Lecture summary. Topic 4. Partial differentiation

Textbook:Ian Jacques “Mathematics for Economics and Business”, 6th Edition (2009). Available in the TTU library.

4.1. Functions of several variables

First-order partial derivatives∂ z∂x

∂ f∂x

f x the partial derivative of f with respect to x;it is found by differentiating f with respect to x, with y held constant

∂ z∂ y

∂ f∂ y

f y the partial derivative of f with respect to y;it is found by differentiating f with respect to y, with x held constant

∂ f∂ x

We read: partial deef by deex

Second-order partial derivativesWhen we differentiate a function of 2 variables, the thing we end up with is itself a function

of 2 variables. That suggests a possibility of differentiation a second time. And there are 4 second-order partial derivatives:

∂2 z∂x2

∂2 f∂ x2

f xx obtained by differentiating f twice with respect to x, with y held constant

∂2 z∂ y2

∂2 f∂ y2

f yy obtained by differentiating f twice with respect to y, with x held constant

∂2 z∂ y∂ x

∂2 f∂ y∂ x

f yx obtained by differentiating first with respect to x and then with respect to y

∂2 z∂x ∂ y

∂2 f∂ x ∂ y

f xy obtained by differentiating first with respect to y and then with respect to x

∂2 f∂ x2

We read: second-order partial deef by deex

f xy ¿ f yx Differentiating with respect to x then y gives the same expression as differentiating with respect to y then x

n first-order partial derivativesWhen we deal with functions of more than 2 variables y= f (x1 , x2 ,…,xn) then we obtain

nfirst-order partial derivatives ∂ f∂ x i (or f i) by differentiating with respect to one variable at a time,

keeping theremaining n-1 variables fixed. The second order partial derivatives are determined in analogues way.

Interpretation of a partial derivativeFor a function of two variables z=f (x , y ):

if x changes by ∆ x and y is fixed then the corresponding change in z satisfies: ∆ z= ∂z

∂ x∆ x

if y changes by ∆ y and x is fixed then the corresponding change in z satisfies: ∆ z= ∂ z

∂ y∆ y

If x and y change simultaneously, the net change in z will be:∆ z= ∂ z

∂ x∆ x+ ∂ z

∂ y∆ y or ∆ z= ∂z∂ x dx+

∂ z∂ ydysmall increment formula

dx ,dy ,dz are differentials1

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Implicit differentiation

iff ( x , y )=constant then dydx

=− f xf y

The technique of finding dy /dx from −f x / f yis calledimplicit differentiation and can be used whenever it is difficult or impossible to obtain an explicit representation for y in terms of x.

Read Textbook, p. 370-381.

4.2. Unconstrained optimizationThe method of finding and classifying stationary points of a function f (x , y ) without

constraints is as follows:Step 1. Solve the simultaneous equations to find the stationary points, (a ,b):

f x ( x , y )=0f y ( x , y )=0

Step 2. If: f xx>0, f yy>0 and f xx f yy−f xy2 >0 at (a ,b) then the function has a minimum at (a ,b) f xx<0, f yy<0 and f xx f yy−f xy2 >0 at (a ,b) then the function has a maximum at (a ,b) f xx f yy−f xy

2 <0 at (a ,b) then the function has a saddle point at (a ,b)

Read Textbook, p. 415-418.

4.3. Constrained optimization. Lagrange multiplierAlgorithm of constrained optimization

To optimize a function z=f (x , y ) subject to constraint φ ( x , y )=M we work as follows:Step 1. Use the constraint φ ( x , y )=M to express y in terms of x.Step 2. Substitute this expression for y into the objective function z=f (x , y ) to write z as a

function of x only.Step 3. Use the theory of stationary points of functions for one variable o optimize z.

From Question 4.3: How do we find stationary points of a function for one variable?Step 1. Differentiate the function, equate the result to zero and solve the equation: f ' ( x )=0Step 2. Differentiate the function a second time and evaluate this second-order derivative at each point.

if f ' ' (a )>0 then f ( x ) as a minimum at x=a if f ' ' (a )<0 then f ( x ) as a maximum at x=a iff ' ' (a )=0 then the point can not be classified using the available information.

Read Textbook, p.430-435.

Lagrange multiplierLagrange multiplier is applied for solving constrained optimization problems. This is a

preferred method, since it handles non-linear constraints and problems involving more than two variables with ease. It also provides some additional information that is useful when solving economic problems.

Algorithm of Lagrange multiplier methodTo optimize an objective function f (x , y ) subject to constraint φ ( x , y )=M we work as follows:Step 1. Define a new function g ( x , y , λ )=f ( x , y )+λ [M−φ(x , y )]

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Dr. Veronika alhanaqtah. mathematics for economics

Here we combine the objective function and constraint into a g function.To do this we first rearrange M−φ(x , y ) and multiply by the scalar λ. This scalar is called Lagrange multiplier. Then we add on the objective function to produce the new function g ( x , y , λ ), which is called the Lagrangian function.

Step 2. Solve the simultaneous equations for the three unknowns, x, yand λ:∂g∂ x

=0

∂ g∂ y

=0

∂g∂ λ

=0

Here we work out the three first-order partial derivatives andequate these tozero to produce a system of three simultaneous equations for the three unknowns, x, yand λ. The point (x, y) is the optimal solution of the constrained problem.

* It is possible to make use of second-order partial derivatives to classify the optimal point. But these conditions are quite complicated (considered in advanced course of Mathematics).

Interpretation of Lagrange multiplierGenerally: the value of λ gives the approximate change in the optimal value of an objective

function due to a 1 unit increase in M. Advanced:

often the Lagrange multipliers have an interpretation as some quantity of interest.For example,  λis the rate of change of the quantity being optimized as a function of the constraint variable;

the optimal value of a Lagrange multiplier has an interpretation as the marginal effect of the corresponding constraint constant upon the optimal attainable value of the original objective function. For example, in economics the optimal profit to a player is calculated subject to a constrained space of actions, where a Lagrange multiplier is the change in the optimal value of the objective function (profit) due to the relaxation of a given constraint (e.g. through a change in income); in such a context λis the marginal cost of the constraint, and is referred to as the shadow price.

Read Textbook, p. 442-447.

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Dr. Veronika alhanaqtah. mathematics for economics