Micro Formula Sheet

Glossary

Info

subscript and superscript used mixed on the slides e.g. $Q^S$ can be found instead of $Q_S$, but it is identical

• $S$ and $D$ … Supply and Demand
  • supply = all $MC$ of all firms added up
• $R$ and $C$ … Revenue and Costs
  • $FC$ … fixed costs
• $Q_x$ … Quantity of x
• $P_x$ … Price of x
• $E_x={\epsilon }_x$ … Elasticity of x
• $X^{\ast }$ … Optimum/Equilibrium of X (Price, Quantity)
  • sometimes used as $X_1^{\ast }$ and $X_2^{\ast }$ in duopoly
• $X^{\ast \ast }$ … another Optimum of X
  • e.g. when comparing competition vs monopoly
• ${\pi }_x$ … Profit of x
  • $\Pi$ … joint profits
• $MX$ … Marginal X (Revenue/Costs/Profit)
  • Marginal Changes
• $AX$ … Average X (Costs, Variable Costs, Revenue)
• $X_T$ … total X (costs, revenues)

Basics

Important

I use capital letters in general formulas or for results, lowercase letters for inputs and when using the formulas. When calculating the equilibrium quantity I use $Q^{\ast }$ When calculating the profit at a quantity I use $\pi (q)$

• Equilibrium: $S=D$
• Elasticity: $\epsilon =E_P=\frac{\partial q/q}{\partial p/p}=\frac{\%\partial q}{\%\partial p}$
  • Marginal Changes
• Cross-Elasticity: $E_{q_1,p_2}=\frac{p_2}{q_1}\frac{\delta q_1}{\delta p_2}$
• $\pi (q)=R(q)-C(q)$

Perfect Competition

• Maximization:
  • ${\pi }^{\prime }=0\Rightarrow MR=MC$
  • ${\pi }^{\prime \prime }<0$
  • $P=AR=AC=MR=MC$
  • at $Q^{\ast }$: $MR=MC$
• Profit: $\pi =(P-AC)\ast Q$
• Supply: $MC\ge AVC$
  • fixed costs included
• long run: $P=MC>AC$ … more competitors enter market, price falls until $P=AC$ and $\pi =0$

Monopoly

• $MR$ … Revenue function with twice the slope → why?
  • $MR=a-2bQ$
• optimal price rule of thumb: $\frac{1}{1+1/E_d}=m$
  • $p=m\ast MC$ … $m$ … markup

Third Degree Price Discrimination

• $\pi (q)=p_1{q}_1+p_2{q}_2+\cdots +p_n{q}_n-C(q_T)$
• $M{R}_1=M{R}_2=\cdots =M{R}_n$

Two-Part Tariffs

• $P$ … per-unit price; $T$ … buy-in price (lump-sum fee)
• $P^{\ast }>MC$ or $P\ast =MC$
• $T\ast$ … consumer surplus of the consumer group with smaller demand $D$
  • $T\ast =\frac{1}{2}(q_D(0)-MC)\ast q_D(MC)$ (linear demand curve → triangle)
• $\pi =N\ast T^{\ast }+(P^{\ast }-MC)(Q_1+Q_2+\cdots +Q_n)-FC$
  • if $P^{\ast }=MC\Rightarrow \pi =N\ast T^{\ast }-FC$
  • $N$ … amount of customers in all groups, $n$ … amount of consumer groups

Oligopoly

Collusion / Cartels

• just use the Monopoly instead of oligopoly formulas

When what?

Competition on (x + y) results in behavior:

• price + sequential = price leadership
• quantity + sequential = quantity leadership (Stackelberg)
• price + simultaneous = Bertrand
• quantity + simultaneous = Cournot

Price Leadership

• firm 1 sets price in anticipation of firm 2’s reaction
• firm 2 takes price of firm 1 as given, adjusts accordingly
• $p2=d+e{p}_1$
• ${\pi }_1=p_1\ast q_1=(a-b{p}_1+c(d+e{p}_1))\ast p_1-C_1$
  • $a,b,c,d,e$ … parameters

Stackelberg

• firm 1 sets output in anticipation of firm 2’s reaction
• firm 2 takes output of firm 1 as given, adjusts accordingly
• $p_2=d+e{p}_1$
• ${\pi }_1=p_1\ast q_1=(a-b{p}_1+c(d+e{p}_1))\ast p_1-C_1$
  • $a,b,c,d,e$ … parameters

Bertrand

• maximized price (same for $P_2^{\ast }$): $P_1^{\ast }(p_2)=\frac{a+b_2{p}_2+c_1{b}_1}{2b_1}$

Cournot

• optimized quantity (same for $Q_2^{\ast }$): $Q_1^{\ast }(q_2)=\frac{1-b{q}_2-c_1}{2b}$
  • reaction curve

Amoroso-Robinson Formula

• $MR=p+p\frac{1}{E_d}$
  • $E_d$ … elasticity of demand (not market elasticity)
• perfect competition: $E_d=0\Rightarrow MR=P$
  • loss of demand with small price change
• market power/monopoly: $-1>E_d>-\infty \Rightarrow MR<P$
  • negatively sloped market demand & marginal revenue curves
• inelastic demand: $-1<E_d<0\Rightarrow MR<0$
  • negative marginal revenue → not optimal!

Lerner Index

$L=\frac{p-MC}{p}=-\frac{1}{E_d}$ $L=0$ → perfect competition $L=1$ → monopoly

Surplus

Consumer Surplus

• area between demand function and given price
• derivative - rectangle
• OR with linear quantity just a rectangle ($\frac{a\ast b}{2}$)

Tax Burden

• on consumers: $\frac{E_s}{E_s-E_d}$
• on sellers: $\frac{-E_d}{E_s-E_d}$