MATHEMATICAL ECONOMICS IN THE EXPLANATION OF ECONOMIC GROWTH IN ECONOMIES WITH ENDOGENOUS AND EXOGENOUS TECHNOLOGICAL CHANGE LA ECONOMÍA MATEMÁTICA EN LA EXPLICACIÓN DEL CRECIMIENTO ECONÓMICO EN ECONOMÍAS CON CAMBIO TECNOLÓGICO ENDÓGENO Y EXÓGENO

Economic growth is a function of the interactions between the different productive factors framed in the economic policy of an economy, in particular, it can be expressed in terms of labour force, productive resources (land, capital) and 1 Universidad Francisco de Paula Santander Doctor en Educación ORCID iD: https://orcid.org/0000-0003-4377-3903 henrygallardo@ufps.edu.co 2 Universidad Francisco de Paula Santander Doctora en Educación ORCID iD: https://orcid.org/0000-0001-8285-2968 mawencyvergel@ufps.edu.co 3 Universidad Francisco de Paula Santander Magister en Ingeniería Civil ORCID iD: https://orcid.org/0000-0003-2682-9880 jhanpierorojas@ufps.edu.co technology, among others. The present work pretends to approximate a model to explain the economic growth in developing economies, for which a model is proposed that explains this growth in function of the referred factors; then production is proposed in function of capital and work and two models are adjusted, one with exogenous technological change and the other that involves technological change in an endogenous manner. The model is developed with a production function with constant substitution elasticity so that it is applicable to both developed and developing economies, since it is to be expected that in developed economies the substitution elasticity is unitary, which would lead

with growth rate n=L(t)/L(t) given exogenously. (1) In equation 1 we include a utility function with elasticity of substitution between present consumption and constant future consumption, which allows us to describe the preferences among them. large, there is low elasticity between present and future consumption, this implies great preference for present consumption and low response of savings to the interest rate with high aversion to risk, which is to be expected in poor economies.
The production in a period t, is determined by the level of technology, A(t), present in that period and by the levels of capital, K(t), and work, L(t), used in that period. Equation 2 describes the level of production, based on the variables described, using a production function with constant substitution elasticity, in which the relative share of each factor in the final product is determined by the distribution parameter and the elasticity of substitution in production is given by where is the substitution parameter The elasticity of substitution is a local measure of substitution is a local measure of the substitution between the factors of production, that is, for a given level of production, measures the proportional change in the use of factors as a result of a proportional change in the marginal rate of labor substitution by capital (TMS), it can also be understood as a measure of the percentage change in the rate of use of the factors of with growth rate =̇( ) ( ) ⁄ given exogenously.
h, of the workers are generated is given by: with growth rate =̇( ) ( ) ⁄ given exogenously.
h, of the workers are generated is given by: with growth rate =̇( ) ( ) ⁄ given exogenously. given by: The optimal trajectory of capital and its exchange rate, , the rate of growth of capital is given by , which implies that the variation of capital is: . If is large parameter (0 ≤ ≤ 1) and the by = 1 (1 + ) ⁄ , given by: given by capital is: and it is expressed as follows: and given by = ( , ).
h, of the workers are generated is given by: with growth rate =̇( ) ( ) ⁄ given exogenously.
. If is large parameter (0 ≤ ≤ 1) and the by = 1 (1 + ) ⁄ , given by: given by capital is: and it is expressed as follows: and given by = ( , ).
h, of the workers are generated is given by: rowth rate =̇( ) ( ) ⁄ given exogenously.
. If is large eter (0 ≤ ≤ 1) and the by = 1 (1 + ) ⁄ , l is: and it is expressed as follows: he workers are generated is given by: with growth rate =̇( ) ( ) ⁄ given exogenously.
. If is large parameter (0 ≤ ≤ 1) and the by = 1 (1 + ) ⁄ , given by: given by capital is: and it is expressed as follows: h, of the workers are generated is given by: enously.
. If is large parameter (0 ≤ ≤ 1) and the by given by: given by capital is: and it is expressed as follows: . If is large arameter (0 ≤ ≤ 1) and the by ven by pital is: and it is expressed as follows: . If is large parameter (0 ≤ ≤ 1) and the by = 1 (1 + ) ⁄ , given by: given by capital is: and it is expressed as follows: . If is large parameter (0 ≤ ≤ 1) and the by = 1 (1 + ) ⁄ , given by: given by capital is: and it is expressed as follows: The growth rate of the economy is obtained from and it is expressed as follows: Note that these rates are constant only in the case that the elasticity of substitution in production is 1.
Its implies that three different cases must be considered (depending on the elasticity of substitution) in order to determine the rate at which per capita consumption must grow in order to satisfy the differential equation and stability of the model is achieved in the long term.
Note that when the elasticity of substitution is less than unity, the growth rate of capital is decreasing and tends to stabilize in the long term.
capital is: and it is expressed as follows: . production is h, of the workers are generated is given by:    The objective now is to determine the trajectories that the per capita consumption c(t) and the production effort u(t) must follow over time in order to maximize the intertemporal utility function at all times. That is, the following problem must be solved: · 1 0 6 ·

MATHEMATICAL ECONOMICS IN THE EXPLANATION OF ECONOMIC GROWTH IN ECONOMIES WITH ENDOGENOUS AND EXOGENOUS TECHNOLOGICAL CHANGE
in which c(t) and u(t) are the control variables and K(t) and h(t) are the state variables.
As in the previous case, the Pontriagyn maximization principle is applied and it is obtained that the optimal solution must satisfy the following necessary conditions: In the margin, goods must be valued equally in their two uses: consumption and capital accumulation. Time must also be valued in its two uses: production and capital accumulation.
The exchange rate of the efficiency price of physical capital must be equal to the discount rate minus the marginal productivity of capital.
The system of simultaneous non-linear differential equations must be given regularity conditions in order to obtain results consistent with economic theory. Under the assumption that the growth rate of per capita consumption is constant over time, it is obtained that the optimal trajectory of the product to capital ratio is constant over time, which implies that the optimal requires that the growth rate of the economy be equal to the growth rate of capital. The optimal capital trajectory must grow at a constant rate given by the sum of the growth rates of labour force and human capital. In the long run, human capital must grow at the same rate as the economy.
The growth of per capita consumption is equal to the growth of technology minus the discount rate plus the growth rate of the labor force, multiplied by the rate of substitution between present consumption and future consumption; which implies that the system grows more to the extent that there is more preference for future consumption.

MATHEMATICS
The production of a country in time period t is determined by the level of technology, A(t), present in that time period and by the levels of capital and labour used in that period. The production level is described by equation (2) which represents a production function with

K(t)>L(T) or tends to infinity if K(t)<L(t).
In this borderline case, the situation is impossible and therefore the curvature of the isocuantas appears at right angles. The CES production function tends to be a Leontief production function.
Second case: when you have to Note that if ( ) → 0, it is obtained that ( ) − → ∞; therefore ( ) and ( ) cannot be zero. Fifth case: when → −1 one has to → ∞; in the exponents of the terms on the left are positive.
Therefore, the isoquantas cut both axes. That is, when and when Fifth case: when α→ -1 one has to in the limit, both exponents of the isoquantas are one. The isoquantas are straight lines. In this case, the production factors are perfect substitutes.

CONCLUSIONS
When the elasticity of substitution in production is greater than 1, the economy will experience high savings rates that in turn push up the interest rate, which leads to a deterioration in the distribution of income causing the rich to appropriate each time a greater proportion of income than the poor. There is also a decrease in the consumption to capital ratio, that is, over time a decrease in the level of consumption is experienced with respect to the level of capital present in the economy.
Conversely, when the elasticity of substitution in production is less than unity, the saving rates will be increasingly low, which leads to an economy