Supplement 2:  A critique of limit pricing
A frequently discussed alternative to the standard monopoly model in the industrial organization literature is the notion of entry deterrence through limit pricing.  This is the notion that the firm might deliberately move away from the standard monopoly (or oligopoly) equilibrium in order to prolong the new lower stream of profits for a longer time.
Consider an industry where the market price (P) is given by the equation
,						(S2.1)
where Q is total output.  The total cost function of the incumbent firm (C_I) is
C_I  = 8 + 4q_I,						(S2.2)
where q_I is the output of the incumbent.  The total cost function of the potential entrant (C_e) is 
C_e = 8 + 4q_e ,						(S2.3)
when q_e > 0;  if q_e = 0 then C_e = 0 (q_e is the output of the potential entrant).  Assuming pure monopoly for the incumbent firm, profit (p) maximization requires MR_I = MC_I
   20 - q_I = 4    q_I* = 16    p_I* = $12    p_I = $120.
If the potential entrant conjectures that the incumbent will hold output constant in the face of new entry (the Bain-Sylos postulate), the marginal revenue facing the potential entrant is
.					(S2.4)
Setting MR_e = MC_e yields the reaction (best response) function for the entrant:
.						(S2.5)
Figure S2.1 shows the set up of an Excel spreadsheet which contains one cell where the hypothetical value of q_I resides (initially filled in with the monopoly output level).  Other cells are then constructed which show the optimal output of the entrant (as given by its reaction function), the total output, the market price, the revenue that accrues to each firm, the costs incurred by each firm, and the total profit of each firm.  The initial set up shows that if the incumbent firm holds its output at the monopoly level, there is room for the entrant to make a positive economic profit in this market.
Figure S2.1:  Evaluating a limit pricing strategy

Note that entry of the new firm drives down the market price from $12 to $8 and reduces the profit of the incumbent from $120 to $56.
Limit pricing theory suggests that instead of letting this happen, the incumbent may hold output at a level higher than the monopoly output so that when the entrant adds its output to the market, the market price falls so low that the entrant cannot earn positive economic profit.  To find the limiting quantity for the incumbent in Excel, the initial scenario in Figure S2.1 was copied to a new location on the worksheet and then the Tools/Goal Seek command was used to find the output of the incumbent that would just drive the potential entrant’s profit to zero.  In Figure S2.1, this means using Goal Seek to set cell H11 to zero by changing cell B10.
Assuming the potential entrant is deterred by the prospect of earning zero economic profit, the incumbent’s 24 units of output result in a market price of $8 and a profit level for the incumbent of $88 [(8 x 24) - (8 + 4 x 24)].  The price of $8 is the limiting price.  Note how the strategy involves a deliberate increase in output on the part of the incumbent from 16 to 24 units.  To avoid entry (and the subsequent reduction of profit to $56), the incumbent makes a strategic sacrifice of profit from the monopoly level of $120 to $88.
For more advanced students, Excel can also be helpful in extending the analysis to an assessment of the credibility of the entry deterrence strategy.  Suppose, for example, that the potential entrant is not deterred by the projection of zero economic profit and enters the market anyway.  If the incumbent continues to hold output at the limiting quantity, note that the limit pricing game scenario shows that the incumbent’s profit falls to $40.  Faced with such a prospect, it will be better for the incumbent to accommodate the entrant and reduce output in the face of new entry.  Knowing that it makes sense for the incumbent to back down, the entrant will ignore the high pre-entry output of the incumbent.  
To demonstrate that it makes sense for the incumbent to back down, consider the third scenario shown in Figure S2.1.  Suppose that instead of holding output at 24 in the face of new entry, the incumbent takes the entrant’s reaction function as given and then chooses an output to maximize its profit.  This is referred to as the incumbent playing a Stackelberg game.  The Stackelberg equilibrium can be determined in Excel using the Tools/Solver command discussed in Section 5 of the paper.  After copying the previous scenario to a new location on the worksheet, the Solver was instructed to maximize incumbent profit (cell H18) by changing incumbent output (cell B18).  Note that the resulting profit for the incumbent ($56) exceeds the level that would be attained at the “limiting” quantity in the post-entry period ($40).  A similar result holds if it is assumed that firms settle into a Cournot oligopoly game in the post-entry period.  The Cournot solution is also illustrated in Figure S2.1 and can be found by copying the previous scenario to another location in the worksheet and then replacing the incumbent’s output (cell B27) by the incumbent’s Cournot reaction function.  Since both firms have identical costs in this example, the incumbent’s reaction function will take the same form as the entrant’s [q_I^* = 16 – (1/2)q_e].  Excel can solve these reaction functions simultaneously provided the iteration option is activated under the Tools/Options…/Calculation menu.  The iteration option is also used in the optimal saving rate application discussed in Section 5 of the paper and the macroeconomic modeling applications discussed in the third and fourth web supplements.
The lack of credibility associated with the limit pricing strategy in this application stems from the lack of any linkage between the pre-entry output of the incumbent and the post-entry profit prospects of the entrant.  This analysis, therefore, can provide a lead-in to a discussion of more modern entry deterrence models which are able to forge a link between current and future strategies.