The model considers 5 economic sectors, it is assumed that each one produces a homogeneous good through a production function nested in four levels. First, the added value of sector j is generated by combining primary factors (labor and capital); next, the demand for the four types of occupation is determined; then the internal production of sector j is assembled, using intermediate goods, and added value y; finally, the country and the imports are combined to obtain the total supply of the good provided by sector j.

Therefore, the optimization process that firms follow to make their decisions is implemented in four levels or stages. In the first stage, company j chooses how much to demand of labor (L_j) and of capital (K_j), minimizing the cost of generating added value (VA_j) subject to the technological restriction, taking as given wages (VA_j) and the capital rate R:

M i n   w j * L J + R * K j s . t .     V A j = A A j K j α j J j 1 - α j f o r   j = 1 ,   2 ,   3 ,   4   a n d   5 . ;

Where AA_j is the coefficient of the value-added function of sector j.

The value-added is generated by combining labor and capital, using a Cobb Douglas technology with constant returns to scale. Thus, substitution is allowed between primary inputs, that is, labor and capital. As a result of this process, the derived demands from factors are obtained as a function of the level of value added and the relative price of labor and capital:

(1) L j = V A j A A j α j 1 - α j - α j w j R - α j

(2) K j = V A j A A j α j 1 - α j 1 - α j w j R 1 - α j

In the next stage, firm j decides how much to hire for each type of occupation ι, (L_ι,j) by minimizing the cost of hiring work subject to the Leontief-type technological restriction with constant returns to scale, taking as given the payment received by each type of occupation (wo_ι)

M i n ∑ ι = 1 4 w o ι * L ι , j s . a . L j = M i n L 1 , j γ 1 , j , … , L ι , j γ i , j , … , L 4 , j γ 4 , j p a r a   j = 1,2 ,   3 ,   4,5 ; e   ι = 1,2 , 3,4 .

Thus, γ_ι,j is the requirement of the type of occupation ι by sector j. The derived demands from economic sector j of each type of occupation are:

(3) L ι , j = γ ι , j * L j

On the other hand, sector j decides how much to demand from intermediate goods provided by itself and other sectors (x_i,j), as well as value added (VA_j), by minimizing the cost of domestic production subject to technological restriction and taking as given the prices of intermediate goods (P_i) and value added (PV_j):

M i n ∑ i = 1 5 P i * x i , j + P V j * V A j s . t . Y j = M i n x 1 , j a 1 , j , … , x i , j a i , j , … , x 5 , j a 5 , j , V A j V j f o r   j = 1,2 ,   3 ,   4,5 ; e   i = 1,2 , 3,4 , 5 .

Where the production of sector j (Y_j) uses intermediate goods and value added in fixed proportions through a Leontief-type function; such that a_i,j is the requirement of the input sold by sector i to produce a unit of the good of sector j and, v_j is the necessary amount of value added per unit of product of sector j. In this sense, the demands for intermediate and value-added goods depend solely on the level of product, they are not affected by relative prices since they are complementary:

(4) x i , j = a i , j * Y j

(5) V A j = v j * Y j

Finally, the sector chooses the level of domestic

(Y_j) and external (M_j) production or imports that minimizes the total cost of offering the good within the country, regardless of its country of origin, subject to technological restriction and taking as given the prices of internal (P_j) and external (PM _j) production:

M i n   P j * Y j + P M j * M j s . t . Q j = B Q j Y j β j M j 1 - β j f o r   j = 1,2 ,   3 ,   4,5 .

Where B_j is the coefficient of the total production function of sector j. We employ the Armington’s (1969) assumption impliying that goods from different countries are imperfect substitutes, thus countries produce and import the same goods but with different qualities. In this way, the total supply of sector j in the region (Q_j) is obtained by combining domestic production (Y_j) and imports (M_j), assuming a Cobb Douglas function with constant returns to scale, which allows a certain degree of substitution between them, this type of function is known as an Armington Aggregator. Thus, the demands for national production and imports are a function of the level of total supply and its relative prices:

(6) Y j = Q j B Q j * β j * P M j 1 - β j * P j 1 - β j

(7) M j = Q j B Q j * β j * P M j 1 - β j * P j - β j

Households in expenditure quintile h make their decisions following a two-level optimization process. At the first level, they choose aggregate consumption C_h and saving (S_h), maximizing their utility subject to their disposable income (ID_h ). It is assumed that the utility functions are of the Cobb Douglas type and they are homogeneous of degree one, and that households take the prices of the aggregate consumer good (PH_h) and of savings (PS) as given:

M a x   U h = C h ω h S h 1 - ω h s . t . I D h = P H h * C h * 1 + T C h f o r   h = 1,2 ,   3 ,   4,5 .

Where the subscript h takes the value of 1 if it is the first quintile of spending, the value of 2 for the second quintile and so on, up to the value 5 that is the richest quintile. Note that each type of household faces its own price level for the aggregate consumer good, since these prices are calculated by weighting the prices of final goods according to their spending patterns, as will be detailed later in the prices section.

Thus, the optimal choices for aggregate consumption and savings are a function of disposable income (ID_h) and prices:

(8) C h = ω h * I D h P H h * 1 + T C h

(9) S h = 1 - ω h * I D h P S

At the next level, they choose how much to consume of each final good (c_i,h), minimizing total consumption expenditure, given the prices of said goods (PQ_i), subject to the aggregate consumption level that was optimal at the first level C_h. It is assumed that the total household consumption h is an aggregate of final goods, with a functional form of the Cobb Douglas type homogeneous of degree one. Hence, the optimization process in the second level is:

M i n ∑ i = 1 5 P Q i * c i , h s . t . C h = Ω h ∏ i = 1 5 c i , h ω i , h f o r   h = 1,2 ,   3 ,   4,5 ; a n d   i = 1,2 , 3,4 , 5 .

Whete 0 ≤ ω_i,h < 1,∑⁵_i=1ω_i,h = 1 and Ω_h is the coefficient of the aggregate consumption function of household in the expenditure quintile h.

The subscript i identifies the final good provided by economic sector i, which are 5 in total. Thus, the optimal levels of consumption in final goods are:

(10) c i , h ω i , h C h P Q i * P H h

The total income of households comes from the payments they receive for being the owners of the productive factors, labor (L_h,l) and capital (K_h), transfers they receive from the government (TR_h) and the net factorial income (including remittances) received by the household in expenditure quintile h (REM_h).

(11) I T h = ∑ l = 1 4 w o l * L h , l + R * K h + T R h + R E M h f o r   h = 1,2 , 3,4 , 5 ;   ι = 1,2 , 3,4 .

Where ι denotes the types of occupation, which are a total of 4 in the model. On the other hand, wo_l is the salary paid to the type of occupation ι; and R is the rent paid to capital.

Families contribute to the government sector paying an income tax for the rent of labor and capital (TH_h); therefore, disposable income is:

(12) I D h = 1 - T H h * I G h + T R h + R E M h

The taxable income (IG_h) is:

(13) I G h = ∑ l = 1 4 w o l * L h , l + R * K h

In this economy, there is only one investment good, which is an aggregate of goods provided by the economic sectors (IDA), and which is demanded for investment by the public sector and the private sector. It is assumed that (IDA) is a Leontief-type function with constant returns to scale meaning that a fixed proportion of goods from sector I (φ_i) is required to be used for investment, such that:

(14) I D A = M i n I 1 φ 1 , … , I i φ i , … , I 6 φ 6

In this form, the demand of each good of sector i to invest (I_i) is decided minimizing the total investment expenditure given the prices of said goods (PQ_i), subject to the level of aggregate investment demand (IDA). The optimization process is:

M i n ∑ i = 1 6 P i * I i s . t . I D A = M i n   I 1 φ 1 , … , I i φ i , … , I 6 φ 6 f o r   i = 1,2 ,   3 , … , 6 .

Hence, the investment demand for products of the sector i is:

(15) I i = φ i * I D A

From the Walras law, in equilibrium investment is equal to total savings, that is the addition of households, government and external savings.

Like investment, the government decides how much to demand for each good in sector i (G_i), minimizing total spending given prices (PQ_i), subject to the level of demand for goods and government services (GDA). The optimization process is:

M i n ∑ i = 1 6 P i * G i s . t . G D A = M i n   G 1 ε 1 , … , G i ε i , … , G 6 ε 6 f o r   i = 1,2 ,   3 , … , 6 .

Hence, the government demand for products of the sector i is:

(16) G i = ε i * G D A

On the other hand, government spending is taken as a proportion of total government receipts.

It is assumed that the economic sectors have a certain degree of market power, so the demand for exports depends on the price of foreign goods in relation to domestic ones:

(17) E X P i = E X P 0 i * P W i P i μ

Where µ is the elasticity of export demand with respect to the relative price of foreign and domestic goods.

The model assumes perfect competition, that is, all the agents of the model make their decisions considering that they cannot affect the prices of products and productive factors. Therefore, consumption prices equal unit expenditure, while production prices equal unit costs. In this sense, the equilibrium prices result from substituting the optimal ones in the respective unit cost and expense functions.

The value-added price of private goods (PV_j) is obtained by substituting the derived demands from primary factors in the respective unit cost functions of generating added value:

(18) P V j = 1 A A j W j α j α j R 1 - α j 1 - α j

Where

(19) W j = ∑ l = 1 4 w o l * γ l , j

The price of domestic production follows the specification of the price-forming equation of a linear model, since a Leontief-type production function was assumed:

(20) P Y j = 1 + T P j * ∑ i = 1 5 a i j * P Y j + v j * P V j

Where TP_j are the taxes on production net of subsidies charged by the government in sector j.

The price of total production of sector j results from introducing the equilibrium levels of domestic and imported production in the unit cost of production:

(21) P Q j = 1 B j P m j 1 - β j 1 - β j P Y j β j β j

The price of the aggregate consumer good (PH_h) faced by each quintile, results from introducing the optimal levels of family consumption in final goods within the unit consumption expenditure:

(22) P H h = 1 Ω h ∏ I = 1 6 P Q i ω i , h ω i , h

The price of the investment good PS is a weighted average of the prices of goods provided by the productive sectors:

(23) P S = ∑ i = 1 6 φ i * P i