Section Summary
In this section, we study simplified monopoly economies and show how a manufacturer maximizes profit.

"Monopoly" means a manufacturer can choose whatever price she wants without fear of competition. Our marketplace will be populated by a large number of individual consumers with no organized cooperation. Their aggregate demand for the product will be determined by the price. If the price is low, then the consumers will buy a great deal of the product. But, as the price rises, the demand will fall. Thus, the demand function will be some sort of decreasing function of price.

Our manufacturer wants to set the price that will maximize her total profit. She cannot simply make the price as high as she wishes because demand will drop. A high price on a few units may well produce less profit than a smaller price on many. On the other hand, a very low price may sell many units but produce no profit because of the costs of manufacturing.

Figure CD-11.1: A linear demand function

Our main simplification is in giving an explicit formula for the number of units of the product that consumers want to purchase or "demand." As an example, we begin by working with the linear demand function, D[p]=1000-500p.

Example CD-11.1 Maximum Profit for Demand D[p]=1000-500p

For this demand function, the maximum possible demand is 1000 units when the product is given away. The maximum price is 2 dollars after which no one wants the product. Thus, we are interested in a compact range of prices .

Now, suppose that the manufacturing cost (i.e., raw materials and labor) for each unit of the product is 20 cents. The unit profit at price p will be (p-0.20) and the total profit T[p] at the price p will be the number of units sold at the price p multiplied by the profit per unit. In general, for any demand function D[p],

Figure CD-11.2: Total profit vs. price

Our manufacturer wants to find the price p that maximizes her profit.

She wants to find the maximum of the function T[p]=(1000-500p)(p-0.20) on the interval [0.20,2]. We already know that the endpoints represent points of zero profit or the mathematically necessary, but economically uninteresting minima. The next step is to differentiate the function T[p]. (We use the Product Rule rather than expand the expression.)

We have already seen that T[0.20]=0 (or T[0]<0) and that T[2.00]=0. There is only one critical point inside (0.20,2) so the Candidates procedure says the max must be at p=1.10, even without the slope analysis or the graph.

The profit maximizing price is $ 1.10, and, at this price, the manufacturer will make a total profit of T[1.10]=405 dollars.

Example CD-11.2 A Parametric Maximum

You will notice an interesting phenomenon in the Exercise CD-11.5.3. Exactly half of the rise in the unit manufacturing cost is passed on to the consumer. Is it true in general that a manufacturer always passes half of a cost increase to her customers? Let us begin by analysing a general linear demand function. That is, use the demand function

Figure CD-11.3: A general linear demand function

Let the constant c represent the unit manufacturing cost for the product. Thus, the profit per unit would be (p-c) and the total profit would be

Solving the optimization problem with the parameters a and b will tell us more than we would have known by solving only numerical examples. The numerical examples suggested that half the cost increases were passed on in the linear demand case, but the parameter solution will show it. Optimization with parameters often yields important insights beyond those that you can see from special numerical cases. The parameters may be confusing so that working a numerical case helps to get started, but frequently they are indispensable for gaining scientific insight.

We want to find the profit-maximizing price as a function of the parameters. The Candidates procedure says to check the endpoints and interior critical points. We are to maximize T[p] over the compact interval [c,a/b], but we know that T=0 at the endpoints p=c and p=a/b. Hence, we calculate

This answers our question for any linear demand function. If our first cost is c₁, our first maximizing price is

What about nonlinear demand functions? See Exercise CD-11.5.4.

1. Suppose that the unit manufacturing cost for the product with demand D[p]=1000-500p rises to 0.30 per item. What price should the manufacturer charge to maximize her profit now? (In the text example she charged 1.10 at a cost of 0.20.) How much of the rise in the manufacturing cost is passed on to the consumer?

2. Suppose the demand function for another product is D[p]=2000-500p. What price should the manufacturer charge to maximize his profit in the following cases?
   • 1. The cost of producing each item is 0.30.
   • 2. The cost of producing each item is 0.50. How much of the rise in the manufacturing cost is passed on to the consumer?

   • Suppose the demand function for a third product is D[p]=1500-100p. What price should the manufacturer charge to maximize profit in the following cases?
     • 1. The cost of producing each item is 0.40.
     • 2. The cost of producing each item is 0.50. How much of the rise in the manufacturing cost is passed on to the consumer?

     • Suppose that the demand for a particular commodity is given by the function
       
       (See Example CD-11.4 for the max-min analysis of this exercise.) Find the profit maximizing price in the following cases:
       • 1. The unit cost of production is c=0.50
       • 2. The unit cost of production is c=1.00
       • 3. The unit cost of production is c=1.50

       • In view of the cases of the previous exercise what can you say about our earlier question on how much of a cost of a materials increase is passed to consumers in the nonlinear case?

         In Chapter 20, we investigate a dynamic model of price adjustment in an economy where producers do not have a monopoly. Instead, they are willing to produce goods in quantities depending on the price. That introduces the supply side.

Many max-min applications are easier to solve using implicit differentiation (first mentioned in Section CD-7.2). Review those simple examples to remind you how to use implicit differentiation and then we can apply the method to max-min with constraints.

Figure CD-11.4: Coordinate Distance

Recall that the formula for the distance between the points with coordinates (x,y) and (a,b) is

This is simply a coordinate expression of the Pythagorean theorem that the length of the hypotenuse is the square root of the sum of the squares of the lengths of the two legs of a right triangle. (English is an awkward way to express this.)

Example CD-11.3 Points on a Circle Nearest a Point

Find the points on the unit circle x²+y²=1 nearest and farthest from (1,1). We will use implicit differentiation in this basic example and ask you to generalize it and check your answer with common sense geometric reasoning in Problem 11.3.

The distance from (x,y) to (1,1) is given by but we will maximize and minimize the square of this distance. The square will be largest and smallest where the distance is, and differentiating the square is much simpler. This makes our problem:

Find

The differential of the square distance is

The differential of the constraint (when ) is

Substituting this into the dy-expression in dD gives

The critical points where (and ) are:

But the point must be on the circle, so we need to have both

Figure CD-11.5: Points on x²+y²=1 nearest and farthest from (1,1)

Example CD-11.4 Endpoints in the Implicit Distance Problem

Once you see the solution to the previous problem on Figure CD-11.5, it is geometrically obvious. However, there are two gaps in the reasoning we used and it is helpful to fill these gaps in a problem that is obvious.

First, we neglected the case y=0 when we computed the differential (since we divided by y.) In Chapter 19 we take a more geometrical approach (that does not require dividing), but, for now, we could simply solve the constraint differential for

The second gap in our reasoning is not just a technical problem. Max-min theory works most effectively in the case where the independent variable of a smooth function runs over a compact interval. In that case, we can guarantee that there is a max and a min and find them by checking endpoints and critical points. Since we have given the problem implicitly, it is a little harder to justify use of this theory. The variable x does range over the compact interval [-1,1], but we do not have the square distance given as a smooth function D=f[x]. We could apply the theory on the top and the bottom of the circle, where D=f[x] may be solved as a continuous function on a compact interval. Those details show that there has to be a max and a min and that the critical values we have found are the only candidates. (A more systematic approach is given in Chapter 19.)

Solving the systems of equations that arise in constrained max-min problems can become quite technical, but do not forget the computer.

Example CD-11.5 Help from the Computer

Find the points on the ellipse

Figure CD-11.6:

SOLUTION:

The distance is given by , but it is easier to extremize the square,

The differential of the squared distance is

The solution is shown on Figure CD-11.7. Notice the perpendicularity of the segments and tangents. See Problem 11.3. This example is also solved using a parametric equation in Problem CD-16.5.1.

Figure CD-11.7: Points on nearest and farthest from (1,1)

In this problem, we know that there is a max and a min because the variables x and y are confined to compact intervals, [-2,2] and [-3,3]. This is a little vague because the distance is not given by an explicit formula, but complete reasoning can be done along the lines of the circle example above. Sometimes it is not so easy to see where the endpoint conditions enter a problem. Here is another constrained maximization problem to illustrate this.

Example CD-11.6 Hidden Endpoints A farmer wants to build a silo with cylindrical sides and a hemispherical top. He has a fixed budget of 10,000 dollars. The sides cost 3 dollars per square foot, but the top costs 9 dollars per square foot because of the labor and materials used in making it spherical. What proportions should the farmer make the silo?

SOLUTION:

Draw a diagram of the silo and let the radius be r and the height h (measured in feet)

Figure CD-11.8: Silo

The area of the cylindrical side is , so the cost of the side is . The area of the hemispherical top is and its cost is . The sum is the total cost, which we set equal to $ 10,000: , and divide both sides by to obtain

The volume of the cylindrical part of the silo is , and the volume of the hemispherical top is , so the total volume is

The critical values, occur at r=0 and h=4r. The answer to the question is that the height should be four times the radius.

This method obscures the role of endpoints somewhat because it does not make the dependence between h and r explicit. We can reason that there are "endpoints" for r if we choose it as the independent variable, however. First, if r=0, the silo has no volume. The limit of V_total as r tends to zero should be zero, but this is not completely obvious because h must tend to infinity in order to hold the cost at 10,000 dollars. However, the microscope comes to our rescue again. The derivative is positive when r is small and positive and h is large and the cost is fixed. This means V_total increases for small values of r and cannot tend to a larger value than the one we found. (Can you write V_total explicitly in terms of r and find the limit of V_total as r tends to zero?)

At the other extreme, if h=0 (so the silo is a hemisphere), you can still spend all the money. Without solving, we can see that there is a maximal value of the radius.

1. Find the dimensions of the rectangle of largest area that can be inscribed in a semicircle of radius r.
   
   Figure CD-11.9: Rectangle inscribed in semicircle

2. A manufacturer wants to design a cylindrical can that will hold 1 liter of liquid (1 liter is 1000.0028 cubic centimeters). The total area of the outside of the can will be the sum of two disks at the top and bottom plus the cylindrical side. Express this in terms of radius and height. What should the dimensions of the can be in order to minimize the total area of the can and, hence, the cost of materials?
   
   Figure CD-11.10: Cylindrical can

3. Electronic Minima
   Find the point closest to (x,y)=(1,1) on each branch of the hyperbola
   
   The computer says that the simultaneous solutions of
   
   are and four complex roots.
   
   Figure CD-11.11:

Sometimes we can find extrema without calculus, and it is a good idea for you to compare a simple example with several methods.

Problem CD-11.1 A Geometric Minimum

• 1. We have a line L and a point p not on the line. Draw an example. The point q on the line L that is closest to p is the one such that the segment pq is perpendicular to L. Add the segment to your figure. Any other point r on L must be farther from p than q. Use basic geometry to show why. (HINT: the Pythagorean Theorem.)
• 2. Suppose the line is given by the equation and the point has coordinates (1,1). Find the coordinates of the point q. HINT: The equation of the line through p and q has slope 2 (the negative reciprocal of the slope of L) and passes through (1,1). Find the equation. Make the algebraic statement that q=(x,y) lies on both lines and solve.
• 3. Use calculus to minimize the square of the distance from (1,1) to a point r=(x,y) on the line .

Figure CD-11.12: The distance from a point to a line

Problem CD-11.2 Microscopic Geometric Minimum

Suppose the point (x,y) minimizes the distance from a point (a,b) to a curve. Why does the segment from (a,b) to (x,y) intersect the curve at right angles? (HINT: Imagine viewing the point (x,y) under a powerful microscope.)

The next exercise follows up on Example CD-11.3.

Problem CD-11.3 Another Geometric Minimum

Let a, b, and r denote constants. Use calculus to show that the points on the circle

of radius r that are nearest and farthest from the point (a,b) lie on the line through the center of the circle and the point (a,b). Notice that the slope of the radial line from (0,0) to a point (x,y) on the circle is y/x and the slope of the line from (a,b) to (x,y) is (y-b)/(x-a). What is the slope of the tangent to the circle at (x,y)? Why does the equation

say that the segment from (a,b) to (x,y) meets the circle at right angles?

We begin with a simple contrived example to illustrate how to work with a strictly symbolic max-min question.

Example CD-11.7 A Parametric Max-Min Problem

The function f[x]=x³+cx has a fixed but unknown parameter c. Find the maximum and minimum of this function for .

First, the function is continuous everywhere because we can compute its derivative, f'[x]=3x²+c, by rules and a get formula valid for all real x. This means that the Extreme Value Theorem applies to f[x] on [-2,2]. The extrema must exist and, by the Critical Points Theorem, must occur either at endpoints or critical points.

Next, we want to know how the shape of the graph depends on the parameter. Compute f'[x]=3x²+c and ask when the graph is level, f'[x]=0:

CASE 1: c>0

The number is complex. There are no real solutions to the equation: "y=f[x] is level" f'[x]=0 .

At x=0, f'[x]=c>0, so the graph y=f[x] is always increasing. This is the easiest case for max-min. Since the function is increasing, f[-2] is minimal and f[2] is maximal for the x-interval [-2,2].

Figure CD-11.13: Examples of y=x³+cx with c>0 and c<0

CASE 2: c<0

The numbers are real - one positive, one negative. The graph y=f[x] is level at these two points and nowhere else.

At x=0, f'[x]=c<0, so the graph is decreasing near zero. When |x| is huge, 3x² is bigger than c, so the graph is increasing and the shape table is "up" before , "over" at , "down" until , "over" at , and up after that.

The candidates for max and min are x=-2, x=+2 and the critical values, if they lie in the interval [-2,2]. Note that .

Figure CD-11.14: Examples of y=x³+cx with c<0 and c<<0

The more interesting question in this specific case, however, is: When is the highest critical value greater than the highest endpoint value? By analyzing the shape table, "up" - "down" - "up", we want to solve and find that this happens when . The maximum of f[x]=x³+cx for is at x=2; for , at for ; and at x=-2 for .

The minimum occurs at the negative value since the function is odd, f[-x]=-f[x], and the interval is symmetric .

Here are some simple exercises on max-min with parameters. Note that some are general cases of previous exercises. We will see other important examples of the use of parameters in the following problems and in the projects.

1. A manufacturer wants to design a cylindrical can that will hold the fixed (but unknown) amount V cubic centimeters. The total area of the outside of the can will be where r is the radius of the cylinder and h is its height. What should the dimensions of the can be in order to minimize the total area of the can and, hence, the cost of materials? (See Exercise CD-11.6.2.)

2. Show that the cylindrical can (cylinder of height h and radius r with top and bottom) of fixed surface area A of maximal volume V has h=2r, the height equal to the diameter.

3. Find the dimensions of the right circular cylinder with the largest surface area that can be enclosed in a sphere of area S. (See Exercise CD-11.3.1.)

4. A farmer wants to enclose a rectangular field with fixed (but unknown) area A square feet with the least possible amount of fencing.
   • 1. What should the dimensions of the field be?
   • 2. What is the minimal perimeter in terms of the parameter A?
   • 3. If p denotes the perimeter of a rectangular field of any dimensions with an area of A, prove the inequality
     
     
   • 4. What is the largest area A of a field with a perimeter that is fixed at a constant p? (See Exercise CD-11.3.1.)

   • A Family of Curves
     The shape of the graph y=f[x]=2cx²-x⁴ for x>0 depends on the value of c. Which values of c give f[x] a positive maximum for ? (HINT: Find the critical values in terms of c. When are they real numbers? Prove your case by making a shape table as in Section CD-9.3.)

   • Find the dimensions of the right circular cylinder with the largest volume that can be enclosed in a sphere of volume V.

   • Find the dimensions of the right circular cylinder with the largest surface area that can be enclosed in a sphere of surface area A.

The arithmetic mean is the usual average

Problem CD-11.4 The Arithmetic and Geometric Means

• 1. Use high school algebra to prove
  
  by squaring both sides and doing some algebra to put the inequality in the form
  
  
• 2. For which positive values of a and b is
  
  
• 3. The inequality is equivalent to
  
  Treat a as a constant and use calculus to minimize the function
  
  and show that the minimum is a when b=a.
• 4. Use calculus to prove
  
  for positive numbers, with equality only if a=b=c. First, this inequality is equivalent to
  
  so treat a and b as parameters and minimize the function
  
  showing that the minimum is greater than ab, unless a=b=c when you get equality.

Resonance of a vibrating system is a maximal response to forcing. Many old cars hum loudly at a speed like 46 mph, but are quieter at both slower and faster speeds. The vibration at 46 mph is maximal, at least for an interval of frequencies (a local maximum). You can hear the resonance.

Cars with worn-out shock absorbers also oscillate. In Chapter 23 we show that an idealized car's front end when forced with the pure sinusoid of frequency ,

Problem CD-11.5 Resonant Frequency of a Linear Oscillator

Find the maximum of for positive forcing frequencies . To simplify the problem, minimize the square of the denominator

by showing that it has critical values at zero and the roots of a simple quadratic equation.

The idealized front end will oscillate on its own at a certain "natural frequency" when 4ms>c². You do not get an interior maximum unless you have the condition 2ms>c². Why is that?

The natural frequency is what you observe when a car with worn shocks goes over a bump and then back onto smooth (unforced) pavement. The car keeps bouncing up and down for a long time. This natural frequency is

How does this compare to the maximal amplitude frequency (where B is minimized, ? In particular, what happens in the particular cases

How do the frequencies compare when c is very small, that is, in the limit as your shocks become completely useless? What happens to the amplitude in this limit?