ECON 321 Assignment 18: Capacity and Entry Deterrence

Two firms both have the following market demand functions

inversedemand = 100-4 ((Subscript[q, 1])+(Subscript[q, 2]))

Working backward, in stage 2, Firm 1 has two different reaction functions depending on which marginal cost is relevant. It jumps between the reaction functions at quantity (Subscript[k, 1]).

profit1Stage2 = (inversedemand-c)(Subscript[q, 1])

FOC1Stage2 = D[profit1Stage2,(Subscript[q, 1])]==0

react1Cost40=Solve[FOC1Stage2/.c->40,(Subscript[q, 1])]/.Rule->Equal//Flatten//First

react1Cost10=Solve[FOC1Stage2/.c->10,(Subscript[q, 1])]/.Rule->Equal//Flatten//First

Meanwhile Firm 2 has a standard profit max problem in quantities with a marginal cost of 30

profit2Stage2 = (inversedemand-30)(Subscript[q, 2])

FOC2Stage2 = D[profit2Stage2,(Subscript[q, 2])]==0

react2=Solve[FOC2Stage2,(Subscript[q, 2])]/.Rule->Equal//Flatten//First

Plotting the reaction functions on a graph with (Subscript[q, 1]) on the horizontal axis requires a inverting the Firm 1 reaction functions, and also a little manipulation to get the curve in the proper format for plotting

inverseReact1Cost40=(Subscript[q, 2])/.Solve[react1Cost40,(Subscript[q, 2])]//First

inverseReact1Cost10=(Subscript[q, 2])/.Solve[react1Cost10,(Subscript[q, 2])]//First

react2Cost30=react2[[2]]

Plot[{inverseReact1Cost10,inverseReact1Cost40,react2Cost30},{(Subscript[q, 1]),0,12},PlotRange->{0,12},PlotLabels->Automatic,PlotTheme->"Monochrome"]

The two potential Cournot equilibria, and the associated operating profits for the firms are

qstarsCost40=Solve[{react1Cost40,react2},{(Subscript[q, 1]),(Subscript[q, 2])}]//Flatten//N

{profit1Stage2/.c->40,profit2Stage2}/.qstarsCost40//N

qstarsCost10=Solve[{react1Cost10,react2},{(Subscript[q, 1]),(Subscript[q, 2])}]//Flatten//N

{profit1Stage2/.c->40,profit2Stage2}/.qstarsCost10//N

Notice that even though Firm 1 behaves as if it has cost 10, it really has cost 40 when you include stage 1, so that’s why there’s still a c->40 even in the second profit calculation.

An Aside: Starting from qstarsCost40, could Firm 1 Push Firm 2 to Produce Less?

Suppose we’re in the situation where firm 1 has cost 40 and firm 2 has cost 30. We just saw that

qstarsCost40

One very simple way to look at this is just to wonder what would happen to Firm 2’s FOC if (Subscript[q, 1])went up. We know that previously we had Firm 2’s FOC at

FOC2Stage2//First

And if firm 2 wants to change this we could check the plot (or the second derivative)

Plot[(FOC2Stage2//First)/.(Subscript[q, 1])->4.16667,{(Subscript[q, 2]),5,8},GridLines->{{6.66667},{}}]

Now if we evaluate that FOC where both (Subscript[q, 1]) and (Subscript[q, 2]) go to the equilibrium quantities, we have

(FOC2Stage2//First)/.(Subscript[q, 1])->4.16667/.(Subscript[q, 2])->6.66667

Well, that’s zero, but of course it’s zero because that’s what we solved for. But what if (Subscript[q, 1]) were bigger, say (Subscript[q, 1])=5?

(FOC2Stage2//First)/.(Subscript[q, 1])->5/.(Subscript[q, 2])->6.66667

Now it’s negative, it says marginal profit is negative. What does firm 2 do if it wants to raise its marginal profit? It produces less.

Plot[(FOC2Stage2//First)/.(Subscript[q, 1])->5,{(Subscript[q, 2]),5,8},GridLines->{{6.66667},{}}]

Deter Entry

Now because there is an entry cost of 90 for Firm 2, we must check whether firm 2’s profits are actually positive depending on what Firm 1 is doing.

netProfit2 = (profit2Stage2/.(react2/.Equal->Rule)//Simplify)-90

So the quantity (Subscript[(OverTilde[q]), 1])that will deter Firm 2 from entering is

q1Deter=Solve[netProfit2==0,(Subscript[q, 1])]//Flatten//N

(Clearly its the smaller solution that is relevant.)

Accommodate Entry

The other possibility for Firm 1 is not to deter entry and just install capacity equivalent to the quantity of a Stackelberg leader. This involves solving the Firm 1 profit max problem substituting in Firm 2’s reaction function

profit1Stage1 = profit1Stage2/.(react2/.Equal->Rule)/.c->40//Simplify

Anticipating Firm 2’s reaction, Firm 1 maximizes

FOC1Stage1=D[profit1Stage1,(Subscript[q, 1])]==0

q1Acc=Solve[FOC1Stage1,(Subscript[q, 1])]//Flatten//First//N

We can plot both of these possibilities on the reaction function diagram

Plot[{inverseReact1Cost10,inverseReact1Cost40,react2Cost30},{(Subscript[q, 1]),0,12},PlotRange->{0,12},GridLines->{{(Subscript[q, 1])/.q1Deter[[1]],((Subscript[q, 1])/.q1Acc)},{}},PlotLabels->Automatic,PlotTheme->"Monochrome"]

So if Firm 1 wants to deter, it should set (Subscript[k, 1])=(Subscript[(OverTilde[q]), 1])so its stage 2 reaction function goes vertical at (Subscript[(OverTilde[q]), 1]), and if it wants to accommodate, it should set (Subscript[k, 1])=(Subscript[q, 1]^S) so its stage 2 reaction function goes vertical at (Subscript[q, 1]^S). It remains to see which is more profitable:

{(profit1Stage1/.q1Deter[[1]]/.(Subscript[q, 2])->0),(profit1Stage1/.q1Acc//N)}

Consumer Welfare

One last question -- which outcome is better for consumers. This is the same as asking which produces the lowest price. Let’s check the Cournot outcomes too for comparison.

{(inversedemand/.qstarsCost40),(inversedemand/.qstarsCost10),(inversedemand/.q1Deter[[1]]/.(Subscript[q, 2])->0),(inversedemand/.(react2/.Equal->Rule)/.q1Acc//N)}

So in fact entry accommodation is better for consumers than if the firms just played Cournot with their respective cost functions including Firm 1’s cost.