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Economic Concepts

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.