-closure, discussed earlier.

Another questionable feature of the above model is the derived definition of the shadow

where εi is the price elasticity of export demand. Since under normal conditions the sign of the latter is negative, the term 1 /εi can be interpreted as a tax rate applied on incomes earned via exports. This solution is well known in international trade theory and they are called optimal tariffs. The theory calls attention to the possibility that the introduction of such tariffs could make price-taking producers behave collectively as a monopoly. Nevertheless, it would not be reasonable to use such an assumption in a macroeconomic resource allocation model.

In document CGE Modelling: A training material (Pldal 72-80)