Economic Concepts

Cost, Revenue and Profit

The following terms are used in discussing the production and sale of a product.

1) The output is to the number of units produced.

2) The cost of producing a commodity depends on many factors.

a) Some costs are incurred no matter what the output. These are the fixed costs.

b) The variable cost are those costs which vary with output. For any given output, the average variable cost is the variable cost divided by output.

c) The total cost is the sum of the fixed cost and variable cost.

_{Total
Cost = Fixed Cost + (Average Variable Cost) x Output}

3) The total revenue from the sale of a good is the selling price multiplied by the number of units sold; this is the total income from sales.

4) The profit is the difference between revenue and cost,

Profit = Revenue - Cost.

5) The break-even point is the point where revenue equals cost, or equivalently profit = 0. Production is profitable only when revenue is greater than cost.

6) The average total cost, (or, briefly, average cost) is the total cost divided by output,

Example 1

If the fixed costs are $100 if the average variable cost is $2, and if the selling price is $2.50 per unit then:

a) the total cost of producing q units is given by the cost function ;

b) the revenue from selling q units is given by the revenue function ;

c) the profit from producing and selling q units is given by the profit function

d) the break even point is determined by solving the equation

This is also illustrated in the figure below.

e) The average cost is _.

Example 2

Sometimes the average variable cost is not constant. Suppliers might give a discount for large orders, which would make the average variable cost decrease as output increases. For example, if the average variable cost is , then this decreases as increases. Assume again that the fixed costs are still $100 and that the selling price is $2.50 per unit

a) The cost function is .

b) The revenue function is .

c) The profit function is

d) The break even point is determined by solving the quadratic equation

We select the positive answer. Note that . See the figure below for a graphical solution.

e) The average cost is

Example 3

On the other hand, increased production might create a shortage of raw materials and so drive up the production costs. In this case, the average variable cost will increase. For example, if the average variable cost is while the fixed costs are still $100 and the selling price $2.50 then

a) the cost function is ;

b) the revenue function is ;

c) the profit function is

d) the break even point is determined by solving the quadratic equation

This has no real solution, and there is no break even point. The graph below is informative. Notice that at a selling price of $2.50, selling more and more products leads to an increase in your loss.

Example 4

Assume that the fixed cost is $1000 and the average variable cost of producing q units is . What should you set as the selling price if you want to break even when output is 800 units?

First, determine the cost function. . The cost of producing 800 units is and the average cost per unit is . Therefore the selling price should be $533.25. The revenue function will be and cost = revenue when q = 800.

Marginal Quantities

The marginal cost is the change in total cost which results from producing one additional unit. When the output is q, the marginal cost is

;

this is the slope of the line between the points and . The derivative ,which is the slope of the tangent line at, gives a good approximation to the exact change in cost, and it is customary to use the derivative to compute the marginal cost.

The marginal revenue is the additional revenue derived from the sale of one additional unit,

As with the cost function we will use the derivative of the revenue function to determine marginal revenue.

The marginal profit is the additional profit derived from the sale of one additional unit,

Again, we will use the derivative of the profit function to determine marginal profit. Note that this is the difference between marginal revenue and marginal cost,

Marginal Profit = Marginal Revenue - Marginal Cost.

Important Observation - If the profit function has a maximum, this occurs when marginal revenue = marginal cost.

We return to the examples. For each, we determine marginal cost, revenue and profit; also, we determine when profit is maximum.

Example 5

If the fixed costs are $100 if the average variable cost is $2, and if the selling price is $2.50 per unit then we determined the cost function is , the revenue from selling q units is revenue function , and the profit function is .

a) The marginal cost is . Notice in this case that this is exactly the same as the quantity and this is the same as the average variable cost per unit.

b) The marginal revenue is . Again, notice in this case that this is exactly the same as the quantity and this is the selling price per unit.

c) The marginal revenue is, and this is the profit per unit.

d) The profit function is increasing and so the profit function has no maximum.

Example 6

If the average variable cost is , the fixed costs are $100 and that the selling price is $2.50 per unit. Then the cost function is , the revenue function is , and the profit function is

a) Compute the marginal cost using the derivative. Thus, . The marginal cost decreases as the output increases. This makes sense since the average variable cost is decreasing.

i) For example, if the quantity produced is 60 units, the actual cost of producing an additional unit is while the marginal cost, computed using the derivative, gives . See the figure below.

ii) If the quantity produce is 80 units then the actual cost of producing an additional unit is while the marginal cost, computed using the derivative, gives . See the figure below.

b) The marginal revenue is the same as in the previous example, .

c) The marginal profit is .

d) Since , the profit functions is always increasing an there is no maximum profit.

Example 7

In this example, the average variable cost is , the fixed costs are $100 and the selling price is $2.50. Then the cost function is , the revenue function is and the profit function is .

a) The marginal cost is .

b) The marginal revenue is the same as previously, .

c) The marginal profit is

d) The profit is maximum when (notice that and so this critical value will produce a maximum). Solving

Therefore profit is maximum when the output is 25. The maximum profit is -93.75. In other words, you are still losing money (profit is negative) but this is the least you would use.

Supply and Demand

A supply curve describes the relationship between the quantity supplied and the selling price. The amount of a good or service that producers plan to sell at a given price during a given period is called the quantity supplied. The quantity supplied is the maximum amount that producers are willing to supply at a given price. Quantity supplied is expressed as an amount per unit of time. For example, if a producer plans to sell 750 units per day at $15 per unit we say that the quantity supplied is 750 unit per day at price $15.

Similarly, the amount of a good or service that consumers plan to buy at a given price during a given period is called the quantity demanded. The quantity demanded is the maximum amount that consumers can be expected to buy at a given price, and it also is expressed as amount per unit of time.

The equilibrium price is the price at which the quantity demanded equals the quantity supplied. The equilibrium quantity. is the quantity bought and sold at the equilibrium price. If the curves are graphed on the same coordinate system, the point of intersection is the equilibrium point, and is where supply equals demand. If the price is below equilibrium there will be a shortage and the price will rise, while if the price is above equilibrium there will be a surplus and the price will fall. If the price is at equilibrium it will stay there unless other factors enter to cause changes.

Example 8

Assume that the supply function is and the demand function is . The breakeven point is found by setting equating the two functions and then solving the resulting equation:

This gives the first coordinate; the second coordinate is (or, using the demand equation, )

Example 9

We make the following assumptions about supply and demand.

The supplier will produce 1000 units when the selling price is $20 per unit and will produce 1500 units if the price is $25 per unit.
Consumers will demand 1500 units when the selling price is $20 per unit but that the demand will decrease by 10% if the price increases by 5%.
Both supply and demand functions are linear.

Determine the supply function, the demand function and the equilibrium point.

1) To determine the supply function, we use a coordinate system and write the equation of the line through the points (1000,20) and (1500,25).

2) For the demand function, one point is (1500,20). If the price increases 5% to $21, the demand will decrease 10% to 1350. Thus the second point is (1350,21) and we can now determine the demand function.




	Home / Review Topics Economic Concepts Cost, Revenue and Profit The following terms are used in discussing the production and sale of a product. 1) The output is to the number of units produced. 2) The cost of producing a commodity depends on many factors. a) Some costs are incurred no matter what the output. These are the fixed costs. b) The variable cost are those costs which vary with output. For any given output, the average variable cost is the variable cost divided by output. c) The total cost is the sum of the fixed cost and variable cost. _{Total Cost = Fixed Cost + (Average Variable Cost) x Output} 3) The total revenue from the sale of a good is the selling price multiplied by the number of units sold; this is the total income from sales. 4) The profit is the difference between revenue and cost, Profit = Revenue - Cost. 5) The break-even point is the point where revenue equals cost, or equivalently profit = 0. Production is profitable only when revenue is greater than cost. 6) The average total cost, (or, briefly, average cost) is the total cost divided by output, . Example 1 If the fixed costs are $100 if the average variable cost is $2, and if the selling price is $2.50 per unit then: a) the total cost of producing q units is given by the cost function ; b) the revenue from selling q units is given by the revenue function ; c) the profit from producing and selling q units is given by the profit function d) the break even point is determined by solving the equation This is also illustrated in the figure below. e) The average cost is _. Example 2 Sometimes the average variable cost is not constant. Suppliers might give a discount for large orders, which would make the average variable cost decrease as output increases. For example, if the average variable cost is , then this decreases as increases. Assume again that the fixed costs are still $100 and that the selling price is $2.50 per unit a) The cost function is . b) The revenue function is . c) The profit function is d) The break even point is determined by solving the quadratic equation We select the positive answer. Note that . See the figure below for a graphical solution. e) The average cost is Example 3 On the other hand, increased production might create a shortage of raw materials and so drive up the production costs. In this case, the average variable cost will increase. For example, if the average variable cost is while the fixed costs are still $100 and the selling price $2.50 then a) the cost function is ; b) the revenue function is ; c) the profit function is d) the break even point is determined by solving the quadratic equation This has no real solution, and there is no break even point. The graph below is informative. Notice that at a selling price of $2.50, selling more and more products leads to an increase in your loss. e) Example 4 Assume that the fixed cost is $1000 and the average variable cost of producing q units is . What should you set as the selling price if you want to break even when output is 800 units? First, determine the cost function. . The cost of producing 800 units is and the average cost per unit is . Therefore the selling price should be $533.25. The revenue function will be and cost = revenue when q = 800. Marginal Quantities The marginal cost is the change in total cost which results from producing one additional unit. When the output is q, the marginal cost is ; this is the slope of the line between the points and . The derivative ,which is the slope of the tangent line at, gives a good approximation to the exact change in cost, and it is customary to use the derivative to compute the marginal cost. The marginal revenue is the additional revenue derived from the sale of one additional unit, . As with the cost function we will use the derivative of the revenue function to determine marginal revenue. The marginal profit is the additional profit derived from the sale of one additional unit, . Again, we will use the derivative of the profit function to determine marginal profit. Note that this is the difference between marginal revenue and marginal cost, Marginal Profit = Marginal Revenue - Marginal Cost. Important Observation - If the profit function has a maximum, this occurs when marginal revenue = marginal cost. We return to the examples. For each, we determine marginal cost, revenue and profit; also, we determine when profit is maximum. Example 5 If the fixed costs are $100 if the average variable cost is $2, and if the selling price is $2.50 per unit then we determined the cost function is , the revenue from selling q units is revenue function , and the profit function is . a) The marginal cost is . Notice in this case that this is exactly the same as the quantity and this is the same as the average variable cost per unit. b) The marginal revenue is . Again, notice in this case that this is exactly the same as the quantity and this is the selling price per unit. c) The marginal revenue is, and this is the profit per unit. d) The profit function is increasing and so the profit function has no maximum. Example 6 If the average variable cost is , the fixed costs are $100 and that the selling price is $2.50 per unit. Then the cost function is , the revenue function is , and the profit function is a) Compute the marginal cost using the derivative. Thus, . The marginal cost decreases as the output increases. This makes sense since the average variable cost is decreasing. i) For example, if the quantity produced is 60 units, the actual cost of producing an additional unit is while the marginal cost, computed using the derivative, gives . See the figure below. ii) If the quantity produce is 80 units then the actual cost of producing an additional unit is while the marginal cost, computed using the derivative, gives . See the figure below. b) The marginal revenue is the same as in the previous example, . c) The marginal profit is . d) Since , the profit functions is always increasing an there is no maximum profit. Example 7 In this example, the average variable cost is , the fixed costs are $100 and the selling price is $2.50. Then the cost function is , the revenue function is and the profit function is . a) The marginal cost is . b) The marginal revenue is the same as previously, . c) The marginal profit is d) The profit is maximum when (notice that and so this critical value will produce a maximum). Solving Therefore profit is maximum when the output is 25. The maximum profit is -93.75. In other words, you are still losing money (profit is negative) but this is the least you would use. Supply and Demand A supply curve describes the relationship between the quantity supplied and the selling price. The amount of a good or service that producers plan to sell at a given price during a given period is called the quantity supplied. The quantity supplied is the maximum amount that producers are willing to supply at a given price. Quantity supplied is expressed as an amount per unit of time. For example, if a producer plans to sell 750 units per day at $15 per unit we say that the quantity supplied is 750 unit per day at price $15. Similarly, the amount of a good or service that consumers plan to buy at a given price during a given period is called the quantity demanded. The quantity demanded is the maximum amount that consumers can be expected to buy at a given price, and it also is expressed as amount per unit of time. The equilibrium price is the price at which the quantity demanded equals the quantity supplied. The equilibrium quantity. is the quantity bought and sold at the equilibrium price. If the curves are graphed on the same coordinate system, the point of intersection is the equilibrium point, and is where supply equals demand. If the price is below equilibrium there will be a shortage and the price will rise, while if the price is above equilibrium there will be a surplus and the price will fall. If the price is at equilibrium it will stay there unless other factors enter to cause changes. Example 8 Assume that the supply function is and the demand function is . The breakeven point is found by setting equating the two functions and then solving the resulting equation: This gives the first coordinate; the second coordinate is (or, using the demand equation, ) Example 9 We make the following assumptions about supply and demand. The supplier will produce 1000 units when the selling price is $20 per unit and will produce 1500 units if the price is $25 per unit. Consumers will demand 1500 units when the selling price is $20 per unit but that the demand will decrease by 10% if the price increases by 5%. Both supply and demand functions are linear. Determine the supply function, the demand function and the equilibrium point. 1) To determine the supply function, we use a coordinate system and write the equation of the line through the points (1000,20) and (1500,25). 2) For the demand function, one point is (1500,20). If the price increases 5% to $21, the demand will decrease 10% to 1350. Thus the second point is (1350,21) and we can now determine the demand function.