Managerial economics theory and practice phần 2 doc

A firm’sprofit function, for example, is equal to the firm’s total revenue function minus the firm’s total cost function.. Related to each of these total concepts are the analytically import

DERIVATIVE OF A FUNCTION

Consider, again, the function

(2.1)

The slope of this function is defined as the change in the value of y divided

by a change in the value of x, or the “rise” over the “run.” When defining

the slope between two discrete points, the formula for the slope may begiven as

(2.6)

Consider Figure 2.12, and use the foregoing definition to calculate the

value of the slope of the cord AB As point B is brought arbitrarily closer

to point A, however, the value of the slope of AB approaches the value of the slope at the single point A, which would be equivalent to the slope of

a tangent to the curve at that point This procedure is greatly simplified,however, by first taking the derivative of the function and calculating its

value, in this case, at x1

The first derivative of a function (dy/dx) is simply the slope of the tion when the interval along the horizontal axis (between x1and x2) is made

func-infinitesimally small Technically, the derivative is the limit of the ratio Dy/Dx

asDx approaches zero, that is,

(2.44)

When the limit of a function as x Æ x0equals the value of the function at

x0, the function is said to be continuous at x0, that is, lim xÆx0 f(x) = f(x0)

dy dx

y x

DD

y x

instanta-Calculus offers a set of rules for using derivatives (slopes) for making

optimizing decisions such as minimizing cost (TC) or maximizing total

profit (p)

RULES OF DIFFERENTIATION

Having established that the derivative of a function is the limit of theratio of the change in the dependent variable to the change in the inde-pendent variable, we will now enumerate some general rules of differenti-ation that will be of considerable value throughout the remainder of thiscourse It should be underscored that for a function to be differentiable at

a point, it must be well defined; that is it must be continuous or “smooth.”

It is not possible to find the derivative of a function that is discontinuous(i.e., has a “corner”) at that point The interested student is referred to the selected adings at the end of this chapter for the proofs of these propositions

POWER-FUNCTION RULE

A power function is of the form

where a and b are real numbers The rule for finding the derivative of a

power function is

(2.45)

where f ¢(x) is an alternative way to denote the first derivative.

Example

A special case of the power-function rule is the identity rule:

Another special case of the power-function rule is the constant-function

rule Since x0= 1, then

Example

SUMS AND DIFFERENCES RULE

There are a number of economic and business relationships that arederived by combining one or more separate, but related, functions A firm’sprofit function, for example, is equal to the firm’s total revenue function

minus the firm’s total cost function If we define g and h to be functions of the variable x, then

dv dx

PRODUCT RULE

Similarly, there are many relationships in business and economics that are defined as the product of two or more separate, but related, func-tions The total revenue function of a monopolist, for example, is theproduct of price, which is a function of output, and output, which is a func-

tion of itself Again, if we define g and h to be functions of the variable x,

S

= 2

dQ dP

D

= -3

QUOTIENT RULE

Even less intuitive than the product rule is the quotient rule Again,

defining g and h as functions of x, we write

Further, let

then

(2.48)

Example

Substituting into Equation (2.48), we have

Interestingly, in some instances it is convenient, and easier, to apply theproduct rule to such problems This becomes apparent when we rememberthat

x x

It is left to the student to demonstrate that the same result is derived by applying the quotient rule.

CHAIN RULE

Often in business and economics a variable that is a dependent variable

in one function is an independent variable in another function Output Q,

for example, is the dependent variable in a perfectly competitive firm’sshort-run production function

where L represents the variable labor, and K0represents a constant amount

of capital labor utilizes in the short run On the other hand, output is theindependent variable in the firm’s total revenue function

where P is the (constant) selling price.

In the example just given, we might be interested in determining howtotal revenue can be expected to change given a change in the firm’s laborusage For this we require a technique for taking the derivative of one func-tion whose independent variable is the dependent variable of another func-

tion Here we might be interested in finding the derivative dTR/dL To find this derivative value, we avail ourselves of the chain rule Let y = f(u) and

u = g(x) Substituting, we are able to write the composite function

The chain rule asserts that

dQ

= ÊË ˆ¯ÊË ˆ¯ =10 2( - 0 5 )=20 - 0 5 =20

dy dx

dy du

du dx

df u du

dg x dx

EXPONENTIAL AND LOGARITHMIC FUNCTIONS

Now we consider the derivative of two important functions–the

expo-nential function and the logarithmic function The number e is the base of the natural exponential function, y = e x The natural logarithmic function is

y= loge x = ln x The number e is itself generated as the limit to the series

(2.50)

To illustrate the practical importance of the number e, suppose, for

example, that you were to invest $1 in a savings account that paid an

inter-est rate of i percent If interinter-est was compounded continuously (see Chapter

12), the value of the deposit at the year end would be

d e

dx e

x x

When more complicated functions are involved, we can apply the chain

rule Suppose, for example, that y = ln x2 Letting u = x2 this becomes

y = ln u The derivative of y with respect to x then becomes

It may also be demonstrated that the result for the derivative of an nential function follows directly from a special relationship that existsbetween the exponential function and the logarithmic function Given the

expo-function x = e y , then y = ln x Moreover, if x = ln y, then y = e x These tions are said to be reciprocal functions When two functions are related in

func-this way, the derivatives are also related; that is, dy/dx = 1/(dx/dy) Using

this rule, we can prove the exponential function rule:

Returning to the earlier discussion of continuous compounding, supposethat the value of an asset is given by

where r is the rate of interest, t time, and D the initial value of the asset.

The rate of change of the value of the asset over time is

That is, the rate of change in the value of the asset is the rate of interest

times the value of the asset at time t.

INVERSE-FUNCTION RULE

Earlier in this chapter we discussed the existence of inverse functions It

will be recalled that if the function y = f(x) is a one-to-one correspondence, then not only will a given value of x correspond to a unique value of y, but

a given value of y will correspond to a unique value of x In this case, the function f has the inverse function g(y) = f-1(y) = x, which is also a one-to-

one correspondence Given an inverse function, its derivative is

D de du

du

dt e

dD dt

dy dx

dy du

du

= ÊË ˆ¯ÊË ˆ¯ =(1 )( )2 = ÊË 12ˆ¯( )2 = 2

dy dx

Equation (2.54) asserts that the derivative of an inverse function is the reciprocal of the derivative of the original function

It will also be recalled that functions with a one-to-one correspondence

are said to be monotonically increasing if x2> x1fi f(x2)> f(x1) Functions

in which a one-to-one correspondence exist are said to be monotonically

decreasing if x2> x1fi f(x2)< f(x1) In general, for an inverse function toexist, the original function must be monotonic In other words, it is not

possible to write x = g(y) = f-1(y) until we have determined whether the function y = f(x) is monotonic.

It is possible to determine whether a function is monotonic by ing its first derivative If the first derivative of the function is positive for all

examin-values of x, then the function y = f(x) is monotonically increasing If the first derivative of the function is negative for all values of x, then the function

y = f(x) is monotonically decreasing.

Problem 2.5 Consider the function

a Is this function monotonic?

b If the function is monotonic, use the inverse-function rule to find

dx/dy.

Solution

a The derivative of this function is

which is positive for all values of x Thus, the function f(x) is a

monoto-nically increasing function

b Because f(x) is a monotonically increasing function, the inverse function

g(y) = f-1(y) exists Thus, it is possible to use the inverse-function rule to

determine the derivative of the inverse function, that is,

It should be noted that the inverse-function rule may also be applied tononmonotonic functions, provided the domain of the function is restricted

For example, y = f(x) = x2is nonmonotonic because its derivative does not

have the same sign for all values of x On the other hand, if the domain of this function is restricted to positive values for x, then dy/dx> 0

Problem 2.6 Consider the function

a Is this function monotonic?

b If the function is monotonic, use the inverse-function rule to find dx/dy.

Solution

a The derivative of this function is

This function is not monotonic, since the sign of dy/dx depends on whether x is positive or negative On the other hand, the derivative is negative for all positive values for x.

b Because the derivative of f(x) is positive for all x> 0, then it is possible

to use the inverse-function rule to determine the derivative of the inversefunction, that is,

for all x> 0

IMPLICIT DIFFERENTIATION

The functions we have been discussing are referred to as explicit tions Explicit functions are those in which the dependent variable is on theleft-hand side of the equation and the independent variables are on theright-hand side In many cases in business and economics, however, we mayalso be interested in what are called implicit functions

Implicit functions are those in which the dependent variable is also tionally related to one or more of the right-hand-side variables Such func-tions often arise in economics as a result of some equilibrium condition that

func-is imposed on a model A common example of an implicit function inmacroeconomic theory is in the definition of the equilibrium level of

national income Y, which is given as the sum of consumption spending C,

which is itself assumed to be a function of national income, net investment

spending I, government expenditures G, and net exports X - M This

equi-librium condition is written

(2.55)

Clearly, any change in the value of Y must come about because of changes

in any and all changes in the components of aggregate demand The totalderivative of this relationship may be written

(2.56)Equation (2.56) is a differential equation We may express the relationship

between consumption expenditures and national income as C = C(Y).

Suppose that the consumption function is well defined and the derivative

dC/dY = C¢(Y) exists, which may be rewritten as

(2.57)

Equation (2.57) may be rewritten as

(2.58)

Suppose that we were specifically interested in the derivative dY/dI It

is possible to find the derivative dY/dI by implicit differentiation ing that a change in I has no effect on G and none on X - M; that is,

Assum-dG = d(X - M) = 0, but does change Y Equation (2.59) reduces to

invest-To implicitly differentiate a function, we treat changes in the two

vari-ables, dY and dI, as unknowns and solve for the ratio of the change in the

dependent variable to the change in the independent variable, which is thederivative in explicit form

TOTAL, AVERAGE, AND MARGINAL

RELATIONSHIPS

Now that we have discussed the concept of the derivative, we are in aposition to discuss an important class of functional relationships There areseveral “total” concepts in business and economics that are of interest to

dY

dI = dC dY

-11

the managerial decision maker: total profit, total cost, total revenue, and so

on Related to each of these total concepts are the analytically importantaverage and marginal concepts, such as average (per-unit) profit and mar-ginal profit; average total cost and marginal cost, average variable cost andmarginal cost, and average total revenue and marginal revenue An under-standing of the nature of the relation between total, average, and marginalrelationships is essential in optimization analysis

To make the discussion more concrete, consider the total cost function

TC = f(Q), where Q represents the output of a firm’s good or service and

dTC/dQ> 0.As we will see in Chapter 6, related to this are two other tant functional relationships Average total, or per-unit, cost of production

impor-(ATC) is defined as ATC = TC/Q Marginal cost of production (MC), which

is given by the relationship MC = dTC/dQ, measures the incremental

change in total cost arising from an incremental change in total output

Clearly, ATC and MC are not the same Nevertheless, these two cost

con-cepts are systematically related Indeed, the nature of this relationship isfundamentally the same for all average and marginal relationships Beforepresenting a formal statement of the nature of this relationship, considerthe following noneconomic example

Suppose you are enrolled in an economics course, and your final grade

is based on the average of 10 quizzes that you are required to take duringthe semester Assume that the highest grade you can earn on any individ-ual quiz is 100 points Thus, if you earn the maximum number of pointsduring the semester, your average quiz grade will be 1,000/10 = 100 Now,suppose that you have taken 6 quizzes and have earned a total of 480 points.Clearly, your average quiz grade is 480/6 = 80 How will your average beaffected by the grade you receive on the seventh quiz? Since the number

of points you earn on the seventh quiz will increase the total number ofpoints earned, we will call the number of additional points earned your marginal grade How will this marginal grade affect your average? Clearly,

if the grade that you receive on the seventh quiz is greater than youraverage for the first six quizzes, your average will rise For example, if youreceive a grade of 90, your average will increase from 80 to 570/7 = 81.4

On the other hand, if the grade you receive is less than the average, theaverage will fall For example, if you receive a grade of 70, your average will decline to 550/7 = 78.6 Finally, if the grade you receive on the next quiz is the same as your average, the average will remain unchanged (i.e.,560/7= 80)

In general, it can be easily demonstrated that when any marginal value

M is greater than its corresponding average A value (i.e., M > A), then A will rise Analogously, when M < A, then A will fall Finally, when M = A, then A will neither rise nor fall In many economic models, when M = A the value of A will be at a local maximum or local minimum These relation-

ships will be formalized in the following paragraphs

total, average, and marginal relationships 73

Consider again the functional relationship in Equation (2.1).

(2.1)Define the average and marginal functions of Equation (2.1) as

Since the value of the denominator in Equation (2.63) is positive, the sign

of dA/dx will depend on the sign of the expression xf ¢(x) - f(x) That is, for the average to be increasing (dA/dx > 0), then [xf¢(x) - f(x)] > 0 This, of course, implies that f ¢(x) > f(x)/x, or M > A For the average to fall (dA/dx

< 0), then [xf¢(x) - f(x)] < 0, or f¢(x) < f(x)/x That is, the marginal must be less than the average (M < A) Finally, for no change in the average (dA/dx

= 0), then [xf¢(x) - f(x)] = 0, or f¢(x) = f(x)/x That is, for no change in the

average, the marginal is equal to the average For the functional ship in Equation (2.1), these relationships are summarized as follows:

relation-(2.64a)

(2.64b)

(2.64c)

Let us return to the example of the total cost function TC = f(Q)

intro-duced earlier Consider the hypothetical total cost function in Figure 2.13,and the corresponding average total cost and marginal cost curves in Figure2.14

In Figure 2.13, the numerical value of ATC is the same as a slope of a ray from the origin to a point on the TC curve corresponding to a given level of output The equation of a ray from the origin is TC = bQ, where b

is the slope of the ray from the origin to a point on the TC curve, which is

xf x f x x

f x x

= = ( )

y=f x( )

where the values where Q1represents the initial value of output and Q2

represents the changed level of output Since the ray passes through

the origin, then the initial values (Q1, TC1) are (0, 0) Setting TC2= TC and

Q2= Q, Equation (2.66) reduces to

(2.66)

Of course, the value of b will change as we move along the total cost curve This is illustrated in Figure 2.13 MC, of course, is the value of the slope of the TC curve and may be illustrated diagrammatically in Figure 2.13 as the slope a line that is tangent to TC at some level of output By comparing the

value of the slope of the tangent with the slope of the ray from the origin,

we are able to illustrate the relationship between MC and ATC in Figure

2.14

Note that output at point A in Figure 2.13, the slope of the tangent (MC),

is less than the slope of the ray from the origin (ATC) Thus, at output level

Q1, MC is less than ATC This is illustrated in Figure 2.14 Now let us move

to point B Note that at Q2the slopes of the tangent and the ray are less

than they were at point A Thus, in Figure 2.14 MC and ATC at Q2are less

than at Q1 Although both MC and ATC have fallen, the slope of the tangent (MC) at Q2is still less than the slope of the ray (ATC) Thus, since MC<

ATC at Q2, then ATC has declined By analogous reasoning, as we move from Q2to Q3, since MC < ATC, then ATC will fall The reader will note that point C in the Figure 2.13 is an inflection point Beyond output level

Q3, the slope of the TC curve (MC) begins to increase Thus, at output level

Q3, marginal cost is minimized Nevertheless, as illustrated in Figure 2.14,

as long as MC < ATC, then ATC will continue to fall.

b ATC TC

Q

b TC Q

total, average, and marginal relationships 75

TC TC

FIGURE 2.13 The total cost curve

and its relationship to marginal and average

total cost.

At output level Q4the slopes of the ray and tangent are identical (ATC

= MC) Thus, at Q4ATC is neither rising nor falling (i.e., dATC/dQ = 0)

After Q4the slope of the tangent not only becomes greater than the slope

of the ray, but the slope of the ray changes direction and starts to increase

Thus, we see that at output level Q5, MC > ATC and ATC are rising These

relationships are illustrated in Figure 2.14

The situation depicted in Figure 2.14 illustrates a U-shaped average total

cost curve in which the MC intersects ATC from below The reader should visually verify that when MC < ATC, even when MC is rising, ATC is falling Moreover, when MC > ATC, then ATC is rising Finally, when MC = ATC, then ATC is neither rising nor falling (i.e., ATC is minimized) In some cases,

the average curve is shaped not like U but like a hill: that is, the marginalcurve intersects the average curve from above at its maximum point Anexample of this would be the relationship between the average and mar-ginal physical products of labor, which will be discussed in detail in Chapter5

PROFIT MAXIMIZATION: THE FIRST-ORDER

CONDITION

We are now in a position to use the rules for taking first derivatives to

find the level of output Q that maximizes p, as illustrated in Table 2.3 sider again the total revenue and total cost functions introduced earlier:

It should be noted in Table 2.3 and Figure 2.11 that profit is maximized(p = 19) at Q = 5 What is more, it should be immediately apparent that if

a smooth curve is fitted to Figure 2.11, the value of the slope at Q= 5 iszero: that is, the profit function is neither upward sloping nor downward

sloping Alternatively, at Q = 5, then dp/dQ = 0 These observations imply

that the value of a function will be optimized (maximized or minimized)where the slope of the function is equal to zero In the present context, the

first-order condition for profit maximization is d p/dQ = 0, thus

(2.68)

This equation is of the general form:

(2.69)

where a = -8, b = 10 and c = -15 Quadratic equations generally admit to

two solutions, which may be determined using the quadratic formula Thequadratic formula is given by the expression:

(2.70)

After substituting the values of Equation (2.68) into Equation (2.70) weget

Referring again to Figure 2.11, we see that the value of p reaches a

minimum and a maximum at output levels of Q = 1 and Q = 5, respectively.

Substituting these values back into the Equation (2.67) yields values of

p = -13 (at Q = 1) and p = 19 (at Q = 5) In this example, therefore, the entrepreneur of the firm would maximize his profits at Q = 5 As thisexample illustrates, simply setting the first derivative of the function equal

to zero is not sufficient to ensure that we will achieve a maximum, since azero slope is also required for a minimum value as well Thus, we need tospecify the second-order conditions for a maximum or a minimum value to

18 126

PROFIT MAXIMIZATION: THE SECOND-ORDER

CONDITION

MAXIMA AND MINIMA

For functions of one independent variable, y = f(x), a second-order dition for f(x) to have a maximum at some value x = x0is that together with

con-dy/dx = f¢(x) = 0, the second derivative (the derivative of the derivative) be

negative, that is,

(2.71)

where f ≤(x) is an alternative way to denote the second derivative.

This condition expresses the notion that in the case of a maximum, theslope of the total function is first positive, zero, and then negative as we

“walk” over the top of the “hill.” Functions that are locally maximum aresaid to be “concave downward” in the neighborhood of the maximum value

of the dependent variable Similarly, the second-order condition for f(x) to have a minimum at some value x = x0, then is

TABLE 2.4 First-order and second-order conditions for

functions of one independent variable.

Maximum Minimum First-order condition

Problem 2.8 Another monopolist has the following TR and TC functions:

Find the p-maximizing output level

To determine whether these values constitute a minimum or a maximum,

we can substitute the values into the profit function and determine theminimum and maximum values directly, or we can examine the values ofthe second derivatives:

For Q1= 4,

(i.e., the second-order condition for a local maximum)

For Q2= 1,

(i.e., the second-order condition for a local minimum)

Substituting Q1= 4 into the p function yields a maximum profit of

INFLECTION POINTS

What if both the first and second derivatives are equal to zero? That is, what if f ¢(x) = f≤(x) = 0? In this case, we have a stationary point, which

is neither a maximum nor a minimum That is, stationary values for which

f ¢(x) = 0 need not be a relative extremum (maximum or minimum) tionary values at x0that are neither relative maxima nor minima are illus-trated in Figures 2.15 and 2.16

Sta-To determine whether the stationary value at x0is the situation depicted

in Figure 2.15 or Figure 2.16, it is necessary to examine the third derivative:

d(d2y/dx2)/dx = d3y/dx3= f¢¢¢ (x) The value of the third derivative for the

p*= - -2 12 4( )+7 5 4 ( )2-Q3= - -2 48 120 64+ - =6

d dQ

15 96

situation depicted in Figure 2.15 is d3y/dx3= f¢¢¢(x) > 0.The value of the third derivative for the situation depicted in Figure 2.16 is d3y/dx3= f¢¢¢(x) < 0.

PARTIAL DERIVATIVES AND MULTIVARIATE

OPTIMIZATION: THE FIRST-ORDER

CONDITION

Most economic relations involve more than one independent

(explana-tory) variable For example, consider the following sales (Q) function of a firm that depends on the price of the product (P) and levels of advertising expenditures (A):

FIGURE 2.15 Inflection point: a

stationary value at x0 that is neither a

maximum nor a minimum.

f(x)

x0

FIGURE 2.16 Inflection point: a

stationary value at x0 that is neither a

maximum nor a minimum.

treating the remaining variables as constants This process, known as taking

partial derivatives, is denoted by replacing d with∂

Example

Consider the following explicit relationship:

(2.74) (where A is in thousands of dollars) Taking first partial derivatives with respect to

P and A yields

(2.75) (2.76)

To determine the values of the independent variables that maximize theobjective function, we simply set the first partial derivatives equal to zeroand solve the resulting equations simultaneously

Example

To determine the values of P and A that maximize the firm’s total sales, Q, set the

first partial derivatives in Equations (2.69) and (2.70) equal to zero.

(2.77) (2.78) Equations (2.77) and (2.78) are the first-order conditions for a maximum Solving these two linear equations simultaneously in two unknowns yields (in thousands of dollars).

Substituting these results back into Equation (2.74) yields the optimal value of Q.

PARTIAL DERIVATIVES AND MULTIVARIATE

OPTIMIZATION: THE SECOND-ORDER

CONDITION8

Unfortunately, a general discussion of the second-order conditions formultivariate optimization is beyond the scope of this book It will be suffi-cient within the present context, however, to examine the second-order con-ditions for a maximum and a minimum in the case of two independentvariables Consider the following function:

y x

y x

vari-f xy=f yx

CONSTRAINED OPTIMIZATION

Unfortunately, most decision problems managers faced are not of theunconstrained variety just discussed The manager often is required to max-imize some objective function subject to one or more side constraints Aproduction manager, for example, may be required to maximize the totaloutput of a given commodity subject to a given budget constraint and fixedprices of factors of production Alternatively, the manager might berequired to minimize the total costs of producing some specified level ofoutput The cost minimization problem might be written as:

Example

The total cost function of a firm that produces its product on two assembly lines is given as

The problem facing the firm is to determine the least-cost combination of output

on assembly lines x and y subject to the side condition that total output equal 20

units This problem may be formally written as

formal proof of this theorem in 1909 using the concept of the limit (see, e.g., Cambridge Tract

No 11, The Fundamental Theorems of the Differential Calculus, Cambridge University Press,

reprinted in 1971 by Hafner Press) According to Silberberg, the result was first published

by Euler in 1734 (“De infinitis curvis eiusdem generis ,” Commentatio 44 Indicis Enestroemiani).

SOLUTION METHODS TO CONSTRAINED

OPTIMIZATION PROBLEMS

There are generally two methods of solving constrained optimizationproblems:

1 The substitution method

2 The Lagrange multiplier method

SUBSTITUTION METHOD

The substitution method involves first solving the constraint, say for x,

and substituting the result into the original objective function Consider,again, the foregoing example

(2.85)Substituting into the objective function yields

(2.86)

(2.87)

In other words, this problem reduces to one of solving for one decision

variable, y, and inserting the solution into the objective function Taking the first derivative of the objective function with respect to y and setting the

result equal to zero, we get

(2.88)

Note also that the second-order condition for total cost minimization isalso satisfied:

(2.89)

Substituting y= 7 into the constraint yields

Finally, substituting the values of x and y into the original TC function

yields:

x= 13

x+ =7 20

d TC dy

LAGRANGE MULTIPLIER METHOD

Sometimes the substitution method may not be feasible because of morethan one side constraint, or because the objective function or side con-

straints are too complex for efficient solution Here, the Lagrange multiplier

method can be used, which directly combines the objective function with

the side constraint(s)

The first step in applying the Lagrange multiplier technique is to firstbring all terms to the right side of the equation.10

With this, we can now form a new objective function called the Lagrangefunction, which will be used in subsequent chapters to find solution values

to constrained optimization:

(2.90)

Note that this expression is equal to the original objective function, sinceall we have done is add zero to it That is,ᏸ always equals f for values of x and y that satisfy g To solve for optimal values of x and y, we now take the

first partials of this more complicated expression with respect to three

unknowns—x, y, and l The first-order conditions therefore become:

orᏸ = f - lg, since one’s choice merely changes the sign of the Lagrangian multiplier.

Note that the values for x and y are the same as those obtained using

the substitution method The Lagrange multiplier technique is more erful, however, because we are also able to solve for the Lagrange multi-plier,l What is the interpretation of l? From Equations (2.91) it can bedemonstrated that the Lagrange multiplier is defined as

pow-(2.92)

That is, the Lagrange multiplier is the marginal change in the maximumvalue of the objective function with respect to parametric changes in thevalue of the constraint.11In the context of the present example,l = -71 saysthat if we relax our production constraint by, say, one unit of output (i.e., if

we reduce output from 20 units to 19 units), our total cost of productionwill decline by $71 It is important to note that because marginal cost is anonlinear function, the value of l may be interpreted only in the neigh-

borhood of Q= 20 In other words, the value of l will vary at differentoutput levels

Problem 2.9 A profit-maximizing firm faces the following constrained

maximization problem:

Maximize:

Subject to:

Determine profit-maximizing output levels of commodities x and y subject

to the condition that total output equals 12 units

Solution Form the Lagrange expression

The first-order conditions are:

This system of three linear equations in three unknowns can be solved forthe following values:

Substituting the values of x and y back into our original objective function

yields the maximum value for profits:

The interpretation of l is that if our constraint is relaxed by one unit, sayincreased from an output level of 12 units to 13 units, the firm’s profits willincrease by $53 Similarly, if output is reduced from say 12 units to 11 units,profits will be decreased by $53 This result is illustrated in Figure 2.18,which shows that the value of l = ∂p/∂k approaches zero as the output con-

straint becomes non binding, that is, as we approach the top of the profit

able because it allowed us to examine relative maxima and minima

Suppose, on the other hand, that we are given the function dy/dx = f¢(x)

= g(x) and wish to recover the function y = f(x) In other words, what tion y = f(x) has as its derivative dy/dx = f¢(x) = g(x)?

func-Suppose, for example, that we are given the expression

From what function was this expression derived? We know from experience that Equation (2.93) could have been derived from each of theexpressions

By examination we see that Equation (2.93) may be derived from thegeneral class of equations

(2.94)

where c is an arbitrary constant The general procedure for finding tion (2.94) is called differentiation The process of recovering Equation (2.94) from Equation (2.93) is called integration.

Equa-In general, suppose that y = f(x), and that

where y = f(x) is referred to as the integral of g(x) If we are given g(x) and wish to recover f(x), the general solution is

(2.95)

The term c is referred to as an arbitrary constant of integration, which may be unknown Since dy/dx = g(x), then

(2.96)Integrating both sides of Equation (2.96), we obtain

is readily apparent upon careful examination because the derivative of the

right is clearly x m.12On the other hand, the integral

may take a while to figure out.

THE INTEGRAL AS THE AREA UNDER A CURVE

The importance of integration stems from its interpretation as the area

under a curve Consider, for example, the marginal cost function MC(Q)

illustrated in Figure 2.19

Marginal cost represents the addition to total cost from producing

addi-tional units of a commodity, Q The process of adding up (or integrating) the cost of each additional unit of Q will result in the total cost of produc- ing Q units of the commodity less any other costs not directly related to the

production process, such as insurance payments and fixed rental payments

Such “indirect” (to the actual production of Q) costs are collectively referred to as total fixed cost TFC Costs that vary directly with output of

Q are referred to as total variable cost TVC Total cost is defined as

From the foregoing discussion, we realize that integrating Equation(2.100) will yield

(2.101)

That is, by integrating the marginal cost function, we will recover the total

variable cost function, with the constant of integration c representing TFC.

This process is illustrated in Figure 2.19

Consider the area beneath MC(Q) in Figure 2.19 between Q1and Q2

Let us denote the value of this area as A Q1ÆQ2 Suppose that we wish to sider the effect of an increase in the value of the area under the curve result-

con-ing from an increase in output from Q2to Q3, where Q3= Q2+ DQ The

value of the area under the curve will increase by DA, where

In the interval Q2 to Q3 there is a minimum and maximum value of

MC(Q), which we will denote as MCm, and MCM, respectively It must bethe case that

This is illustrated in Figure 2.20 as the shaded rectangle Thus, estimatingthe value of DA by using discrete changes in the value of Q results in an

approximation of the increase in the value of the area under the curve.How can we improve upon this estimate of DA? One way is to divide

DQ into smaller intervals This is illustrated in Figure 2.21.

Taking the limit as DQ Æ 0 “squeezes” the difference between MC mand

MC M to its limiting value MC(Q) Thus,

limD DD

FIGURE 2.20 Approximating the increase in

the area under a curve.

As noted, A and TC(Q) can differ only by the value of some arbitrary constant c, which in this case is TFC.

Consider, now, the area under the marginal cost curve from Q1to Q3.The total cost of production over that interval is

(2.102)

Problem 2.10 Suppose that a firm’s the marginal cost function is MC(x)

= 50x + 600.

a Find the total cost function if total fixed cost is $4,000

b What is the firm’s total cost of producing 5 units of output?

c What is the firm’s total cost of producing from 2 to 5 units of output?

tionships are very often expressed as functions In mathematics, a functional

relationships of the form y = f(x) is read “y is a function of x.” This tionship indicates that the value of y depends in a systematic way on the value of x The expression says that there is a unique value for y for each value of x The y variable is referred to as the dependent variable The x variable is referred to as the independent variable.

rela-Functional relationships may be linear and nonlinear The distinguishing

characteristic of a linear function is its constant slope; that is, the ratio of

the change in the value of the dependent variable given a change in thevalue of the independent variable is constant The graphs of linear func-

tions are straight lines With nonlinear functions the slope is variable The graphs of nonlinear functions are “curved.” Polynomial functions constitute

a class of functions that contain an independent variable that is raised tosome nonnegative power greater than unity

Two of the most common polynomial functions encountered in

eco-nomics and business are the quadratic function and the cubic function.

Many economic and business models use a special set of functional tions called total, average, and marginal functions These relations are espe-cially useful in the theories of consumption, production, cost, and marketstructure In general, whenever a function’s marginal value is greater thanits corresponding average value, the average value will be rising Wheneverthe function’s marginal value is less than its corresponding average value,the average value will be falling Whenever the marginal value is equal tothe average value, the average value is neither rising nor falling

rela-Many problems in economics involve the determination of “optimal”solutions For example, a decision maker might wish to determine the level

of output that would result in maximum profit In essence, economic mization involves maximizing or minimizing some objective function, whichmay or may not be subject to one or more constraints Finding optimal solu-

opti-tions to these problems involves differentiating an objective function, setting

the result equal to zero, and solving for the values of the decision variables.For a function to be differentiable, it must be well defined; that is, it must

be continuous or “smooth.” Evaluating optimal solutions requires an uation of the appropriate first- and second-order conditions There are generally two methods of solving constrained optimization problems: the

eval-substitution and Lagrange multiplier methods.

Integration is the reverse of differentiation Integration involves

recov-ering an original function, such as a total cost equation, from its first ative, such as a marginal cost equation The resulting function is called anindefinite integral because the value of the constant term in the originalequation, such as total fixed cost, cannot be found by integrating the firstderivative Thus, the integral of the marginal cost equation is the equationfor total variable cost Integration is particularly useful in economics whentrying to determine the area beneath a curve

CHAPTER QUESTIONS

2.1 Economic optimization involves maximizing an objective function,which may or may not be subject to side constraints Do you agree with thisstatement? If not, why not?

2.2 For an inverse function to exist, the original function must be onic Do you agree? Explain

monot-2.3 What does it mean for a function to be well defined?

2.4 The inverse-function rule may be applied only to monotonic tions Do you agree with this statement? If not, why not?

func-2.5 Suppose that a firm’s total profit is a function of output [i.e., p =

f(Q)] To maximize total profits, the firm must produce at an output level

at which M p = dp/dQ = 0 Do you agree? Explain.

2.6 Suppose that a firm’s total profit is a function of output [i.e., p =

f(Q)] Marginal and average profit are defined as M p = dp/dQ and Ap = p/Q Describe the mathematical relationship between total, marginal, and

average profit

2.7 Maximizing per-unit profit is equivalent to maximizing total profit

Do you agree? Explain

2.8 Describe briefly the difference between the substitution andLagrange multiplier methods for finding optimal solutions to constrainedoptimization problems

2.9 The Lagrange multiplier is an artificial variable that is of no tance when one is finding optimal solutions to constrained optimizationproblems Do you agree with this statement? Explain

2

x+ + = 1y z

2.3 Find the first derivatives and the indicated values of the derivatives.

2.5 The total cost function of a firm is given by:

where TC denotes total cost and Q denotes the quantity produced per unit

of time

a Graph the total cost function from Q = 0 to Q = 100.

b Find the marginal cost function

c Find the marginal cost of production from Q = 0 to Q = 100.

2.6 The total cost of production of a firm is given as

where TC denotes total cost and Q denotes the quantity produced per unit

of time

a Graph the total cost function from Q = 0 to Q = 200.

b Find the marginal cost function

c Find the average cost function

d Graph in the same diagram average and marginal costs of production

from Q = 0 to Q = 200.

2.7 Here are three total cost functions:

a Determine for each equation the average variable cost, average cost,and marginal cost equations

b Plot each equation on a graph

c Use calculus to determine the minimum total cost for each equation

2.8 The market demand function for a commodity x is given as

where Q denotes the quantity demanded and P its price.

a Find the average revenue function (i.e., price as a function of quantity)

b Find the marginal revenue function for a monopolist who produces

Q.

c Graph the average revenue curve and the marginal revenue curve

from Q = 1 to Q = 100.

2.9 A firm has the following total revenue and total cost functions:

a At what level of output does the firm maximize total revenue?

b Define the firm’s total profit as p = TR - TC At what level of output

does the firm maximize total profit?

c How much is the firm’s total profit at its maximum?

2.10 Assume that the firm’s operation is subject to the following duction function and price data:

pro-where X and Y are two variable input factors employed in the production

of Q.

a In the unconstrained case, what levels of X and Y will maximize Q?

b It is possible to express the cost function associated with the use of X and Y in the production of Q as TC = 3X + 6Y Assume that the firm

has an operating budget of $250 Use the Lagrange multiplier

tech-nique to determine the optimal levels of X and Y What is the firm’s

total output at these levels of input usage?

c What will happen to the firm’s output from a marginal increase in the operating budget?

2.11 Evaluate the following integrals:

a Ú(8x2+ 600)dx

b Ú(5x + 3)dx

c Ú(10x2+ 5x - 25)dx

2.12 Suppose that the marginal cost function of a firm is

The firm’s total fixed cost is 10

TR=21Q Q- 2

Q=300 30- ( )P

a Determine the firm’s total cost function.

b What is the firm’s total cost of production at Q= 3?

SELECTED READINGSAllen, R G D Mathematical Analysis for Economists New York: Macmillan, 1938.

——— Mathematical Economics, 2nd ed New York: Macmillan, 1976.

Brennan, M J., and T M Carroll Preface to Quantitative Economics & Econometrics, 4th ed Cincinnati, OH: South-Western Publishing, 1987.

Chiang, A Fundamental Methods of Mathematical Economics, 3rd ed New York: Hill, 1984.

McGraw-Draper, J E., and J S Klingman Mathematical Analysis: Business and Economic Applications, 2nd ed New York: Harper & Row, 1972.

Fine, H B College Algebra New York: Dover, 1961.

Glass, J C An Introduction to Mathematical Methods in Economics New York: McGraw-Hill, 1980.

Henderson, J M., and R E Quandt Microeconomic Theory: A Mathematical Approach, 3rd

ed New York: McGraw-Hill, 1980.

Marshall, A Principles of Economics, 8th ed London: Macmillan, 1920.

Purcell, E J Calculus with Analytic Geometry, 2nd ed New York: Meredith, 1972.

Rosenlicht, M Introduction to Analysis Glenview, Ill: Scott, Foresman, 1968.

Silberberg, Eugene The Structure of Economics: A Mathematical Analysis, 2nd ed New York: McGraw-Hill, 1990.

Youse, B K Introduction to Real Analysis Boston: Allyn & Bacon, 1972.