Mathematıcal Economıcs (ENG) — Ünite 5 Soru-Cevap

Optimization is the general term used for processes involving a search for a maximum (in the context of a maximization process) or a minimum (in the context of a minimization process) among a number of alternatives.

An optimization problem calls for finding the best choice among several alternatives.

Both maximum and minimum values are
extreme magnitudes (extremums or extrema)
that are distinctly different from comparable
magnitudes observed. Maximums (maxima)
or minimums (minima) or, shortly, optimums
(optima) can be local or global.

An optimal value
is a local or relative optimum if it has the highest
(maximum) or the lowest (minimum) value within
a subset of all possible values. An optimal value is
a global or absolute optimum if it has the highest
(maximum) or the lowest (minimum) value within
the set of all possible values.

Domains of functions used in economic analysis are usually restricted since economists typically deal
with choice variables that take only non-negative values (or a subset thereof ).

The purpose of the first derivative test is to find critical values of choice variables that will potentially
allow the objective function to reach its maximum or minimum value. Critical values in this context are
those values of choice variables at which the first derivative of the function is equal to 0 or does not exist.
Those values are candidates for maximizing or minimizing the value of the optimization object when
inserted into the objective function as arguments.

Geometrically, the first derivative of a function (such as f ) with a single argument (such as x) evaluated
at a particular value of x corresponds to the slope of the function’s graph.

To find the critical values of x, its first derivative is set to zero.

Setting the first derivative equal
to zero to find the critical values of x, therefore, amounts to finding the values of x at which the slope of the
function is zero. A zero slope, in turn, means that the function is neither increasing nor decreasing at that
point. Alternatively, it must have just completed its ascent or descent to reach a maximum or a minimum.

In general, for a critical value x* of the choice variable, if y* = f (x*) is greater
than all y+/- = f (x* ∓ ε) for arbitrarily small values of real number ε > 0, then y*
is a local maximum.

If y* = f (x*) is smaller than all y+/- = f (x* ∓ ε) for arbitrarily
small values of real number ε > 0, then y* is a local minimum.

We can generalize the argument, in fact, and note that if the sign of the first derivative (or the slope)
changes from negative to positive around a critical value of the choice variable, then that critical value
corresponds to a local minimum. Alternatively, if the sign of the first derivative (or the slope) changes
from positive to negative, then, the critical value at which the switch of sign occurs corresponds to a local
maximum.

Since the second derivative f ‘‘(x) ≡ y‘‘ measures the rate of change in the slope dy dx of the function, it can be expressed as y '' ≡ dy '/ dx ≡ d( dy/ dx ) dx ≡ d2 y /dx2 .

So, for a critical value of the choice variable at which f ‘(x*)=0 or does not exist, the secondderivative
test will point to
• the existence of a local maximum valued y* = f (x*) at x = x* if f ‘‘(x) ≡ y‘‘ <0
• the existence of a local minimum valued y* = f (x*) at x = x* if f ‘‘(x) ≡ y‘‘ <0
• an indeterminacy about the status of the value of the objective function f (x*) at x = x*
if f ‘‘(x) ≡ y‘‘=0 –in which case one must resort to the more general nth order derivative
test described shortly

For a critical value x* of the choice variable at which f ‘(x*)=0 , if the second derivative f ‘’(x*) is also
zero, one must continue taking the higher-order derivatives until coming across a non-zero value. If
the first non-zero derivative turns out to be the nth derivative such that f (n)(x*)≠0, then the nth-order
derivative test will conclude that
• y*= f (x*) will be a local maximum if n is an even number and f (n)(x)<0
• y*= f (x*) will be a local minimum if n is an even number and f (n)(x)>0
• there exists an inflection (saddle) point if n is an odd number
an indeterminacy about the status of the value of the objective function f (x*) at x = x* if f ‘‘(x) ≡ y‘‘=0
–in which case one must resort to the more general nth order derivative test described shortly.

This function has two first-order partial derivatives with respect to x and w, measuring the rate of change in z resulting from a tiny change in one decision variable at a time, holding the other constant.

fx ≡ ∂ f (x,w)/ ∂x ≡ ∂z ∂x ≡ zx

fw ≡ ∂ f (x,w) /∂w ≡ ∂z ∂w ≡ zw

Given z = f (x, w), the total differential dz of z showing the total change
in z is given by
dz=fx dx+fwdw
The first product on the right-hand side (fxdx) signifies the contribution
of the total change in x to dz, whereas the second product (fwdw) signifies
the contribution of the total change in w to dz.

The so-called Lagrange-multiplier method is the most common method used in economics to solve
problems like this constrained utility maximization Named after Italian mathematician Joseph Louis
Lagrange, Lagrange-multiplier is a popular tool used to convert constrained optimization problems into
unconstrained optimization problems, allowing them to be solved using the techniques already developed
for solving unconstrained optimization problems. The Lagrange-multiplier method is only applicable to
problems with equality constraints. It is based on the idea that if the optimal solution is to occur on the
constraint itself, then the information contained in the constraint could be incorporated into the objective
function to end up with an unconstrained optimization problem that will give the same solution as the
original constrained optimization problem.