A stylised CGE model based on problem 2.2-2

The previous section has paved the way for reformulating the optimum conditions of the studied optimal resource allocation problem into a set of equilibrium conditions. We will use the equations characterizing the optimum conditions, revised in the way suggested above, organized into the same 26 blocks as before. We will simply tell the story of the origin of these conditions using the language and terminology of general equilibrium theory, and following its logic, rebuild the model using the typical building blocks of the CGE models.

It will be ease to identify the equations, and the following summary table will help the reader to recall the meaning of the 27 blocks of core variables used. We call them core variables, because the 26 blocks of equations define yet not a complete stylised CGE model, and we will have to introduce further equations and variables to complete the model.

Endogenous variables of the core model

xj (composite) production levels pih

producers' prices of domestic sales xjh

production supplied on domestic markets pja

average producers' prices

zj volume of exports pim

domestic prices of imports sjd

Share coefficients of domestic sales piwe

world market prices of exports sje

share coefficients of exports pie

domestic prices of exports xihm

(composite) supply on domestic markets pihm

average users' prices net of taxes

mi volume of imports pic

consumers' prices sih

share coefficients of domestic supply pcv consumers' price index sim

share coefficients of imports pjb

prices of capital goods sicv

structure of variable consumption qj cost of capital ycv level of variable consumption wj cost of labour

l_j labour input coefficients w general level of wages k_j capital input coefficients π ^net rate of return on capital

v exchange rate PRIMARY GOODS, DOMESTICALLY PRODUCED AND IMPORTED COMMODITIES

There are two types of commodities: n kinds of goods produced in n sectors, and two kinds of primary resources (labour and capital) with exogenously given supply (L0 and K0). It is assumed that all users of the various commodities minimize their costs. Domestic products compete with imported commodities of the same sort in each sector, and that they are imperfect substitutes of each other, making up a special composite commodity (the so-called Armington assumption). The joint use value, that is, the volume of the domestic/import composite commodity of sectoral origin i is given as

(P3) xihm domestically produced and imported component, respectively.

Domestic outputs as well as imported goods are assumed to be composed of homogenous commodities themselves. xihm

is a linear homogeneous function and all users to minimize their cost, therefore each user will use domestic and imported goods of the same variety in the same proportion: mi/xih closed forms, by the following unit demand functions (homogeneous of degree zero):

(P17) sih

are the component prices, given in the cost minimizing exercise, and (P11’) pim

The demand for sectoral goods is always given in terms of the domestic/import composite commodity. In the case of linear homogeneous production (aggregation) functions, the only prices, which are compatible with the assumption of profit maximization and equilibrium, are such that are equal to their minimal cost. Therefore, the unit price of the domestic/import composite good of sector origin i (pihm

) must also be equal to their minimal combine cost (the value of the Lagrange multiplier in the cost minimizing problem):

(P13) pihm

THE REPRESENTATION OF PRODUCTION AND EXPORTS

Production is organized into n sectors, each producing jointly two commodities of the same sector origin, close, but not perfect substitutes for each other. One is supplied on domestic, the other on foreign markets. The joint level of the output in sector j is given by the set of

(P2) x_j = x_j(x_j^h, z_j), j = 1, 2, ... , n, (E7)

linear homogeneous transformation (aggregation) function, and it can be interpreted as measuring the volume of a special home/export composite good. One unit of such composite commodity can be achieved by various combinations of the two goods, which must satisfy the condition x_i(s_i^d, s_i^e) = 1.

The home/export composite commodities are produced by means of domestic/import composite commodities (a_ij fixed unit input coefficients), and primary resources (l_j and k_j variable unit input coefficients). Various combinations of labour and capital can provide the same level of capacity, defined by f_j(K_j, L_j), linear homogeneous (constant return to scale) production functions. The scale of these functions is set in such a way that the production of one unit home/export composite good requires (at least) one unit of the composite factor:

f_j(l_j, k_j) = 1, j = 1, 2, ... , n.

The overall production function and capacity constraint in sector j takes, thus, the following form: solve a revenue maximizing exercise taking the prices offered on the two markets (p_j^h and p_j^e) as given. We assume that these problems, too, have always unique solutions, and the revenue maximizing compositions, at unit level of the outputs, can be expressed in closed forms, by the following supply functions (homogeneous of degree zero):

(P19) s_j^d = s_j^d(p_j^h, p_j^e), j = 1, 2, ... , n, (E8) (P20) s_j^e = s_j^e(p_j^h, p_j^e), j = 1, 2, ... , n, (E9) where p_j^h and p_j^m are the component prices, given in the cost minimizing exercise, and

(P12’) p_j^e = (1+τje

)·v·p_j^we, j = 1, 2, ... , n, (E10)

and the world market price of exports is given by the following inverse demand functions:

(P21) p_j^we = p_j^we(z_j), j = 1, 2, ... , n. (E11) Based on the above forms, the ratio of exports to domestic supply can be defined as follows:

(P4) z_j = x_j^h·s_j^e/s_j^d, j = 1, 2, ... , n, (E12)

The revenue achieved in sector j by selling one unit of output in optimal proportions can be expressed as

(P10) p_j^a = p_j^h·s_j^d + p_j^e·s_j^e, j = 1, 2, ... , n, (E13) which is equal to the value of the Lagrange multiplier in the revenue maximizing problem, and it defines the unit price of the domestic/export composite output in sector j (p_j^a).

Similarly, producers choose such combination of labour and capital that minimizes their joint cost. The labour and capital demand per unit of output in sector j is, thus, determined by a cost minimization exercise, in which the unit costs (prices) of labour and capital are given by the following forms:

(P25’) w_j = (1+τjw

)·w·d_j^w, j = 1, 2, ... , n, (E14)

(P15’) q_j = p_j^b·(r_j^a + π^·djπ), j = 1, 2, ... , n. (E15) Here again, it is assumed that functions f_j are all well-behaved and the cost minimizing demand for labour and capital at unit level of the output can be expressed as functions (homogeneous of degree zero) of the factor prices:

(P23’) l_j = l_j(w_j, q_j), j = 1, 2, ... , n, (E16)

(P24’) k_j = k_j(w_j, q_j), j = 1, 2, ... , n. (E17)

The unit cost of the output in sector j will be, thus, as follows:

p_j^ac = Σ_i (1+τiu

)⋅p_i^hm⋅a_ij + w_j⋅l_j + q_j⋅k_j, j = 1, 2, ... , n,

where the value of l_j and k_j is given by the above unit level factor demand functions. Apart from the producers’ taxes and revenues, the unit price of the domestic/export composite output in equilibrium must be equal to its cost, therefore the following equilibrium pricing condition must hold:

(P9’) p_j^a = (1+τjx

)⋅

(

^Σi (1+τiu

)⋅p_i^hm⋅a_ij + w_j⋅l_j + q_j⋅k_j

)

^, j = 1, 2, ... , n. (E18) THE REPRESENTATION OF FINAL DEMAND

Final demand for the domestic/import composite commodities will be grouped into the following components: (variable) private consumption, public consumption, replacement and net investments and the rest in the following way:

FD_i = s_i^cv·y_cv + g_i·y_g +

Σ

j b_ij·(y_j^br + y_j^bn) + c_i⁰.

In the current specification of the model, only the variable part of private consumption (y_cv and s_i^cv variables) and the commodities used for replacement investments (y_j^br variables) will be treated as endogenous variable. The category rest includes, thus, committed consumption as well in addition to the change in stock, if we employ, as usual, a utility function of Stone–

Geary type to explain private consumption decisions.

The levels of the replacement investments in the different sectors (y_j^br) will be defined by the depreciation of the capital used, that is, y_j^br = r_j^a·k_j·x_j. The equilibrium prices of the sector

specific capital goods (p_j^b), which appear in the definition of the cost of capital, are equal to the costs of their formation via the investments, that is,

(P14’) p_j^b =

Σ

i(1+τib

)·p_i^hm·b_ij, j = 1, 2, ... , n. (E19) Consumption decisions are assumed to imitate that of a representative household, having a (linear homogeneous) utility function, which is defined over the set of variable consumption unique and yields the following demand functions:

(P22’) sicv

where the set of the consumers’ prices are given as:

(P26’) p_i^c = (1+τic

)·p_i^hm, i = 1, 2, ... , n. (E21)

The optimal consumption basket will be the numeraire, and its price index is given as (P16’)

Σ

ipic

·sicv

= pcv (= 1), (E22)

where pcv is equal to the value of the Lagrange multiplier used in the above cost minimization exercise, and it will be set to be equal to one.

MARKET CLEARING CONDITIONS AND THE CURRENT ACCOUNT

The above definitions of the behavioural rules and optimal decisions determine the supply and demand of the various goods represented in the model. We can now formulate the market

The last equation represents the current account (the balance of trade). If its balance, de is exogenously set, as in this specification, it behaves as a special commodity (foreign currency) with limited external supply. Therefore, the price associated with it (v) reflects the scarcity of this resource relative to the other primary resources and the final goals, as we have seen it in the programming version of the same model. The only variable part of final demand that can adjust to the production capacity determined by the technology and the available stock of resources (labour, capital and currency, in this case) is variable personal consumption. As we have seen it in the programming version of the model, the utility function of the representative households takes in fact the place of a welfare function to be maximized.

In document CGE Modelling: A training material (Pldal 87-92)