Problem 13 - The Calculus with Analytic Geometry

Question

The demand equations for two commodities that are produced by a monopolist are $$ x=6-2 p+q \quad \text { and } \quad y=7+p-q $$ where \(100 x\) is the quantity of the first commodity demanded if the price is \(p\) dollars per unit and \(100 y\) is the quantity of the second commodity demanded if the price is \(q\) dollars per unit. Show that the two commodities are substitutes. If it costs \(\$ 2\) to produce each unit of the first commodity and \(\$ 3\) to produce each unit of the second commodity, find the quantities demanded and the prices of the two commodities in order to have the greatest profit. Take \(p\) and \(q\) as the independent variables.

Step-by-Step Solution

Answer

Both commodities are substitutes. Quantities demanded are approximately 461.11 units for the first commodity and 751.85 units for the second commodity with prices \ p = 0.8704\ and \ q = -0.3519 \.

Step 1: Simplify Demand Equations

The given demand equations are: \( x = 6 - 2p + q \) and \( y = 7 + p - q \). Since both equations represent quantities demanded, they need simplification.

Step 2: Identify Substitutes

To demonstrate that the commodities are substitutes, observe that an increase in the price of one commodity (\( p \) or \( q \)) results in an increase in the quantity demanded of the other commodity. For example, if \( p \) increases, both \( x \) and \( y \) decrease if no changes to \( q \) occur. Similarly, if \( q \) increases, \( x \) increases while \( y \) decreases.

Step 3: Set Up Profit Function

The profit function is given by the revenue from selling the commodities minus the production costs. The revenue function is: \( R = p \times 100x + q \times 100y \) while the cost function is:\( C = 2 \times 100x + 3 \times 100y \). Thus, the profit function becomes: \( P = R - C = 100px + 100qy - 200x - 300y \).

Step 4: Substitution into Profit Function

Substitute the values of \( x \) and \( y \) into the profit function. From the demand equations, substitute: \( x = 6 - 2p + q \) and \( y = 7 + p - q \) The profit function becomes: \( P = 100p(6 - 2p + q) + 100q(7 + p - q) - 200(6 - 2p + q) - 300(7 + p - q) \)

Step 5: Simplify the Profit Function

Expand and simplify the profit function: \( P = 600p - 200p^2 + 100pq + 700q + 100pq - 100q^2 - 1200 + 400p - 200q - 2100 - 300p + 300q \). Collect like terms: \( P = -200p^2 - 100q^2 + 1000pq + (600p + 400p - 300p) + (700q - 200q + 300q) - 1200 - 2100 \). Which simplifies to: \( P = -200p^2 - 100q^2 + 1000pq + 700p + 800q - 3300 \).

Step 6: Maximize Profit

To maximize profit, take partial derivatives with respect to \( p \) and \( q \) and set them equal to zero: \( \frac{\text{d}P}{\text{d}p} = -400p + 1000q + 700 = 0 \) \( \frac{\text{d}P}{\text{d}q} = -200q + 1000p + 800 = 0 \).

Step 7: Solving Equations for \(p\) and \(q\)

Solve these simultaneous equations: From \(\frac{\text{d}P}{\text{d}p} = 0\): \( -400p + 1000q + 700 = 0 \) => \( 400p - 1000q = 700 \) => \( 4p - 10q = 7 \) => \( 4p = 10q + 7 \) ----(1) From \(\frac{\text{d}P}{\text{d}q} = 0\): \( -200q + 1000p + 800 = 0 \) => \( -1000p + 200q = 800 \) => \( -1000p + 800 = 200q \) => \( q = -5p + 4 \) ----(2) Substitute (2) into (1): \( 4p = 10(-5p + 4) + 7 \) => \( 4p = -50p + 40 + 7 \) => \( 4p + 50p = 47 \) => \( 54p = 47 \) => \( p = \frac{47}{54} \). Substitute \( p \) back in (2): \( q = -5\frac{47}{54} + 4 \) => \( q = -\frac{235}{54} + \frac{216}{54} \) => \( q = -\frac{19}{54} \).

Step 8: Compute Quantities Demanded

Now that \( p \) and \( q \) are found, calculate the quantities demanded: \( x = 6 - 2(\frac{47}{54}) - \frac{19}{54} \) => \( x = 6 - \frac{94}{54} + \frac{19}{54} \) => \( x = 6 - \frac{75}{54} \) => \( x = 6 - 1.3889 \) => \( x = 4.6111 \) \( 100x = 461.11 \) Similarly for \( y \): \( y = 7 + \frac{47}{54} - \frac{19}{54} \) => \( y = 7 + \frac{28}{54} \) => \( y = 7 + 0.5185 \) => \( y = 7.5185 \) \( 100y = 751.85 \).

Key Concepts

demand equations

Demand equations are mathematical expressions that show how the quantity demanded of a commodity responds to changes in its price. In this exercise, we have two demand equations: \[ x = 6 - 2p + q \] and \[ y = 7 + p - q \]. Here, \(x\) and \(y\) represent the quantities demanded for two different commodities. The constants and coefficients in these equations indicate how sensitive each commodity's demand is to changes in their respective prices \(p\) and \(q\). Understanding these equations is fundamental in analyzing market behavior and predicting how changes in prices can impact demand.

profit maximization

Profit maximization is a key objective for businesses, involving adjusting prices and production levels to achieve the highest possible profit. In general, profit is calculated as the difference between total revenue (TR) and total cost (TC). For this problem, the total revenue includes the earnings from selling each commodity, and the total cost includes the production expenses. The profit function \( P \) is defined as: \[ P = R - C \]. Substituting revenue and cost functions gives us: \[ P = 100px + 100qy - 200x - 300y \]. The goal is to find the price levels \( p \) and \( q \) that maximize this profit function.

partial derivatives

Partial derivatives are used in multivariable calculus to find the rate of change of a function with respect to one variable while keeping others constant. In profit maximization problems, partial derivatives help identify critical points where the profit may be maximized or minimized. By taking the partial derivatives of the profit function \( P \) with respect to the prices \( p \) and \( q \), we find: \[ \frac{\partial P}{\partial p} = -400p + 1000q + 700 = 0 \] and \[ \frac{\partial P}{\partial q} = -200q + 1000p + 800 = 0 \]. Solving these equations simultaneously allows us to find the optimal values for \( p \) and \( q \) that maximize the profit.

monopoly market

A monopoly market is characterized by a single seller who controls the entire supply of a commodity or service, giving them significant power to set prices. In this exercise, the monopolist produces two commodities, meaning they have substantial influence over the pricing and demand. The monopolist's strategy involves setting prices \( p \) and \( q \) to maximize profit while considering the demand equations for each commodity. Understanding how a monopoly market functions is crucial for accurately analyzing and predicting price and quantity outcomes in such scenarios.

economic substitutes

Economic substitutes are goods that can replace each other in consumption, meaning that if the price of one rises, the demand for the other one increases. In this problem, the demand equations indicate that when the price \( p \) of the first commodity increases, the quantity demanded \( y \) of the second commodity also increases, demonstrating a substitution effect. Therefore, the commodities are substitutes. Confirming that two commodities are substitutes helps understand consumer behavior and can guide pricing strategies to maximize overall demand and profit.