1. Optimization of Multivariate Functions
Chapter 15

2. 15.2 Maximum and Minimum Values
Conditions for a maximum
Consider a function z=f(x,y)
P is the point of maximum at
Section APB is obtained by considering an iso-y section at
Maximizing function
results in us finding
Recall the first- and second-order conditions: at P.
Repeat the analysis for

3. Conditions for a Maximum
The necessary conditions for a maximum of z=f(x,y) to be reached at :

4. Conditions for a Minimum
The necessary conditions for a maximum of z=f(x,y) to be reached at :

5. Saddle Points
At S, we have a maximum on both iso-x and iso-y sections, but considering section ESF reveals this is not a maximum point.
Point S is called saddle point.

6. Sufficient Condition for an Extremum
In addition to the first- and second-order conditions, an additional condition is imposed to exclude the possibility of a saddle point:

7. 15.3 Saddle Points
Definition. A stationary point (i.e. such that at ) is called a saddle point if in a vicinity of z=f(x,y) may be greater and less than
A sufficient condition for a saddle point:
Note 1. This is a sufficient, but not a necessary condition: some functions exist with a saddle point where the condition above is not satisfied.
Note 2. We do not know what happens if

8. Saddle Point: an Example
At point S, an iso-y section has a local minimum (section CD).
At the same point S, an iso-x section has a local maximum (section AB).
For this type of a stationary point to be saddle point, the sufficient condition is:
has an opposite sign with

9. Iso-Sections for Saddle Points
At z=20, the iso-z line has two branches that are tangent to each other at point S.
Point S is a pass—the lowest point on the ridge separating A in one valley and B in another valley.

10. Inflection-Type Saddle Points
At point P, both iso-x and an iso-y sections have an inflection point at P.
The following function has this type of saddle point:

11. An Example Consider a function
Its contours are given in the following graph:
We can show that z=7/3 is a local maximum.

12. Matrices: a Digression
Definition. A table of (real) numbers is called a matrix.
Example.
Notation.
Element of a matrix.
i = row number
j = column number
A matrix with two rows and two columns.
A matrix with three rows and two columns:
Definition. Consider matrix The expression is called a determinant of matrix A.
Note. Determinants are defined for a matrix of any dimension given the number of rows is equal to the number of columns.

13. A Function and its Hessian
Definition. Consider a function z=f(x,y). A matrix of its second-order partial derivatives is called this function’s Hessian:
Note. Note that denoting the Hessian becomes

14. Extrema, Saddle Points, and Hesisans
Consider a determinant of a Hessian pertaining to function z=f(x,y):
The sufficient condition for a maximum or a minimum says that a stationary point must be such that at that point
The sufficient condition for a saddle point will be
In case , we need to consider higher-order Hessians.

15. 15.4 Total Differential
The distance JK is a linear approximation to J’K.
JK=(JK/PK)PK, where and
We can make the approximation error JK-J’K as small as we wish by reducing dx down to a very small number.
Similarly, LM is a linear approximation to L’M, and it’s equal to

16. Simultaneous Changes in X and Y
What happens to the function’s value if x changes by dx and y changes by dy?
If x changes by dx, we move from P to J’.
When y after that changes to dy, we move from J’ to Q.
The total change will be J’K+RQ.
Since dx is small, we can approximate the exact change J’K+RQ by
Definition. A function’s differential at is given by

17. Partial Differential
Definition. The partial differential of z=f(x,y) with respect to x is defined as ,
and the partial differential of z=f(x,y) with respect to y is defined as
Partial differentials are linear approximations to changes in the function z’s value in case only one of the (two) variables changes.
Partial differentials can be thought of as differentials of the iso-x or iso-y sections.

18. Generalization to Many Variables
Consider a function
Its total differential at a particular point is given by
This total differential approximates the change in z when all independent variables change simultaneously.

19. Example 15.2 Consider a function z=xy.
The functions returns the area of a rectangle with sizes and .
Suppose the size increases by dx, and size increases by dy.
The total area of the rectangle increases by A+B+C.
The area of rectangle A is equal to and the area of B is
The true change in the area is equal to
Since , so dz is approximating A+B+C with error

20. 15.5 Differentiating a Function of a Function
Suppose we are given a function , and
Whenever x changes, it produces two effects on z: a direct effect working through , and an indirect effect working through
The direct effect is measured by the partial derivative
The indirect effect is measured by
Total differential approximates the total change in z as follows:
Dividing both sides by dx we obtain the rule for differentiating a function of a function:

21. Total Derivative
Total derivatives, as opposed to the partial derivatives, take account of both direct and indirect effects of a change in an independent variable.
Total derivatives are indiscriminate from the partial derivatives if all arguments of a function are independent variables.
Definition. For any function z=f(x,y) where y=g(x), the total derivative of function f is defined as follows:
Note 1. We write because x is a independent variable, and when we differentiate y with respect to x, we do not hold any other variable constant.
Note 2. The two derivatives and convey very different meanings: the total derivative captures all effects of a change in x, the partial derivative only captures the direct effect of a change in x.

22. 15.6 Marginal Revenue
Re-cap: total revenue is defined as TR=pq where p=f(q), the inverse demand function.
The differential of total revenue is a sum of two influences: a change in the quantity (sell one unit more), and the associated change in the price (price goes down by law of demand):
The direct effect of a change in quantity: pdq.
The indirect effect of a change in quantity:
The total differential approximates the true change in total revenue when quantity changes by dq:
Indirect effect
Divide both sides by dq:
Direct effect

23. Total Revenue and Elasticity
Consider the total derivative of total revenue:
This derivative has a name of marginal revenue. Rearranging, we obtain:

24. 15.7 Differentiating an Implicit Function
Consider an implicit function y=y(x) defined by
Create a new variable:
The total differential of this function is:
Suppose we change x and y in such a way as to leave z constant at the same level z=0.
In this case we jump from one point on the curve described by to another point described by the same equation z=0.
If we produce such a jump, dz=0, and implies
The expression above represents the slope of implicit function given by everywhere except y=0. Why?

25. Derivative of an Implicit Function
In case of an implicit function defined by a relationship f(x,y)=0, the derivative of y(x) at a particular point is given by:
Note is in the nominator, and the minus sign is essential.
In case of three or more variables when y=y(x) is given by F(x,y,v)=0:

26. 15.8 Slope of an Iso-z Section
Consider a function z=f(x,y).
Its iso-z section is given by
This equation defines an implicit function y=h(x) described
by a relationship
Applying the formula for implicit function differentiation, we obtain:
Differentiation of an implicit function given by is equivalent to finding a slope of an iso-z section.

27. 15.9 Shifts Between Iso-z Sections
A change from a point on iso-z section corresponding to to another iso-z section corresponding to results in a change in the function’s value that can be approximated by total differential:
In case z changes exogenously, it triggers changes in x and y of the size dx and dy, respectively.
For any given change dz, there can be many combinations of the corresponding changes dx and dy.

28. Production Function
Consider a production function f(K,L) and its isoquant defined by f(K,L)=60.
The slope of this isoquant at any point is given by the theorem of implicit function differentiation:
Definition. The slope of an isoquant is called the marginal rate of (technical) substitution.
Note. It is easy to see that the MRS is equal to the negative of the ratio of the two marginal products, i.e.

29. Isoquant Slope and Marginal Products
Starting at R, we reduce the amount of capital we use by dK.
We offset the negative impact of this decrease by hiring more labor dL so that the total output is unchanged at Q=60: we are staying on the same isoquant!
The decrease in output due to losing dK is given by a partial differential:
The increase in output due to more labor is given by another partial differential:
To stay on the same isoquant, the two partial differential must be equal, and

30. Convexity of an Isoquant
We know that the slope of an isoquant is given by
To see whether an isoquant is convex or concave, we must look at the sign of the first derivative of the slope, i.e. at
By using the chain rule, we obtain:
It is important to understand that is a total derivative, and so is
and

31. Utility Functions
Definition. The slope of an indifference curve is called the marginal rate of substitution.
For the indifference curves to be convex, it is enough to require that the marginal product of X falls relative to the marginal product of Y as we move down the indifference curve.
This is derived from the satiation property.

32. 15.14 Macroeconomic Equilibrium
Consider the following macroeconomic model:
Y = GDP
C = Private consumption
I = Investment
G = Government purchases
X = Trade balance (exports minus imports)
This relationship can be rewritten in terms of implicit functions:
The total differential is

33. Marginal Propensity to Consume
Consider the following consumption function:
Definition. The partial derivative of the consumption function with respect to income
is called marginal propensity to consume.
MPC tells us what fraction of an additional dollar of income becomes consumption.
Note. 1-MPC is the fraction of an additional dollar that is saved, and is called marginal propensity to save.
We assume the following:

34. Keynesian Multiplier Total differential of income
Partial differential of the consumption function
Re-write total differential of income using partial differential of consumption function
Express total differential of income as a function of the Keynesian multiplier
If government consumption and private consumption are exogenous
Total derivative becomes partial derivative because we are assuming exogeneity of G and C.

35. IS Curve
Suppose investment I is no longer exogenous, and depends on the interest rate r:
The partial differential of the investment function is
Substitute the consumption and investment functions into
to obtain
Rewrite as
Definition. An implicit function Y(r) defined by is called the IS curve.
The IS curve is consistent with equilibrium: it is a locus of combinations of Y and r such that the demand for goods by households C, firms I, government G and foreigners X equals actual production Y.

36. Slope of the IS Curve
Consider total differential of the relationship defining the IS curve:
Assuming exogeneity of G and X, this becomes
Dividing both sides by dY, one obtains the slope of the IS curve:
The slope of the IS curve is negative because the MPC is between zero and one, and because investment decreases with the interest rate.

37. Shifts in the IS Curve
Changes in exogenous G or X or both shift the IS curve around the graph.
Assume, for instance, dG>0 and dX=0. The total differential equation then becomes:
Suppose that the interest rate does not change, i.e. dr=0:
This is telling us how GDP would react if the government increased its purchases, holding interest rate and net exports constant: the GDP would increase.
This is a basis for the government’s policy to increase its purchases when an economy is in a recession.

38. Shifts in IS Curve
is giving us the horizontal distance between JJ’ and PP’.