What Is Marginal Average Cost?

To calculate the marginal average cost, we need to first calculate the average cost,

\displaystyle \overline{TC}\left( Q \right)=\frac{TC\left( Q \right)}{Q}

where TC(Q) is the total cost to produce Q units. Once we have the the average cost function, the marginal average cost is simply its derivative,

\displaystyle \text{Marginal Average Cost}=\overline{TC}{{\,}^{\prime }}\left( Q \right)

Example Suppose the total cost (in thousands of dollars) to produce Q units is

\displaystyle TC\left( Q \right)=\frac{9Q-5}{7Q+2}

a. Find the the average cost of producing 40 units.

b. Find the average cost function \displaystyle \overline{TC}\left( Q \right).

c. Find the marginal average cost function \displaystyle \overline{TC}{{\,}^{\prime }}\left( Q \right).

d. Find and interpret \displaystyle \overline{TC}{{\,}^{\prime }}\left( 40 \right).

m212_marg_ave_cost_1

m212_marg_ave_cost_2

m212_marg_ave_cost_3The marginal average cost is simply the slope of the tangent line to the average cost\displaystyle \overline{TC}{{\,}^{\prime }}\left( Q \right). The slope has vertical units of thousands of dollars per unit and horizontal units of units. So the rate has units of thousands of dollars per unit per unit. This means that if the production were to increase by one, the average cost would drop by 0.0007701 thousand dollars per unit or 0.7701 dollars per unit. This means that the average cost is decreasing…probably a good thing for the bottom line.

How Do You Maximize Revenue Using Elasticity?

There are several ways we can define elasticity E. Each indicates how the quantity demanded changes as the price is changed. In the examples below, we’ll utilize elasticity defined as

\displaystyle \text{E}\approx \frac{P}{Q}\,\frac{dQ}{dP}

With this definition, the revenue is maximized when E = -1.

Example 1 Suppose the demand function is defined by

\displaystyle Q=30-\frac{P}{5}

where Q units are demanded at a price of P dollars each.

a. Find an expression for the elasticity in terms of P.

b. Find the quantity at which the revenue is maximized.

m212_elasticity_1

Example 2 Suppose the demand function is defined by

\displaystyle Q=40-\frac{P}{4}

where Q units are demanded at a price of P dollars each.

a. Find an expression for the elasticity in terms of P.

b. Find the quantity at which the revenue is maximized.

m212_elasticity_2

Example 3 Suppose the demand function is defined by

\displaystyle Q=39300-7{{P}^{2}}

where Q units are demanded at a price of P dollars each.

a. Find an expression for the elasticity in terms of P.

b. Find the quantity at which the revenue is maximized.

m212_elasticity_3

How Do You Find the Economic Order Quantity?

Although many textbooks use a restrictive formula to find economic order quantity or economic lots size, you can use tables to come up with more general formulas.

Here are more examples from class.

Problem 1 A restaurant has an annual demand for 1200 bottle of wine. It costs $1 to store one bottle for a year and $5 to place an order. Orders are made when the inventory of wine is zero. If each bottle costs $15, find the optimum number of bottles per order.

The strategy above is correct, but there is a math mistake in the last board…do you see their error?

Problem 2 If the restaurant in Problem 3 orders wine when the inventory is half of an order size, find the optimum number of bottles per order.

This problem can’t be done with the formula given in Help Me Solve This. The change in when the order is made (when the inventory drops to half the order size) leads to a different expression for the storage costs. However, once this pattern is established the cost function can be maximized as before.

What Is Marginal Average Cost?

In each problem below, the average cost function by dividing the cost function by the variable representing the quantity. For a cost function C(Q), the average cost function is

\displaystyle \overline{C}(Q)=\frac{C(Q)}{Q}

The marginal average cost function is the derivative of the average cost function.

Problem 1 Suppose the total cost function for a product is

\displaystyle TC(Q)=\frac{3Q+1}{Q+2}\text{ hundred dollars}

where Q is the number of units produced.

    1. Find the average cost of producing 20 units.
    2. Find the average cost function.
    3. Find the marginal average cost function.
    4. Find and interpret the marginal average cost when 20 units are produced.

This means that each of the 20 units costs an average of .1386 hundred dollars or $13.86.

In this board they have used the fact that dividing by Q is the same as multiplying by 1/Q.

Although it is OK to leave the derivative unsimplified, they need to put in 20. So it is best to do some algebra before putting in the value. Since -0.006 is the slope of the tangent line on the average cost function, the units on it is hundreds of dollars per unit per unit:

\displaystyle \frac{-0.006}{1}\frac{\frac{\text{hundreds of dollars}}{\text{unit}}}{\text{unit}}

This means that if production is increased by 1 unit, the average cost will drop by 0.006 hundred dollars per unit.

Problem 2 Suppose the total cost function for a product is

\displaystyle C(x)=\frac{30x^{2}+500}{x+2}\text{ thousand dollars}

where x is the number of units produced.

    1. Find the average cost of producing 10 units.
    2. Find the average cost function.
    3. Find the marginal average cost function.
    4. Find and interpret the marginal average cost when 10 units are produced.

This value tells us that if production is increased by 1 unit, the average cost will drop by 0.3472 thousand dollars per unit or $347.2 per unit. Had they rounded one more decimal place, we would have had this number to the nearest penny.

Problem 3 Suppose the total cost (in thousands of dollars) to produce Q units is

\displaystyle TC\left( Q \right)=\frac{9Q-5}{7Q+2}

a. Find the the average cost of producing 40 units.

b. Find the average cost function \displaystyle \overline{TC}\left( Q \right).

c. Find the marginal average cost function \displaystyle \overline{TC}{{\,}^{\prime }}\left( Q \right).

d. Find and interpret \displaystyle \overline{TC}{{\,}^{\prime }}\left( 40 \right).

The marginal average cost is simply the slope of the tangent line to the average cost\displaystyle \overline{TC}{{\,}^{\prime }}\left( Q \right). The slope has vertical units of thousands of dollars per unit and horizontal units of units. So the rate has units of thousands of dollars per unit per unit. This means that if the production were to increase by one, the average cost would drop by 0.0007701 thousand dollars per unit or 0.7701 dollars per unit. This means that the average cost is decreasing…probably a good thing for the bottom line.

What is Elasticity?

Here are several problems classes have worked out to calculate the elasticity E,

\displaystyle E=\frac{P}{Q} \frac{dQ}{dP}

In this formula, \displaystyle \frac{dQ}{dP} is the derivative of the demand function when it is given as a function of P. Here are two examples the class worked.

Problem 1 Suppose the quantity demanded by consumers in units is given by Q=5000-5P where P is the unit price in dollars.

    1. Find the elasticity of demand with respect to price when P = 200.
    2. Find the quantity at which revenue is maximized.

This means that a 1% increase in price results in a 0.25% drop in the quantity demanded. The demand is inelastic and the price increase results in an increase in revenue.

Problem 2 Suppose the quantity demanded by consumers in units is given by Q=100-\frac{P}{2} where P is the unit price in dollars.

    1. Find the elasticity of demand with respect to price when P = 110.
    2. Find the quantity at which revenue is maximized.

This means that a price increase of 1% will lead to a 1.22% drop in demand, demand is elastic and the price increase results in a drop in revenue.

Problem 3 Suppose the quantity demanded by consumers in units is given by Q=500-0.1{{P}^{2}} where P is the unit price in dollars.

    1. Find the elasticity of demand with respect to price when P = 100.
    2. Find the quantity at which revenue is maximized.

An increase of 1% in price results in a drop in demand of 0.041%…demand is inelastic so the increase will result in an increase in revenue.