Hrvatsko društvo ekonomista

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January 2005. ::: Vol.56 No.01-02

Ante Puljić
Ilko Vrankić

FROM TOTAL DIFFERENTIAL TO MAXIMAL ECONOMIC PROFIT Izvorni znanstveni članak The intention of this article is to explain in the most concise and simple way the optimization of the unconstrained function. This paper thus first explains the critical point term, the point in which the total differential of the function is equal to zero and in which the value of the function momentarily neither grows nor falls. After establishing the stationary point, sign of the second order differential, which takes the quadratic form, determines whether the function in the critical point has relative maximal or relative minimal value. If the Hessian matrix of such envisioned quadratic form in critical point is negative definite, then total second order differential in this point is less than zero and the function in the critical point has relative maximal value, and if that matrix is positive definite, then the function in critical point has relative minimal value. Special attention is given to the clarification that the matrix in question is negative defi nite when the values of its leading principal minors of odd order are less than zero and values of leading principal minors of even order are greater than zero, while the matrix is positively defi nite when values of all of the leading principal minors are greater than zero. Unconstrained function optimization is applied to two economic profit models, on the model that is built on the assumption that the function of total minimal economic costs is known and on the model built on the assumption that the production function is known. In the former model the decision about critical production quantity that maximizes economic profit is being made, while in the latter model the decision about critical quantity of factors of production which maximize economic profit is being made. In the paper we prove that optimization of thus formulated models of economic profit lead to equal maximal economic. total differential; stationary point; function’s extreme value; concavity and convexity; quadratic form; Hessian matrix; positive/negative definite quadratic form; critical production quantity; critical production factors quantity; economic costs; econo Puni tekst (Hrvatski) Str. 3 - 38 (pdf, 589.23 KB)

FROM TOTAL DIFFERENTIAL TO MAXIMAL ECONOMIC PROFIT

Izvorni znanstveni članak

The intention of this article is to explain in the most concise and simple way the optimization of the unconstrained function. This paper thus first explains the critical point term, the point in which the total differential of the function is equal to zero and in which the value of the function momentarily neither grows nor falls. After establishing the stationary point, sign of the second order differential, which takes the quadratic form, determines whether the function in the critical point has relative maximal or relative minimal value. If the Hessian matrix of such envisioned quadratic form in critical point is negative definite, then total second order differential in this point is less than zero and the function in the critical point has relative maximal value, and if that matrix is positive definite, then the function in critical point has relative minimal value. Special attention is given to the clarification that the matrix in question is negative defi nite when the values of its leading principal minors of odd order are less than zero and values of leading principal minors of even order are greater than zero, while the matrix is positively defi nite when values of all of the leading principal minors are greater than zero. Unconstrained function optimization is applied to two economic profit models, on the model that is built on the assumption that the function of total minimal economic costs is known and on the model built on the assumption that the production function is known. In the former model the decision about critical production quantity that maximizes economic profit is being made, while in the latter model the decision about critical quantity of factors of production which maximize economic profit is being made. In the paper we prove that optimization of thus formulated models of economic profit lead to equal maximal economic.

total differential; stationary point; function’s extreme value; concavity and convexity; quadratic form; Hessian matrix; positive/negative definite quadratic form; critical production quantity; critical production factors quantity; economic costs; econo

Puni tekst (Hrvatski) Str. 3 - 38 (pdf, 589.23 KB)