marginal cost is the change in total cost which results
from producing one additional unit. When the output is q,
the marginal cost is
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
marginal revenue is the additional revenue derived
from the sale of one additional unit,
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,
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.
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.
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 .
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
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.
marginal revenue is, and this is the profit per unit.
profit function is increasing and so the profit function has no
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
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.
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.
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.
marginal revenue is the same as in the previous example, .
marginal profit is .
, the profit functions is always increasing an there
is no maximum profit.
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 .
marginal cost is .
marginal revenue is the same as previously, .
marginal profit is
profit is maximum when (notice that and so this critical value will produce a
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
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.
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, )
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.
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).
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