ECON 321 Assignment 12: Switching Costs

Stage 2: Existing Market Shares (Subscript[α, A]) and (Subscript[α, B])already determined

First, we need to set up utility functions for consumer type x buying from firm 1 and firm 2. I’m also going to add on that the utilities are possibly different for the two firms, so (Subscript[r, A]) for firm A and 1 for firm B. Find the indifferent consumer and then demand.

{vA2=(Subscript[r, A])-x-pA2,vB2=1-(1-x)-pB2}/.{(Subscript[r, A])->1}

(stage2xhat=Solve[vA2==vB2,x]//Flatten//First)/.{(Subscript[r, A])->1}

({qA2=(1-(Subscript[λ, n]))(Subscript[α, A])+(Subscript[λ, n])x/.stage2xhat,qB2=(1-(Subscript[λ, n]))(Subscript[α, B])+(Subscript[λ, n])(1-x)/.stage2xhat})/.{(Subscript[r, A])->1}

Firm 1 and 2’s Profits, adding  the possibility that the firms have different costs, but only looking at the output where the costs are the same, are

({profitA2=(pA2-(Subscript[c, A]))qA2,profitB2=(pB2-(Subscript[c, B]))qB2})/.{(Subscript[r, A])->1,(Subscript[c, A])->c,(Subscript[c, B])->c}

({FOCA2=D[profitA2,pA2]==0,FOCB2=D[profitB2,pB2]==0}//Simplify)/.{(Subscript[r, A])->1,(Subscript[c, A])->c,(Subscript[c, B])->c}

Now solve simultaneously to get the Nash equilibrium prices and profits

(pstars2=Solve[{FOCA2,FOCB2},{pA2,pB2}]//Flatten//Simplify)/.{(Subscript[r, A])->1,(Subscript[c, A])->c,(Subscript[c, B])->c}

({{qstarA2,qstarB2},{profitstarA2,profitstarB2}}={{qA2,qB2},{profitA2,profitB2}}/.pstars2//Simplify)/.{(Subscript[r, A])->1,(Subscript[c, A])->c,(Subscript[c, B])->c};

Stage 1

In stage 1, consumers decide to sign up for firm A or B with rational expectations about facing switching costs in stage 2. Then find the indifferent consumer.

{vA1=(Subscript[r, A])-x-pA1+(1-(Subscript[λ, n]))((Subscript[r, A])-x-pA2)/.pstars2,vB1=1-(1-x)-pB1+(1-(Subscript[λ, n]))(1-(1-x)-pB2 )/.pstars2}/.{(Subscript[r, A])->1,(Subscript[c, A])->c,(Subscript[c, B])->c}

(stage1xhat=Solve[vA1==vB1,x]//Flatten//First)/.{(Subscript[r, A])->1,(Subscript[c, A])->c,(Subscript[c, B])->c}

To find demand for firm 1 and firm 2, we need the alpha shares to satisfy both that they are determined by the indifferent consumer and that the indifferent consumer is rational in the sense of expecting the correct alphas.

(q1=Solve[{(Subscript[α, A])==x/.stage1xhat,(Subscript[α, B])==(1-x)/.stage1xhat},{(Subscript[α, A]),(Subscript[α, B])}]//Simplify//Flatten)/.{(Subscript[r, A])->1,(Subscript[c, A])->c,(Subscript[c, B])->c}//FullSimplify

{profitsA1,profitsB1}={(pA1-(Subscript[c, A]))(Subscript[α, A]),(pB1-(Subscript[c, B]))(Subscript[α, B])}

So total profits, period 1 plus period 2, are

(totalprofitsA=(profitsA1/.q1 )+(profitstarA2/.q1))/.{(Subscript[r, A])->1,(Subscript[c, A])->c,(Subscript[c, B])->c}//FullSimplify

(totalprofitsB=(profitsB1/.q1 )+(profitstarB2/.q1))/.{(Subscript[r, A])->1,(Subscript[c, A])->c,(Subscript[c, B])->c}//Expand//FullSimplify

The first order conditions for a profit max at stage 1 are

({FOCA1,FOCB1} = {D[totalprofitsA,pA1]==0,D[totalprofitsB,pB1]==0})/.{(Subscript[r, A])->1,(Subscript[c, A])->c,(Subscript[c, B])->c}//FullSimplify

Equilibrium prices and market shares in period 1 are:

(pstars1=Solve[{FOCA1,FOCB1},{pA1,pB1}]//Flatten)/.{(Subscript[r, A])->1,(Subscript[c, A])->c,(Subscript[c, B])->c}//FullSimplify

(αstars=q1/.pstars1//Flatten)/.{(Subscript[r, A])->1,(Subscript[c, A])->c,(Subscript[c, B])->c}//Simplify

And now we can sub these back into the period 2 prices to see what they will be in the equilibrium of the whole game.

(pstars2/.q1/.pstars1)/.{(Subscript[r, A])->1,(Subscript[c, A])->c,(Subscript[c, B])->c}//Flatten//Simplify

Calibrating this Model to Dropbox Versus Box

Based on (admittedly very casual) research, Dropbox enterprise costs $45 per month for 3 users and Box enterprise costs $51 per month for 3 users. Based on even more casual research, Dropbox market share is 26% and Box market share is 23%.

Assumption 1 : Treat this as a two firm market:

{αDropbox=(26/(26+23)),αBox=(23/(26+23))}//N

Assumption 2: Assume the prices and market shares we observe are from Stage 1, with the presumption that both companies will try to lock in businesses and then upsell them later once switching costs are high. Scale the prices to set Dropbox’s price = 2.

{pDropbox=45,pBox=51}//N

Based on our assumptions, we want pstars1 and αstars to come out equal to the above quantities. That’s 4 equations. Which is good news, because we have 3 unknowns: (Subscript[r, A]),c, (Subscript[λ, n]). Before we just solve away, let’s look at these equations to make sure they look good:

eqpA=(pA1/.pstars1)==pDropbox//Simplify

eqpB=(pB1/.pstars1)==pBox//Simplify

eqαA=((Subscript[α, A])/.αstars)==αDropbox//Simplify

Seem ok, so let’s go solve them all simultaneously to get the parameters

paramsCloud=Solve[{eqpA,eqpB,eqαA},{(Subscript[r, A]),(Subscript[c, A]),(Subscript[c, B]),(Subscript[λ, n])}]//Flatten//N

We have one free parameter, (Subscript[λ, n]),which we could do research on or just set equal to a reasonable value.

paramsCloud/.(Subscript[λ, n])->.3

pstars1/.paramsCloud/.(Subscript[λ, n])->0.3

αstars/.paramsCloud/.(Subscript[λ, n])->0.3

OK, now the big payoff is that we should be able to predict Stage 2 prices and market shares!

pstars2/.αstars/.paramsCloud/.(Subscript[λ, n])->0.3

{{qstarA2,qstarB2},{profitstarA2,profitstarB2}}/.αstars/.paramsCloud/.(Subscript[λ, n])->.3