Conclusions: towards the computable general equilibrium models

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.

2.2.4. Conclusions: towards the computable general equilibrium models

By means of introducing auxiliary variables and mathematically equivalent alternative forms, the conditions of optimal resource allocation can be rearranged into an alternative specification. We have chosen such an alternative set of equations, which will be easy to compare with the input-output volume and price models. Their comparison reveals the real nature of not only the optimal resource allocation model chosen, but also of the typical computable equilibrium models, the structure and the underlying logic of which is basically the same. The system will consist of 20n + 5 unknowns and equal number of equations. The variables and parameters of the derived equation system can be classified in the following way:

Endogenous variables: real (volumes) xj, xihm

, xjh

, zj, mi, (5n)

(20n+5) real (structural) sih

, sim

Potential (endogenous or exogenous) variables: y_g, y_j^bn, c_i⁰, p_i^wm, L₀, K₀ and d_e. Parameters: a_ij, b_ij, r_j^a, g_i and the parameters of the various functions.

We will group the conditions characterizing the optimal solution of the problem NLP-2.2-2 similarly into three categories:

A) balances and definitions of volume categories (x_j, x_i^hm, x_j^h, z_j, m_i),

B) balances and definitions of price categories (p_i^h, p_j^a, p_i^m, p_i^e, p_i^hm, p_j^b, q_j, p_cv, w, ρ^{, v),} C) definitions of structural parameters s_i^h, s_i^m, s_j^d, s_j^e, p_i^we, l_j, k_j, s_i^cv.

A)BALANCES AND DEFINITIONS OF THE VOLUME CATEGORIES (5n + 3):

We can further divide this category into two subgroups. (On the left hand side we will assign list numbers to the equation blocks, whereas, on the right hand side, endogenous variables, that will make it easier to count the number of unknowns and equations, and check the regularity of the equation system.)

A1) Sectoral commodity balances and their components (5n):

(P1) xihm

This subgroup defines an extended input-output volume model, in which both imports and exports are treated as endogenous variables by setting them proportional to domestic supply.

We have discussed earlier an almost identical linear version of this model, in which we had xj = xjh unknowns and equations, is a well determined macro-model. Under normal conditions would expect it to have a unique solution, thus, the values of xjh

, zi and mi would also be determined in it. Consequently, the value of the expressions on the right hand side of following balance equations (labour, capital and foreign currency demand) would be also determined by them:

A2) Balances of the primary resources (3):

(P6)

Σ

j l_j·x_j = L₀ (w, L₀)

(P7)

Σ

j k_j·x_j = K₀ (ρ^{, K}0)

(P8)

Σ

i

(

^piwm

· m_i − piwe

·z_i

)

^{= d}e, (v, d_e)

where p_i^we, l_j, k_j would also be constant parameters in a conventional input-output model.

The natural closure of the extended input-output system, defined by the parameters and unknowns, and equations (P1) – (P8), would be to treate L₀, K₀ and d_e as endogenous demand variables (this is why we have also assigned these variables to the equations). The extended input-output model with such a closure would be final demand driven. We have discussed the closure possibilities of such a model in connection with the applied linear input-output volume models. We have pointed out that one can not expect that the above equation system, in which two or more of the supply constraints would be fixed, would have a sensible solution.

We could choose, in general, only one out of L₀, K₀ and d_e as exogenous supply variable and turn, in exchange, the level of some final demand (e.g., y_cv, as in the optimal resource allocation model) into endogenous variable. With such a change, the demand driven model would become supply driven, as the optimal resource allocation model. In the latter model all the three, that is, L₀, K₀ and d_e were considered to be fixed (exogenous supply variables). This is why we would assign w, ρ and v as complementary variables in the optimal resource allocation model to the same equations, in which their values reflect their relative scarcity.

B)PRICE IDENTITIES AND EQUATIONS (7n+1):

This block contains shadow price identities and equations, derived from the dual optimum conditions. Most of them can be interpreted and take the form of cost-price definitions, as we have discussed already. well defined extended input-output model in terms of the 7n+1 number of unknowns, pih

, pja

C)EQUATIONS OF STRUCTURAL VARIABLES AND WORLD MARKET EXPORT PRICES (8N):

The last block contains equations that define the required proportions between various volume and price variables, which were also derived mainly from the dual optimality conditions. The coefficients setting these proportions would be typically fixed in a linear input-output or programming model of resource allocation. The constraint defining the inverse export demand

function seems to behave like a cuckoo’s egg among the other variables. But it could be rearranged into such an alternative form that would also define a proportion, namely that of the domestic export to the offer of foreign competitors.

As we have demonstrated, these constraints can be interpreted as setting flexible upper and lower bounds on specific variables to confine their departure from their observed values into reasonable ranges. In addition to that these proportion variables link together the first two, the volume and price blocks of equations, which would be completely independent from each other, were these proportions exogenously fixed, as in the input-output models.

(P17) s_i^h = s_i^h(p_i^h, p_i^m) (s_i^h)

(P18) s_i^m = s_i^m(p_i^h, p_i^m) (s_i^m)

(P19) s_j^d = s_j^d(p_j^h, p_j^e) (s_j^d)

(P20) s_j^e = s_j^e(p_j^h, p_j^e) (s_j^e)

(P21) p_i^we = p_i^we(z_i) (p_i^we)

(P22) s_i^cv = c_i^v(p₁^hm, p₂^hm, ... , p_n^hm) (s_i^cv)

(P23) l_j = l_j(w, q_j) (l_j)

(P24) k_j = k_j(w, q_j) (k_j)

COUNTING EQUATIONS AND VARIABLES

We have listed above 20n+4 equations altogether, whereas there are 20n+5 variables. It seems as if the equation system is yet not fully determined. Checking the variables assigned to the constraints listed, we can see that the only variable, which does not have a counterpart equation, is the level of variable (private) consumption, ycv. It can be shown that the rest of the equations will uniquely define its value by force of Walras’s law.

One could introduce one more variable, ev as the expenditure spent on variable consumption and add one more equation, the expenditure constraint, to the system in one of the following forms

pcv·ycv =

Σ

ipihm

·sicv

·ycv = ev or

Σ

ipihm

·sicv

·ycv = ev. (ycv)

In such a specification ev could be taken as the undefined variable, whose value is set by Walras’s law, as in the case of the Johansen model.

In any case, there would be one more unknown than equations. One can, however, easily check, here again, that all equations are independent of the general level of the price and value terms (i.e., homogenous in prices), therefore we can fix the price level by setting the value of one them, for example set pcv =

Σ

ipihm

·sicv

= 1, as in the programming model, or ev = 1, as the Johansen model). That would make the equations system well determined (regular).

SUMMARY AND CONCLUSIONS

It is worth summing up what we have demonstrated. We have seen, first of all, that the feasibility conditions of the resource allocation problem are nothing but the conventional macroeconomic accounting identities: balance requirements described in terms of

supply-demand equations for sectoral commodities, primary resources and foreign currency (see, balance of trade). They are the same conditions that should appear in any nation-wide model, especially in those, which are built upon the input-output tables.

The number of the equations defined by these constraints is relatively small compared to the number of the potential unknowns. For example, in the case of NLP-2.2-2, the primal (physical) resource allocation constraints consisted of 4n + 3 equations, expressed in terms of 8n variables (ignoring y_cv and its definition, which could have been put directly into the objective function). Unlike, thus, in an input-output model, there will be a large degree of freedom left by the primal resource constraints and balance requirements. (In the case of the input-output models we reduced the degree of freedom to zero by fixing the value of many potential variables exogenously). In other words, the set of feasible resource allocation patterns will be quite large. We try to reduce this set to a single point by optimizing an appropriate welfare function over it. If there is a single solution, the model can be used for comparative static exercises, that is, compare solutions received by assigning different values to specific exogenous variables.

We have demonstrated that the above ‘regularisation’ of the set of feasible allocation patterns, i.e., its representation by just one salient point, is equivalent with complementing the set of primal variables and equations with appropriate dual variables and constraints, which would together define a regular equation system. The numerical solution of the optimizing model would, however, not be able to replicate the observed values of the variables, even if the benchmark data set were consistent with the implicitly assumed optimizing behaviour. For the simple reason that the necessary conditions of the optimal solution are not all reflecting the actual rules of accounting, especially not in the case of the pricing rules, which ignore taxes and subsidies. The programming approach corresponds to the world of perfect competition.

It is important to note in this connection that that the majority of the dual variables and conditions follow quite closely the conventional accounting principles. So, if one changed the specification of the unrealistic dual conditions, he could achieve both goals, i.e., the feasible set would become ‘regularised’ and the solution of the equation would replicate the observed values of the endogenous variables. As a matter of fact, as we will soon show it, this is exactly the purpose of model calibration in the case of the computable general equilibrium models.

(Calibration means the adjustment of parameter values of the model until the output from the model matches an observed set of data.)

What makes the general equilibrium approach feasible is the way we introduced another set of dual variables and constraints (various proportions), which so as to set flexible upper and lower bounds on specific variables. As we have seen, they could be derived from the optimizing behaviour of representative agents (producers, consumers, foreign buyers) put in charge to make analogous decisions. This is in fact their usual interpretation based on neoclassical economics. Our above demonstration should have, however, convinced the reader that it would be more proper to view these dual equations as describing the behaviour of the structural variables, rather than of some mysterious representative agents.

This would not contradict the fact that we borrowed concepts and tools for our macro model from microeconomic theories. It will not make the indisputably macro model, built up from macroeconomic aggregates, into a microeconomic construct either. What justifies the use of microeconomic rules of behaviour in a genuinely macroeconomic model is that they provide a convenient way to link together, and at the same time limit the movement of certain macroeconomic variables, as we have pointed out (see, flexible bounds).

2.3. The concept and the main building blocks of the CGE models

In document CGE Modelling: A training material (Pldal 80-85)