ECON 321 Assignment 20: Compatibility and Networks

Utility

A type θ consumer receives the following utilities from the two networks:

utilityA=θ + (1/4)((Subscript[β, A])+(Subscript[q, A])+γ*(Subscript[q, B]))-(Subscript[p, A])

utilityB=θ + (1/4)(γ*((Subscript[β, A])+(Subscript[q, A]))+(Subscript[q, B]))-(Subscript[p, B])

Equal Quality Adjusted Prices

The first equilibrium equation in this model says that the only way for both networks to get any customers is if their quality-adjusted prices are the same

pHatA = -utilityA+θ

pHatB = -utilityB+θ

Marginal Consumer

The θ type of the marginal consumer has to equal the quality adjusted price pHat. There are 1-pHat consumers with higher utility than this, so the total number of new consumers is

pHat=1-((Subscript[q, A])+(Subscript[q, B]))

Demand

The demand for each network is given by the simultaneous solution of the above 3 equations. In particular, we are going to use the Cournot solution concept, so we really want the inverse demand.

{inverseDemandA,inverseDemandB}={(Subscript[p, A]),(Subscript[p, B])}/.Solve[{pHatA==pHat,pHatB==pHat},{(Subscript[p, A]),(Subscript[p, B])}][[1]]

Network Profit Max and Reaction Functions

{profitA,profitB}={(inverseDemandA-(Subscript[c, A]))(Subscript[q, A]),profitB=(inverseDemandB-(Subscript[c, B]))(Subscript[q, B])}

{FOCA,FOCB}={D[profitA,(Subscript[q, A])]==0,D[profitB,(Subscript[q, B])]==0}

{reactA,reactB}={Solve[FOCA,(Subscript[q, A])],Solve[FOCB,(Subscript[q, B])]}/.Rule->Equal//Flatten

Equilibrium Quantities, Prices, and Profits

qstars=Solve[{reactA,reactB},{(Subscript[q, A]),(Subscript[q, B])}]//Flatten

{profitstarA,profitstarB} ={profitA,profitB}/.qstars//Simplify

Consumer Surplus at Equilibrium

Demand is 1-pHat, and pHat=(Subscript[q, A])+(Subscript[q, B]) = Q. Thus, consumer surplus is the area under demand where Q goes from 0 and qstarA+qstarB

CS=Integrate[1-Q,{Q,0,((Subscript[q, A])+(Subscript[q, B]))/.qstars}]//Simplify

Standardization Game

Suppose the firms choose to use technology a or technology b, where a is preferred by A and b is preferred by B. When firms use their preferred technology, their costs are c=0, and otherwise their costs are c. There are 4 possible outcomes:

Compatibility Technology a: A uses a and B uses a

{aaA,aaB}={profitstarA,profitstarB}/.{(Subscript[c, A])->0,(Subscript[c, B])->c,γ->1}

Compatibility Technology b: A uses b and B uses b

{bbA,bbB}={profitstarA,profitstarB}/.{(Subscript[c, A])->c,(Subscript[c, B])->0,γ->1}

Incompatibility Technology ab: A uses a and B uses b

{abA,abB}={profitstarA,profitstarB}/.{(Subscript[c, A])->0,(Subscript[c, B])->0,γ->0}

Incompatibility Technology ba: A uses b and B uses b

(This is stupid!)

{baA,baB}={profitstarA,profitstarB}/.{(Subscript[c, A])->c,(Subscript[c, B])->c,γ->0}

Example Nash Equilibrium

(({{\[Placeholder], a, b}, {a, {aaA,aaB}, {abA,abB}}, {b, {baA,baB}, {bbA,bbB}}}))/.{(Subscript[β, A])->0.1,c->.1}//MatrixForm

Conditions For Nash Equilibrium

aa=aaA≥baA&&aaB≥abB

bb=bbA≥abA&&bbB≥baB

ab=abA≥bbA&&abB≥aaB

ba=baA≥aaA&&baB≥bbB

RegionPlot[{aa,bb,ab,ba},{c,0,.5},{(Subscript[β, A]),0,.5},PlotLegends->"Expressions",PlotStyle->14]