Problem 40 - Applied Calculus: For Business, Economics, and the Social and Life Sciences

Question

CONSUMER EXPENDITURE The demand for a certain commodity is \(D(x)=-50 x+800 ;\) that is, \(x\) units of the commodity will be demanded by consumers when the price is \(p=D(x)\) dollars per unit. Total consumer expenditure \(E(x)\) is the amount of money consumers pay to buy \(x\) units of the commodity. a. Express consumer expenditure as a function of \(x\), and sketch the graph of \(E(x)\). b. Use the graph in part (a) to determine the level of production \(x\) at which consumer expenditure is largest. What price \(p\) corresponds to maximum consumer expenditure?

Step-by-Step Solution

Answer

Consumer expenditure is \(E(x) = -50x^2 + 800x\). Maximum expenditure occurs at \(x = 8\) units or 3200 dollars spent when price \(P = 400\) dollars per unit.

Step 1: Identify the demand function

The demand function is given as: \[ D(x) = -50x + 800 \]

Step 2: Express consumer expenditure

Consumer expenditure, E(x), is the product of the number of units x and the price per unit D(x). Therefore, \[ E(x) = x \times D(x) = x \times (-50x + 800) \]

Step 3: Simplify the expenditure function

Multiply out the terms: \[ E(x) = -50x^2 + 800x \]

Step 4: Sketch the graph of \(E(x)\)

To sketch the graph of \(E(x) = -50x^2 + 800x\), note that it is a downward-opening parabola. Calculate key points like the vertex: The vertex of a parabola given by \(ax^2 + bx + c\) occurs at \(x = -\frac{b}{2a}\). Here, \(a = -50\) and \(b = 800\): \[ x = -\frac{800}{2(-50)} = 8 \] This means the parabola reaches its maximum at \(x = 8\).

Step 5: Find the maximum expenditure

Substitute \(x = 8\) back into E(x) to determine the maximum expenditure: \[ E(8) = -50(8)^2 + 800(8) \] \[ E(8) = -50(64) + 6400 \] \[ E(8) = -3200 + 6400 = 3200 \] So, the maximum consumer expenditure is 3200.

Step 6: Determine the price at maximum expenditure

Substitute \(x = 8\) back into the demand function \(D(x)\) to determine the corresponding price: \[ P = D(8) = -50(8) + 800 \] \[ P = -400 + 800 \] \[ P = 400 \] So, the price at maximum expenditure is 400 dollars.

Key Concepts

demand function

In economics, the demand function shows the relationship between the quantity demanded of a commodity and its price. In simpler terms, it tells us how many units of a commodity consumers are willing to buy at different prices.

• For our example, the demand function is given by \((D(x) = -50x + 800)\). Here, \(x\) represents the number of units, and \(D(x)\) is the price per unit.
• This equation indicates that as the number of units \(x\) increases, the price \(D(x)\) decreases.

Mathematical equations are often used to express demand functions because they provide a precise way to understand consumer behavior.
Observing the equation, \((D(x) = -50x + 800)\), we notice it has a negative coefficient for \(x\), which tells us that there is an inverse relationship between price and quantity demanded. This is consistent with the law of demand, which states that, all else being equal, as the price of a good increases, the quantity demanded decreases.

consumer expenditure function

Consumer expenditure is the amount of money consumers spend to buy a specific number of units of a commodity. The expenditure function represents this expenditure mathematically. In our example: \((E(x) = x \times D(x))\).

• The expenditure function tells us how much total money consumers will spend for \(x\) units, given the price \(D(x)\).
• Substituting the demand function \((D(x) = -50x + 800)\) into the expenditure function gives us: \((E(x) = x(-50x + 800))\).

Simplifying this, we get: \((E(x) = -50x^2 + 800x)\).
This quadratic equation allows us to calculate consumer expenditure for any given number of units. It also provides insights into how expenditure changes with different levels of production.

maximum expenditure

Maximum expenditure occurs at the point where the total amount of money spent by consumers is the highest. For quadratic functions, this is typically at the vertex of the parabola.

• Given \((E(x) = -50x^2 + 800x)\), we find the maximum expenditure by determining the vertex of the parabola.
• The vertex formula for a parabola \((ax^2 + bx + c)\) is: \((x = -\frac{b}{2a})\).

Plugging in the values \((a = -50)\) and \((b = 800)\), we get: \((x = -\frac{800}{2(-50)} = 8)\).
Substituting \(x = 8\) into the expenditure function: \((E(8) = -50(8)^2 + 800(8) = 3200)\).
Thus, the maximum expenditure is 3200.

price determination

The price corresponding to the maximum expenditure is found by substituting the \(x\)-value at maximum expenditure back into the demand function.

• From our previous calculations, we determined that maximum expenditure occurs at \(x = 8\).
• We then substitute \(x = 8\) into \(D(x)\)

Giving us \((D(8) = -50(8) + 800)\).
Therefore, \((D(8) = 400)\).
Thus, the price at maximum consumer expenditure is 400 dollars.

parabolic graph in economics

In economics, a parabolic graph often represents relationships where one variable changes at a non-linear rate with respect to another variable.

• For our example, the expenditure function \((E(x) = -50x^2 + 800x)\) results in a downward-opening parabola.
• This shape indicates that there is a maximum point on the graph, after which any increase in \(x\) will result in a decrease in \(E(x)\).

By plotting this graph, we can visualize how consumer expenditure changes with different quantities of the commodity. The peak of the parabola represents the maximum expenditure point, giving valuable insights into consumer behavior and spending patterns.
These graphs help economists and businesses make informed decisions about production levels and pricing strategies.