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Chapter VII

Applications of Differential Calculus

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Profit Maximization:

• The most direct way to approach profit maximization occurs when the firm knows the functional relationship relating profit to the quantity of sales.

• The general functional statement is

PR = f(Q)

where PR represents profit in dollars and Q represents the quantity of goods sold.

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Example:

A firm knows that the relationship between its weekly sales Q and weekly profit PR is expressed by the function:

PR = - 0.002Q2 + 10Q – 4000

The firm wants to determine the profit-maximizing level of weekly sales.

A possible optimum sales point is determined by setting the first derivative equal to 0 and solving for Q:

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PR = - 0.002Q2 + 10Q – 4000

d(PR)/dQ = - 0.004Q + 10

= 0 = - 0.004Q + 10

0.004Q = 10

Q = 2500

The second-derivative test is applied to confirm that Q = 2500 is a maximum point on the profit function:

d2(PR)/dQ2 = - 0.004

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• As the second derivative is equal to a negative constant, so a sales level of 2500 units per week will maximize profit.

• The profit corresponding to this sales level can be determined by substituting Q = 2500 into the profit function:

PR = - 0.002Q2 + 10Q – 4000

= (- 0.002)(2500)2 + 10(2500) – 4000

= $8500

• Thus, the maximum profit for the firm is $8500 per week, achieved by selling 2500 units per week.

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Profit Maximization

Principles of Marginal Analysis:

• Business decisions often are based on marginal analysis.

• For example, should an additional worker be hired, or should an additional unit of output be produced?

• These concepts can be applied to finding the profit-maximizing level of output for a firm.

• The change in total production cost resulting from production of an additional unit of output is the marginal cost of that unit.

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• The marginal cost of a unit of output is the value of the first derivative of the total cost function at that point.

Total Cost

Marginal Cost

Output

Cost

Fig. 1: A total cost function and corresponding marginal cost function

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Example 2:

• A typewriter manufacturer has the following total cost function relating total cost TC to quantity sold Q:

TC = 0.05Q3 – 0.2Q2 + 17Q + 7000

• Total cost is measured in dollars and quantity is measured in number of typewriters.

• This is a typical total cost function with fixed costs equal to $7000.

• The marginal cost function is equal to the first derivative of the total cost function which is found as follows:

MC = d(TC)/dQ = 0.15Q2 – 0.4Q + 17

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• For example, at an output of 40 typewriters, marginal cost MC is:

MC(40) = (0.15)(40)2 – (0.4)(40) + 17

240 – 16 + 17 = $241

• The above result implies that the additional cost of 40th typewriter is $241.

• Marginal revenue MR is the change in total revenue from selling one more unit.

• Total revenue TR is equal to the price of the product times the number of units sold P X Q and marginal revenue is the first derivative with respect to Q of the total revenue expression.

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• In some cases, firms do not have direct knowledge of their total revenue function but know only the expression for demand, stating price as a function of the quantity sold, P = f(Q).

• A demand function of this form can be considered a per unit or average revenue AR function.

• This is because

TR = P X Q

P = TR/Q = per unit or average revenue

• Knowing only the demand function of the form P = f(Q), the total revenue function can be found by multiplication of the demand function by Q.

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Quantity

Total revenue

Average revenue

Marginal revenueQM

Pric

e

$

Quantity

(a)

(b)

Fig.

3:

Tota

l rev

enue

, ave

rage

rev

enue

, and

mar

gina

l re

venu

e fu

nctio

ns.

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• In Fig. 3, total revenue is a quadratic function and both the average and marginal revenue functions are linear with negative slope coefficients.

• Marginal revenue is equal to zero at the quantity where total revenue is maximized, QM on Fig. 3.

Marginal Analysis and Profit Maximization:

• In attempting to obtain maximum profit, a firm wants to maximize total, not marginal or final unit profits.

• If the revenue gained by selling the item (marginal revenue) exceeds the marginal cost of its production, sale of the good will increase the total profit.

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• If the marginal cost exceeds marginal revenue for an item, its sale will reduce total profit and the item should not be offered for sale.

Marginal cost

Marginal revenue

Q*Quantity

$

Fig. 4: Marginal revenue and marginal cost function.

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• Over the relevant range of production options for the firm, marginal revenue is a decreasing or constant function of quantity sold and marginal cost is an increasing function of the quantity sold.

• In Fig. 4, at points less than Q*, total profit can be increased by the sale of additional units as the additional revenue obtained from each unit exceeds the additional cost of that unit.

• At quantity levels greater than Q*, in contrast, profits are decreased for each additional unit sold.

• Total profit is maximized at the quantity point corresponding to the equality between marginal revenue and marginal cost.

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Example 3:

For the following total cost function TC and demand function P = f(Q), determine the quantity and price which will maximize profit. Also, determine this profit.

The total cost and demand functions, where Q represents quantity of typewriters in each function, are

TC = 0.05Q3 – 0.2Q2 + 17Q + 7000

P = 557 – 0.2Q

Solution: To solve the problem, the marginal revenue and marginal cost functions are found and set equal to each other.

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TR = P X Q

= 557Q – 0.2Q2

MR = d(TR)/dQ = 557 – 0.4Q

TC = 0.05Q3 – 0.2Q2 +17Q + 7000

MC = d(TC)/dQ = 0.15Q2 – 0.4Q + 17

• The next step includes equating the two marginal functions and solving for the profit-maximizing quantity Q*:

MR = MC

557 – 0.4Q = 0.15Q2 – 0.4Q + 17

Q2 = 3600

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Q = Q* at the point of maximum profit

Q* = 60

• Therefore, to maximize total profit the firm should sell 60 typewriters per week.

• Selling less than 60 will lead to foregone profit and selling more than 60 will lead to a loss being incurred on each additional unit incurred.

• Price corresponding to the quantity point of 60 is found by substitution of Q = 60 into the demand function

P = 557 – 0.2Q = $545

• A price of $545 per typewriter will permit 60 typewriters to be sold and lead to maximization of profit.

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Total revenue TR = 557Q – 0.2Q2 = $32, 700

Total cost TC = 0.05Q3 – 0.2Q2 + 17Q + 700

= $18, 100

Profit = total revenue – total cost

= 32, 700 – 18, 100 = $14, 600

• It is determined that maximum profit per week at a price of $545 and a quantity of 60 typewriters is $14, 600.

• The same result can be obtained by computing a single profit function consisting of total revenue minus total cost and solving for price and quantity at the maximum profit.

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Class Assignment:

1. For the total cost TC and total revenue TR function, find the corresponding marginal cost and marginal revenue function.

a. TC = 0.03Q3 + 5Q2 – 12Q + 400

b. TR = Q3 + 0.2Q2 + 6Q

2. The total revenue TR and total cost TC functions for a certain brand of desk-top computer are as follows:

TR = - 3Q2 + 216Q

TC = 0.08Q3 – 3Q2 + 120Q + 200

where Q represents the number of desk-top computers.

a. Develop a profit function for the computer.

b. Find the quantity of computers corresponding to the maximum profit point.

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Cost Analysis

Finding a Firm’s Shut-Down Price:

• In contrast to fixed costs such as facility rent, variable costs such as labor and utilities are those elements of production costs which change as output changes.

• Knowledge of the dollar value of minimum average (per unit) variable cost is useful to a firm.

• If product price (average revenue) does not meet or exceed average variable cost, the firm should stop production.

• This does not imply that the firm goes out of business because the fixed costs (i.e., rent) must continue to be paid.

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• If price does not meet or exceed the average variable cost and production continues, the firm loses not only its fixed costs but also a part of the variable cost on every unit produced.

• When product price falls below average variable cost, the firm will minimize its losses by ceasing production.

• The price corresponding to minimum average variable cost specifies the lowest possible price that the firm can face in the market and economically continue production.

• Frequently, this price is referred to as the firm’s shut-down price.

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Average Variable Cost

Average revenue

Q1

P1“Shut-Down”

Price

Price

Quantity

Fig. 5: The “Shut-Down” price for a firm.

• Fig. 5 shows a quadratic average variable cost curve.

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• At its minimum point (Q1, P1), a demand (average revenue) function touches the average variable cost curve.

• The price point P1 corresponds to the lowest price a firm will accept for its product and continue production of the good.

Example 4:

The following is the total cost function TC for voltage regulators used by various industries. Here, total cost is measured in dollars, and it is a function of the quantity of voltage regulators produced Q:

Total cost TC = Q3 – 24Q2 + 230Q + 500

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• The constant term 500 represents fixed cost and is not relevant in the present analysis.

• The remainder of the function expresses total variable cost TVC:

TVC = Q3 – 24Q2 + 230Q

Average variable cost AVC is a per unit measure of TVC:

AVC = TVC/Q = Q2 – 24Q + 230

• To minimize AVC, the first derivative is computed and set equal to zero:

d(AVC)/dQ = 2Q – 24 = 0

Q = 12

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• The quantity point of 12 is a minimum point as the second derivative is equal to +2.

• At a quantity of 12 voltage regulators, the firm will reach minimum average cost.

• The dollar value of AVC at this quantity is found by substituting Q = 12 into the AVC function, and this is comparable to the lowest acceptable price to the firm. The price is $86.

AVC (12) = (12)2 – (24)(12) + 230 = $86

Shut-down price = $86

• The firm should cease production if the voltage regulators is less than $86.

• If the price is less than $86, the firm will minimize its lossesby stopping production and meeting only its fixed-cost obligations.

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Diminishing Marginal Returns of Inputs:

• The use of inputs (labor, machinery, management, etc.) often is associated with the principles of diminishing marginal returns.

• It indicates that the additional output because of the use of additional units of a resource increases up to a point thereafter declines.

• The decline in marginal returns occurs because inefficiencies are encountered as resource use expands.

• At the point where the marginal returns of an input stop increasing and start to decrease, marginal returns are at their maximum point.

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• When marginal returns from inputs are increasing, the marginal cost of output is decreasing.

• Decreasing marginal returns from inputs are associated with an increasing marginal cost of output.

• Consequently, the point of change between increasing and decreasing marginal returns and decreasing and increasing marginal cost is of importance to the operation of a firm.

• If total cost TC is a function of the quantity produced Q, and the quantity produced Q is a function of a resource input R, then:

Total cost function: TC = f(Q)

Production function: Q = f(R)

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• A production function is a quantitative relationship between an input factor or factors and the final output of a good or service.

• Marginal cost MC is the first derivative with respect to quantity of the total cost function.

MC = f’(Q)

• The following explains the relationship between decreasing and increasing marginal returns to inputs and increasing and decreasing marginal cost:

1. If dQ/dR > 0 then d(MC)/dQ < 0

(increasing marginal returns to inputs)

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2. If dQ/dR < 0 then d(MC)/dQ > 0

(decreasing marginal returns to inputs)

3. If dQ/dR = 0 then d(MC)/dQ = 0

Example 5:

A manufacturer of electric hot-water heaters knows the total cost function for the production process is

1. TC = 0.1Q3 – 12Q2 + 800Q + 2100

where TC represents total cost in dollars and Q represents the quantity of electric hot-water heaters.

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Find the quantity of hot-water heaters corresponding to the point of minimum marginal cost.

To determine this quantity, the marginal cost function is derived as follows:

2. MC = d(TC)/dQ = 0.3Q2 – 24Q + 800

To find a stationary point on this function, the first derivative is set equal to zero and the result is solved for Q. This is shown in steps 3, 4, and 5.

3. d(MC)/dQ = 0.6Q – 24

4. d(MC)/dQ = 0 = 0.6Q – 24

5. 0.6Q = 24

Q = 40

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6. d2(MC)/dQ2 = 0.6

Step 6 demonstrates that the point Q = 40 represents minimum marginal cost since the second derivative is positive at that point.

The marginal cost at Q = 40 is

7. MC = 0.3Q2 – 24Q + 800

MC(40) = 0.3(40)2 – 24(Q) + 800 = 320

• The point Q = 40 with a marginal cost of $320 represents minimum marginal cost.

• At this point, returns from use of input are at their highest level.

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• At outputs higher than 40, diminishing marginal returns from use of input, and a corresponding increase in the marginal cost of output, take effect.

• Thus, Q = 40 represents an output of maximum marginal returns from use of input and minimum marginal cost of producing electric hot-water heaters.

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Marginal Productivity Analysis

• The basic marginal productivity measurement is referred to as the marginal physical product of a resource use.

• Marginal physical product is measured in physical output, not dollars.

• It is an expression of the additional output resulting from the application of an additional unit of input, such as one more laborer or one more hour of machine time.

• It is found by evaluating the instantaneous rate of change (first derivative) of the production function at a particular level of input use.

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• The first derivative of the production function with respect to one input is the marginal physical product of that input.

• The marginal physical product of a resource multiplied by the market value of the output produced by that resource yields a dollar figure representing the monetary value of the input to the firm.

• The concept of marginal physical product is an important component of models concerned with wages, machinery usage, and other questions of resource allocation.

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Example 6:

For a steel manufacturer using a particular process, output per day in tons of steel Y is expressed as a function of labor-hours X1 and machine-hours X2:

Y = X11/4 X2

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Find the expression for the marginal physical product of each input. Also, find the marginal physical product of each at an application of 16 labor-hours and 81 machine-hours.

Solution:

Marginal Physical Product of Labor:

1. MPP(X1) = Y/ X1 = X11/4[ (X2

3/4)/ X1] + X2

3/4[ (X11/4)/ X1]

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Y/ X1 = X11/4(0) + X2

3/4 (1/4) X1-3/4

= (¼)X1-3/4X2

3/4

Marginal Physical Product of Machinery:

2. MPP(X2) = Y/ X2 = X11/4[ (X2

3/4)/ X2] + X2

3/4[ (X11/4)/ X2]

Y/ X2 = X11/4(3/4)X2

-1/4 + X23/4(0)

= (3/4)X11/4X2

-1/4

1. MPP(X1) at X1 = 16 and X2 = 81:

MPP(X1) = (¼)X1-3/4X2

3/4 = 27/32 of a ton of steel

2. MPP(X2) at X1 = 16 and X2 = 81:

MPP(X2) = (¾)X11/4X2

-1/4 = ½ of a ton of steel

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• Thus, for 16 labor-hours and 81 machine-hours, the marginal physical product of labor is 27/32 ton of steel and the marginal physical product of machinery is ½ ton of steel.

• Each amount represents the incremental change in steel production in response to a 1-unit change in the respective input at the stated points.

• For example, if labor-hours are increased from 16 to 17 and machinery is held constant at 81 hours per day, steel production will increase by 27/32 ton of steel per day.

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Class Assignments:

1. For each total cost function TC, find the average variable cost function and shut-down price.

a. TC = 0.6Q3 – 24Q2 + 410Q +1500

b. TC = 0.004Q3 – Q2 + 80Q + 1600

2. For each marginal cost function MC, find the quantity Q corresponding to minimum marginal cost. Find marginal cost at this point and show that the point is a minimum. State the importance of this point to the analysis of a firm’s production process.

a. MC = 0.1Q2 – 6Q + 170

b. MC = 3Q2 – 180Q + 2900

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3. For the function given below, find the marginal physical product of X1 at X1 = 8 and X2 = 4 and marginal physical product of X2 at X1 = 5 and X2 = 4.

Y = 0.03X13 – 0.4X1X2 + 0.6X2

1/2

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Elasticity of Demand

• In business, “elasticity” is a term of general application used to express the change in one variable in response to a given change in a second variable.

• Some of the most common elasticity measurements refer to changes in the demand for a good or service caused by various other factors, and this group is referred to as elasticity of demand measurements.

Examples:

• Income Elasticity of Demand – It is the percentage change in the quantity demanded of a good in response to a percentage change in consumer income.

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• Cross-Price Elasticity of Demand – It is the percentage change in the quantity demanded of one good in response to a percentage change in the price of a second good.

Price Elasticity of Demand:

• It evaluates the percentage change in the quantity demanded of a good in response to a percentage change in the price of the good.

Price elasticity of demand for X =

Percentage change in the quantity demanded of X

Percentage change in the price of X

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• The price elasticity of demand can be computed at a single point on the demand function or over a range of the function, yielding the point price elasticity of demand and the arc price elasticity of demand, respectively.

• Since differential calculus is used to determine the rate of change of a function at a point, so point price elasticity of demand will be discussed here.

Point Price Elasticity of Demand –

Price elasticity of demand for X =

(ΔQX/Qx)/(ΔPX/Px) = (ΔQX/ΔPX)(PX/QX)

where QX and PX refer to the original quantity and price points, respectively.

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• In the above expression, ΔQX/ΔPX can be stated dQ/dP because ΔPX approaches the limit of zero at a point.

Point price elasticity of demand (εp) = (dQ/dP)(P/Q)

• When demand function is of the form P = f(Q), dQ/dP is obtained by the inverse function rule, i.e., dQ/dP = 1/(dP/dQ), to determine the point price elasticity of demand (εp).

εp = {1/(dP/dQ)}(P/Q)

Example 7:

The demand, or average revenue, function for television is

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P = 450 – 0.4Q

where P is expressed in dollars and Q represents the number of televisions demanded by consumers.

A firm wants to determine the point price elasticity of demand at a quantity of 120 televisions.

Solution:

First: The price corresponding to Q = 120 is

P = 450 – 0.4Q = 450 – 0.4(120) = $402

Therefore, the coordinates of the relevant point on the demand curve are Q = 120, P = $402

Second: The rate of change at the above point must

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be determined by use of the inverse function rule:

P = 450 – 0.4Q

dP/dQ = - 0.4

dQ/dP = 1/(dP/dQ) = - 2.5

Third: The point price elasticity of demand εp is computed at Q = 120:

εp = (dQ/dP)(P/Q) = (- 2.5)(402/120) = - 8.375

• At this point, the point price elasticity of demand indicates that a 1 percent change in price (either an increase or decrease) from $402 will result in an 8.375 percent change in the quantity demanded in a direction opposite to the price change.

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• As a demand curve specifies an indirect relationship between price and quantity demanded, the sign on the point price elasticity of demand is always negative.

• By considering the absolute value of the point price elasticity of demand, |εp|, the categories can be described as shown below:

1. If |εp| < 1, demand at that point is inelastic with respect to price.

2. If |εp| > 1, demand at that point is elastic with respect to price.

3. If |εp| = 1, demand at that point is unitary with respect to price.

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• Knowledge of the price elasticity of demand is useful to a firm in computing the “sensitivity” of sales to price changes.

• Necessities such as bread or household electricity exhibit inelastic demand; thus, their price sensitivity is low.

• Luxury items like jewelry tend to have a high price sensitivity (elastic demand).

• Therefore, it is important for firms to understand a product’s elasticity of demand when developing pricing and marketing strategies.

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Other Elasticity Measurements:

• For any dependent variable Y and independent variable X, a general statement of the elasticity of Y with respect to X at a point of Y is stated as follows.

• For Y = f(X), the percentage change in Y in response to a percentage change in X at (x, y) is expressed as

(dY/dX)(x/y)

where x and y represent the coordinates of the point under consideration.

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• At times, it is necessary to find the elasticity associating the dependent variable with one independent variable in a function with several independent variables.

• If Y is the dependent variable and Xi is the particular independent variable used in the elasticity figure, the elasticity of Y with respect to Xi at the point (xi, y) is stated as follows: Percentage change in Y in response to a percentage change in Xi at (xi, y) is:

( Y/ Xi)(xi/y)

where xi represents the value of the variable Xi at the point under consideration.

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• For the computation of the above, all other Xi must be held constant at specific stated values.

Example 8:

The weekly sales of home computers in a city is expressed by the following function:

Qc = 30 – 0.02P + 2.1I

Where Qc = weekly sales of home computers

P = average retail price of home computers

I = average family income in thousands

A firm wants to determine the income elasticity of demand at an average family income of $20, 000 (I = 20) and average retail price of $600 (P = 600).

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Solution:

• The firm wants to determine the percentage change in computer sales in response to a percentage change in income at an income level of $20,000 holding price constant at $600.

1. Qc = 30 – 0.02P + 2.1I

2. Find Qc at I = 20 and P = 600

Qc = 30 – 0.02P + 2.1I

= 30 – (0.02)(600) + (2.1)(20)

= 30 – 12 + 42 = 60

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3. Find (∂Qc/∂I)I/Qc:

(∂Qc/∂I)(I/Qc) = (2.1)(20/60) = 42/60 = +0.7

4. Income elasticity of demand at an income of $20,000, holding price constant at $600, is equal to +0.7.

• This shows that a 1 percent increase (decrease) in family income will increase (decrease) home computer sales by 0.7 percent. This relation is valid if the average price is held constant at $600.

Example:

Find the point price elasticity of demand at a price $600 (P = 600) and an income level of $20,000

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per year (I = 20) for the following demand function for home computers Qc:

1. Qc = 30 – 0.02P + 2.1I

= 30 – (0.02)(600) + (2.1)(20) = 60

2. Point price elasticity of demand is

(∂Qc/∂P)(P/Qc) = (- 0.02)(600/60) = -12/60 = - 0.2

• This indicates that at a constant income level of $20,000 and constant price of $600, the demand is inelastic. Thus, the percentage change in sales will not be as great as the percentage change in price at this point of the function.

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Class Assignment:

Find the point price elasticity of demand at each specified point:

1. P = 32 – 0.004Q at Q = 5000

2. P = 70 – 0.02Q at Q = 800