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Wize University Calculus 1 Textbook > Applications of Differentiation for Business & Econ

Marginal Concepts In Economics

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Cost, Revenue, and Profit

We can use our knowledge of calculus to solve many problems in economics.

Given the following:
  • xx, the number of items produced
  • C(x)C(x), the cost function, which represents the total cost of producing xx items
  • p(x)p(x), the price function, which represents the selling price for one item

Revenue

The revenue function R(x)R(x) represents the total revenue from the production of xx items, and is defined to be the number of items sold times the price of each:
R(x)=x⋅p(x)\boxed{\quad R(x)=x\cdot p(x) \quad}

Profit

Then the profit function P(x)P(x) is defined to be the difference between revenue and cost of production:
P(x)=R(x)−C(x)\boxed{\quad P(x)=R(x)-C(x) \quad}

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Average Change Functions

An average quantity is usually determined by taking all of the function values and dividing by how many we have. More specifically, an average function is constructed by diving the original function by xx, which represents the "number of items".

Average Cost is the cost function divided by xx: Cave=C(x)xC_{ \text{ave}}=\dfrac{C(x)}{x}

Average Revenue is the revenue function divided by xx: Rave=R(x)xR_{ \text{ave}}=\dfrac{R(x)}{x}

Average Profit is the profit function divided by xx: Pave=P(x)x P_{ \text{ave}}=\dfrac{P(x)}{x}

Marginal Functions

In business, a marginal quantity represents the instantaneous rate of change with respect to the number of items produced, xx

Marginal Cost is the derivative of the cost function: C′(x)C'(x)

Marginal Revenue is the derivative of the revenue function:R′(x)R'(x)

Marginal Profit is the derivative of the profit function: P′(x)P'(x)
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Example: Marginal Cost

Suppose that a company’s total cost (in dollars) to carry out a production of xx units of its product is given by

C(x)=x2512+10x+100C(x)=\displaystyle\frac{x^{2}}{51^{2}}+10x+100

a) Find the average cost for the first 51 units produced.

b) Find the marginal cost for the first 5151 units produced.


Part a)

Average cost is defined: Cave=C(x)xC_{ \text{ave}}=\dfrac{C(x)}{x}


C(x)x=x2512+10x+100x=x512+10+100x\dfrac{C(x)}{x}=\dfrac{\displaystyle\frac{x^{2}}{51^{2}}+10x+100}{x}=\displaystyle \frac{x}{51^{2}}+10+\frac{100}{x}


Cave(51)=51512+10+10051C_{ \text{ave}}(51)=\displaystyle \frac{51}{51^{2}}+10+\frac{100}{51}


=151+10+10051=\dfrac{1}{51}+10+\dfrac{100}{51}

=61151≈11.98/item=\dfrac{611}{51}\approx11.98/\text{item}

Which is the average cost of producing all the items produced so far (all 51)

Part b)

Marginal cost = the derivative of the cost function.

Differentiate and evaluate at x=51x=51 :

C′(x)=2x512+10C'(x)=\displaystyle\frac{2x}{51^{2}}+10

⇒2512 (51)+10\Rightarrow \displaystyle\frac{2}{51^{2}}\ (51)+10

=251+10=\displaystyle\frac{2}{51}+10

=51251≈10.04/item=\displaystyle\frac{512}{51}\approx 10.04/\text{item}

Which is the cost for producing one item at this moment (after 5151 items have already been produced).

If the selling price of xx items is given by the function p(x)=502+xp(x)=\dfrac{50}{2+x} , what is the marginal revenue?





Given that the price of a commodity, in dollars, can be modeled as p(x)=12−xp(x)=12-x , and the cost of production for one item can be modeled as: C(x)=x+1C(x)=x+1, find the following:

a) The marginal profit for the 3rd item sold

b) When should the company stop producing?



Extra Practice
Suppose Thor's Hammer Makers increases its output from 20 to 25 hammers. Its total cost increase from $500 to $750. What is the company's average rate of change of the cost of producing a hammer? Enter a whole number.
If the price of pizza is p, and the number of pizzas sold is q the pizza shop will observe the following relation 4p2+q2=324p^2+q^2 = 32. What price should the pizzas be sold at in order to maximize revenue (r), r = pq.
Law of Demand
A manufacturing company produces a variety of products, but prices them at $3 per unit. Every month, the company sells 25,000 units and for every $0.5 increase in price, the number of products sold decreases by 1,500. There is a flat maintenance cost of $100,000 for the factory, and each unit requires $1 to make. However, the Province of BC subsidizes the company $20,000 every month (i.e. the company gets $20,000 every month, no matter how many products they sell).
a) Find the linear demand equation of these products. Use pp for unit price and qq for monthly demand.
b) Find the monthly cost function C(q)C(q) as a function of qq .
Law of Demand
Producing qq motherboards has a cost function of C(q)=4q2125+60q+112500.C(q)=\frac{4q^2}{125}+60q+112500. The price at which these motherboards can be sold is $120, when 10,000 units are sold. For every extra 1,000 units sold, the price of the motherboard drops by $8.
a) Find the linear demand of motherboards. Use pp for unit price, and qq for weekly demand.
b) Find the weekly revenue function R(q)R(q) as a function of qq .
Law of Demand
The Regno: Custom Millwork Co. manufactures and sells custom furniture. For desks, the monthly demand is 40 units at a price of $800. When the cost of the desk increases by $50, the number of desks sold decreases by 2. Assume that it costs $300 to produce each desk and that the fixed monthly cost is $20,000.
a) Find the linear demand equation for the Regno: Custom Millwork's desk. Use pp for the unit price and qq for the monthly demand.
b) Find the monthly revenue function R(q)R(q) as a function of qq .
Marginal Concepts in Economics
The total cost of producing x pairs of shoes is C(x)=0.5x3+x2C\left(x\right)=0.5x^3+x^2
The total revenue of producing x pairs of shoes is R(x)=x4+0.5x3+2x2R(x)=x^4+0.5x^3+2x^2
a) Determine the average cost function and evaluate it at 4 pairs of shoes.
The price of BYC Colognes is $100. The firm’s cost function is C(q)=2q2+450C\left(q\right)=2q^2+450. Find the profit-maximizing quantity and the profit.
Find qqwhere the average cost is at its minimum.
Given the supply curve S(x)=4x+5S(x) = 4x + 5 the ratio of the change in input over the change in output is a constant. This statement is:
A company manufactures and sells xx electric drills per month. The monthly cost is defined as C(x)=68000+40xC(x)=68000+40x, and the price-demand relationship is given as p=190−x20p=190-\dfrac{x}{20}. Find the production level that results in the maximum profit.
The selling price for xx objects is given by the function p(x)=1005+xp(x)=\dfrac{100}{5+x} . Find the marginal revenue.
Q:\bf{Q:} Given the price of a pizza pp , and the number of pizzas sold qq. We know that q=f(p)q=f(p). Given f(10)=16f(10)=16 and f′(10)=5f'(10)=5. If the revenue as a function of price is denoted by R(p)R(p), what is R′(10) ?R'(10)\ ?
If the relationship of price and demand is governed by p=100−4q2p=100-4q^2, find the revenue as a function of qq. When the revenue is maximized, how many items are sold?
Marginal Concepts in Economics
Given that the price of a commodity, in dollars, can be modeled as p(x)=12−xp(x)=12-x , and the cost of production for one item can be modeled as: C(x)=x+1C(x)=x+1, find the following:
a) The revenue function
b) The revenue generated from selling 3 items
You plan on putting away $10,000 today in a special compound savings account. Your bank account pays 9% per year. After how many whole years will you have at least $19,000?
Concepts in Economics: Revenue and Profit Functions
A company selling DVD's finds out that the cost of production and the revenue can be modeled by the following functions: C(x)=200+100x4C(x) = 200 + 100\sqrt[4]{x} and R(x)=120+90xR(x) = 120 + 90\sqrt{x} . Determine:
a) The average cost, revenue and profit functions.
b) The rate at which the average profit is changing when 5 DVD's are sold.
Concepts in Economics: Supply and Demand Functions
The supply and demand functions are given by s(x)=x2−20xs(x)=x^2-20x and d(x)=1200−30xd(x)=1200-30x. Find the market equilibrium point.
Marginal Concepts in Economics
If the selling price of xx items is given by the function p(x)=502+xp(x)=\dfrac{50}{2+x} , what is the marginal revenue?
Q:\bf{Q:} If the price function is p=200−83q2p=200-\dfrac{8}{3}q^2 , find the revenue as a function of the number of items qq. When the revenue is maximized, how many items are sold?
Q:\bf{Q:} A small manufacturer wholesales leather jackets to a number of specialty stores. The monthly demand from these stores for the jackets is described by the demand equation
p=400−80qp=400-80q
Here pp is the wholesale price, in dollars per jacket, and qq is the monthly demand, in thousands of jackets. Note that the demand equation makes no sense if q≥5q ≥ 5. The manufacturer's marginal cost function is given by the equation
Q:\bf{Q:} A music DVD company sells a DVD for $40. Let qq be the number of items. They sell 36000 units for this price per year. Marketing studies show that for each dollar increase they sell 600 fewer DVDs. The DVDs cost $15 each to make with a fixed cost of $100,000 for the company.
(a) Find price-demand relationship.
(b) Find the profit function.