TOTAL COST (TC)
Total cost is all the cost of producing a certain amount of products, including fixed costs (FC) and variable costs (VC).

TC = FC + VC

FIXED COST (FC)
Fixed cost is the costs a firm incurs in the short run for all fixed inputs, regardless of the level of output.

 

VARIABLE COST (VC)
Variable cost is the costs incurred by the firm for all variable inputs. VC changes as the firm changes its level of output.

 

AVERAGE FIXED COST (AFC)
Average fixed cost is the cost calculated by dividing the fixed cost (FC) by the number of units produced (Q).

AFC = \frac{FC}{Q}

 

AVERAGE VARIABLE COST (AVC)
Average variable cost is the variable cost per unit, calculated by dividing the variable cost (VC) by the number of units produced (Q).

AVC = \frac{VC}{Q}

 

AVERAGE COST (AC)
Average cost is the total cost per unit, calculated by dividing the total cost (TC) by the number of units produced (Q).

AC = \frac{TC}{Q} = AFC + AVC

MARGINAL COST (MC)
Marginal cost is the additional cost of making a single extra unit above the number already planned, or the change in total costs per unit change in output.

MC = \frac{\Delta TC}{\Delta Q}

The marginal cost at Q = Q0 is the slope of the tangent line to the graph of TC at Q = Q0.
Those who have studied calculus can apply the formula: MC = \frac{dTC}{dQ}

 

TOTAL REVENUE (TR)
Total revenue is the total amount received in exchange for goods or services, which is equal to price (P) times quantity (Q).

TR = P ⋅ Q

 

AVERAGE REVENUE (AR)
Average revenue is the revenue from one unit of product sold, calculated as the total revenue (TR) divided by the number of units sold (Q).

AR = \frac{TR}{Q}

MARGINAL REVENUE (MR)
Marginal revenue is the income from selling a single extra unit more than the existing number of sales, i.e. in addition to the existing total revenue.

MR = \frac{\Delta TR}{\Delta Q}

The marginal revenue at Q = Q0 is the slope of the tangent line to the graph of TR at Q = Q0.
Those who have studied calculus can apply the formula: MR= \frac{dTR}{dQ}.

 

PROFIT (π)
The excess of total revenue (TR) over all explicit costs and implicit costs. (In a specific context, it is the excess of total revenue over total cost.

π = TR – TC

 

BREAK-EVEN POINT (BEP)
Break-even point is the output level at which a firm’s total revenue equal its total costs. Equivalently, break-even point is the output level at which the total profit is zero. BEP is the solution of the following equation in Q.

TR = TC

 

EQUILIBRIUM
Equilibrium is the market condition where the quantity of a commodity that consumers are willing to purchase exactly equals the quantity producers are willing to supply. Geometrically, the equilibrium occurs at the intersection of the demand and supply curves. The price (P) and quantity (Q) at which equilibrium occurs are called the equilibrium price and equilibrium quantity, respectively.

To find the equilibrium price, solve the equation in P:  QD = QS

To find the equilibrium quantity, solve the equation in Q: PD = PS

Notes:
PD = demand equation in Q
PS = supply equation in Q
QD = demand equation in P
QS = demand equation in P

 

Example 1
The fixed costs at a company are $2,000 annually. The main product has revenue of 8 cents per unit and variable cost of 6 cents per unit. Determine a) the total cost function in Q, b) the total revenue function in Q, c) the annual profit if 300,000 units are sold in a year, d) the break-even quantity per year

Answer
Suppose that the firm sells Q units of the products.
The total variable cost (VC) is $0.06Q. Thus, the total costs function is: TC = FC + VC = $(2,000 + 0.06Q).
The total revenue (TR) function is: TR = P⋅Q = $ 0.08 Q.
TC(300,000) = $2,000 + $(0.06⋅300,000) = $2,000 + $18,000 = $20,000
TR(300,000) = $(0.08⋅300,000) = $24,000
Annual profit = π = TR – TC = $24,000 – $20,000 = $4,000.
At break-even, TR = TC.
0.08 Q = 2,000 + 0.06 Q
0.02 Q = 2,000
Q = \frac{2,000}{0.02} = 100,000
Therefore, the break-even is 100,000 units (per year).

 

Example 2
The demand and supply functions of a product were QD = 9 – P2 and QS = P2 + 2P – 3, respectively. Suppose that the government imposed a per unit sales tax $1 on the producer. Find: a) the quantity demanded before the taxation was implemented, b) the quantity demanded after the taxation was implemented, c) the proportion of the per-unit tax paid by each consumer, d) the total amount of tax received by the government.

Answer
Before-tax equilibrium: QD = QS
9 – P2 = P2 + 2P – 3
2P2 + 2P – 12 = 0
P2 + P – 6 = 0
(P+3)(P-2) = 0
P + 3 = 0 or P – 2 = 0
P = -3 or P = 2
As the price cannot be negative, select P = 2.
By substituting P = 2 into QD = 9 – P2, we have Q = 9 – 22 = 5.
Therefore, 5 units of the products had been demanded before the taxation was implemented.
Taxation shifted the supply curve upwards. As the tax is t = $1/unit, the supply function became QS’ = (P – 1)2 + 2(P – 1) – 3 = P2 – 4.
After-tax equilibrium: QD = QS’
9 – P2 = P2 – 4
2P2 = 13
P2 = 6.5
P= - \sqrt{6.5} or P= \sqrt{6.5}
Since P cannot be negative, select P= \sqrt{6.5}.
By substituting P= \sqrt{6.5} into QD = 9 – P2, we get Q = 9 - (\sqrt{6.5})^2=9 - 6.5 = 2.5.
Therefore, 2.5 units of the product were demanded after the taxation was implemented.
The unit market price of the product raised from $2 to $√6.5 ≈ $2.55. The increase in price, i.e. $0.55 was paid by the consumers. Consequently, the proportion paid by every consumer is \frac{\$ 0.55}{\$ 1} \cdot 100 \% = 55 \%.
After the tax was implemented, the quantity demanded was 2.5 units. The government received $1 for each unit sold. Hence, the total amount of tax received by the government was T = 2.5 ⋅ $1 = $2.5.

 

Example 3
The total cost (in dollars) of producing Q units of product is given by TC(Q) = 10,000+90 Q–0.05Q2. a) Find the exact cost to produce the 101th unit of the product, b) Find the marginal cost at a production level of 100 units and interpret the result.

Answer
The total cost to produce 100 units is TC(100) = $(10,000 + 90⋅100 – 0.05⋅1002) = $18,500.
The total cost to produce 101 units is TC(101) = $(10,000 + 90⋅101 – 0.05⋅1012) = $18,579.95.
So, the exact cost to produce the 101th unit of the product is ($18,579.95 – $18,500) = $79.95.
Marginal cost at a production level of Q0 = the slope of the tangent line to the graph of TC at Q = Q0.
As the total cost is $18,500 when 100 units are produced, the tangent line has an equation of the form TC – 18,500 = m(Q – 100), or equivalently TC = mQ + (18,500 – 100m). Since it is a tangent line, the discriminant of the quadratic equation 10,000 + 90 Q – 0.05 Q2 = mQ + (18,500 – 100m) must be zero. The quadratic equation can be rewritten as: 0.05 Q2 + (m – 90)Q + (8,500 – 100m) = 0. Its discriminant is D = (m-90)2 – 4(0.05)(8,500 – 100m) = m2 – 160m + 6,400. Since D = 0, it follows that m2 – 160m + 6,400 = 0, which is equivalent to (m – 80)2 = 0. The root of this equation is m = 80. Therefore, the marginal cost at a production level of 100 units is $80/unit. At a production level of 100 units, the total production cost is increasing at the rate of $80/unit. [Comparing this to the previous result, i.e. $79.95, it can be easily seen that marginal cost (MC) at Q = 100 is an approximation to the exact cost to produce the 101th unit of the product.]

 

Example 4
Given the total cost function TC = $ (2Q2 – 24 Q + 102), a) find the production level at which the total cost is minimum. b) what is the minimum total cost? c) find the average cost and average fixed cost at the production level. If one extra unit of the product is made, what additional cost will the firm incur?

Answer
TC = $ (2Q2 – 24 Q + 102) is a quadratic function of the form of f(x) = ax2 + bx + c with a = 2, b = -24, and c = 102. In order that the total cost reaches its minimum value, it must hold that Q=- \frac{b}{2a}= - \frac{-24}{2 \cdot 2} = 6.
The minimum total cost is TC(6) = $ (2⋅62 – 24⋅6 + 102) = $ (72 – 144 + 102) = $30.
The average cost is AC = \frac{TC}{Q} = \frac{\$ 30}/(6 \: units) = \$ 5/unit.
From TC = $ (2Q2 – 24 Q + 102), we conclude that the fixed cost is FC = $102. As a consequence, the average fixed cost is AFC = \frac{FC}{Q} = \frac{\$ 102}{6 \: units} = \$ 17/unit.
If one extra unit of the product is made, the additional cost incurred will be TC(7) – TC(6). Since TC(7) = $ (2⋅72 – 24⋅7 + 102) = $ 32, the additional cost is ($32 – $30) = $2.

 

Example 5
A firm collected the total revenue expressed as TR = 16,000Q + 480Q2 – 60Q3. Find the maximum average revenue and the value of Q at which the average revenue attains its maximum value.

Answer
Note that AR = \frac{TR}{Q} = \frac{16,000Q+480Q^2-60Q^3}{Q} = 16,000 + 480Q - 60Q^2.
It is a quadratic function of the form f(x) = ax2 + bx + c with a = -60, b = 480, and c = 16,000.
The discriminant is D = b2 – 4ac.
The maximum AR is \frac{D}{-4a} = \frac{b^2-4ac}{-4a} = c - \frac{b^2}{4a} = 16,000 - \frac{480^2}{4(-60)} =16,000+960 = 16,960.
The maximum AR will be attained if Q = - \frac{b}{2a} = - \frac{480}{2(-60)} =4.

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