Wolfram Alpha:

```Marginal Revenue and Cost
-------------------------

Marginal revenue is the change in the total revenue that arises
when the quantity sold is incremented by one unit, that is, it is
the revenue generated by selling one more unit of a good.

Marginal cost is the change in the total cost that arises when
the quantity produced is incremented by one unit, that is, it is
the cost of producing one more unit of a good.

Total Revenue = TR = P*Q
Marginal Revenue = MR = dTR/dQ

Total Cost = TC = Fixed Cost + Unit Cost*Q
Marginal Cost = MC = dTC/dQ

Where P is the price and Q is the quantity sold/produced.

Profit Maximization
------------------

MR < MC => profit decreases.

MR > MC => profit increases.

MR = MC => profit maximized.

Supply and Demand Functions
---------------------------

Linear case:

P
|
+P'      /
|\      /S = m'Q + P'
| \    /
|  \  /
|   \/
|   /\
|  /  \
| /    \
|/      \D = -mQ + P
+P       \
|
------------- Q

Equilibrium:

-mQ + P = m'Q + P'

Production Functions
--------------------

A production function relates physical output of a production
process to physical inputs or factors of production.  There can
be a number of different inputs to production, but they are
generally designated as either capital or labor. (Technically,
land is a third category of factors of production, but it's not
generally included in the production function except in the
context of a land-intensive business).  Production functions
can be short run or long run.  In the short run, the amount
of capital that a factory uses is generally thought to be fixed.
The reasoning is that firms must commit to a particular size
of factory, office, etc. and can't easily change these decisions
without a long planning period.  Therefore, the quantity of labor,
L, is the only input in the short-run production function.  In the
long run a firm has the planning horizon necessary to change
not only the number of workers but the amount of capital as
well, since it can move to a different size factory, office, etc.
Therefore, the long-run production function has two inputs
that be changed capital, K, and labor, L.

Example 1
---------

Consider a monopolist where TC = 20 + 10Q + 0.3Q2
and MC = 10 + 0.6Q.  The demand function for the firm's goods
is P = 160 - 0.5Q.  The firm optimizes by producing the level
of output that maximizes profit or minimizes loss. If the firm
uses a uniform pricing strategy, determine the number units of
output, the price charged and the total profit.

TR = PQ = 160Q - 0.5Q2

MR = dTR/dQ = 160 - Q

For maximum profit MR = MC,  Therefore,

160 - Q = 10 + 0.6Q

∴ Q = 94

P = 160 - 0.5Q = 160 - 0.5*94 = 113

TC = 20 + 10Q + 0.3Q2 = 20 + 10*94 + 0.3*942 = 3611

Profit, Π = TR - TC

= 160*94 - 0.5*942 - 3611

= 7011

Example 2
---------

Suppose a firm is producing a level of output such that
MR = MC. The firm is selling its output at a price of \$8
per unit and is incurring average variable costs of \$6 per
unit and average total costs of \$7 per unit. Given this
information, determine if the firm is operating at a profit
that could be increased by producing more output.

MR = MC

TR = 8Q

∴ MR = ΔTR/ΔQ

TC = 7Q

∴ MC = ΔTC/ΔQ

∴ MR > MC

Example 3
---------

Suppose a firm's production function is Q = 0.4K0.5L0.5.
Its level of capital is fixed at 100 units, the price of
labor is PL = \$4 per unit and the price of capital PC = \$8
per unit. Given this information, calculate the firm's
short run production function.

K = 100

Q = 0.4K0.5L0.5 = 0.4*1000.5*L0.5

= 4L0.5

Given the same information, calculate the firm's total cost
function.

TC = VC + FC

= PL*L +  PK*K

= 4L + 8*100

L0.5 = Q/4 ∴ L = Q2/16

∴ TC = (Q2/4) + 800
```