**Econ 243 Class 05 of Sept 21, 2015**

**Algebra: Our basis linear model**

• assume complications away!

= if fixed costs don’t affect decisions on the margin, assume FC = 0

= assume demand is linear [since we typically observe a modest range of prices & quantities in the real world, that’s often innocuous]

» computers = empirical work however often finds using logged variables convenient.

= assume MC = c constant [horizontal]

→ all this gives us a simple graphical framework and simple algebra

• algebra

→ figuring out profit-maximizing output, quantity, profits to facilitate comparisons among strategic decisions

→ figuring out links to demand elasticities to tie to real-world data and decisions

• p = a – bq leads to

→ MR = a – 2bq [same “p” intercept, twice as steep so intersects midpoint of q-axis]

→ p = (a+c)/2

→ q = (a-c)/2b

→ π = (a-c)^{2}/4b

• elasticities: needed for empirical work (including antitrust cases but also corporate pricing and marketing strategies)

= demand is inelastic below the midpoint (0 at the intercept with the “q”- axis)

» firms ** never** want to be in that region: higher “p” raises total revenue, lowers costs [q falls so cq falls while fixed costs remain fixed] and hence raises total profits

= so we only care about the upper part of the curve

= MR = d(pQ)/dQ = p + Q dp/dQ [chain rule].

» p falls when Q rises so the second term is always negative: MR is always less than p

= MR = p (1 + Q/p dp/dQ) or p(1 + dp/p / dQ/Q) and dp/p / dQ/Q = – 1/|ε|

» the sign change is because by convention we (and the Martin text) use | ε | though a computer will calculate it as negative. this practice was to make life easier for typesetters and readers. it no longer makes sense; live with it.

→ hence MR = p (1 – 1/ε)

— and clearly negative marginal revenue is bad ←→ we only consider cases of elastic demand ε > 1.

= this also leads to a simple formula for the price-cost margin (in percentage terms):

→ (p-c)/p = 1/ε so when demand is highly elastic (our wheat farmer) the margin is close to zero, while as we move down the demand curve [draw!!] margins rise in percentage terms towards 100% [*no, but in many businesses a gross margin of 60% is not unusual, though it is often offset by high fixed or sunk costs*]

• LOOK THROUGH THE GRAPHS IN THE BOOK TO MAKE SURE YOU FOLLOW THEM.