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\begin{center}



 \begin{Large}



 {\bf   A New Model of a Kinked Demand Curve    }  \\



\end{Large}



July 12, 2004 \\



\bigskip



Tilman Klumpp and Eric Rasmusen \\







\bigskip



{\it Abstract}



\end{center}



  \vspace*{-12pt}



 Various models generate a kinked demand curve. The common intuition is wrong,

though.



\bigskip



\begin{small}



\noindent

 \hspace*{20pt} Klumpp:  Assistant Professor, Department of

Economics, Indiana University,

  Office: (812)

xxx9.  Fax: 812-xxx  sss@indiana.edu. http://ssss.  //







\noindent

 \hspace*{20pt} Rasmusen:  Indiana University Foundation Professor, Department

of Business

Economics and Public Policy, Kelley School of Business, Indiana University,

BU456, 1309 E.  10th Street, Bloomington, Indiana, 47405-1701. Office: (812)

855-9219.  Fax: 812-855-3344 Erasmuse@indiana.edu. http://www.rasmusen.org.  //









This papers:  http://www.rasmusen.org/papers/kinked-demand-rasmusen.pdf.



\noindent

 We thank  xxx for comments.



\end{small}





\newpage





\noindent

  {\bf 1.  Introduction}



 The main implication of a kinked demand curve is that if marginal cost changes

a little, the price a firm charges will not change.



The original rationale for the kinked demand curve is from Sweezy-- that

competitors match decreases in price,  but not increases. That doesn't make

sense. Page 154 or maybe page 256 of   Tirole's book tries to model this,   I

think.  An oligopoly model, with focal points, maybe, that gets the result.

Might be rather contrived.



A second reason is the   direct one that demand just happens to have a kinked

shape. We analyze that below.



 Salop has a circular location  article in which marginal or fixed cost  can

rise, and prices will fall because of exit. The paper   generates a kinked

demand curve, but for the  opposite of the  expected reason. In his model, small

increases just lose existing customers to not buying at all. Small reductions

reqeuire luring competitors form other firms. Big reductions wipe out the

nearest competitor entirely.  This model also works on an interval, and rather

more realistically.



 A final  model is one of  switching costs. Small changes just change quantities

of existing customers; big price changes change who is a customer. This model

has been in the air, but not formalized. We will do that below.









\bigskip

\noindent

  {\bf 2.  A Simple Model}



What if demand just happens to be kinked? There is no law saying that demand

curve need to be linear. A kinked shape might be the result of consumer

preferences. Such a demand curve would result if the consumers willing to pay

the highest price also have elastic demand, and the consumers willing to pay

only a low price have inelastic demand. This is a special case, but not an

unrealistic special case.



 First, we might  jump straight to the kinked demand curve:



 \begin{equation} \label{e100}

\begin{array}{ll}

p(q) & = \alpha - \beta q \; if  \;  q \leq q_k\\

  & =  \gamma - \delta q \; if  \;  q > q_k\\

\end{array}

\end{equation}

 with $\beta < \gamma$, so demand slopes down more gently at higher prices,  and

$\alpha - \beta q_k = \gamma - \delta q_k$, so the demand curve is continuous at

the kink, $q_k$.



 This demand curve generates the following marginal revenue curve (add here).

The two curves are shown in Figure 1 (add later).





Second, we could look at a somewhat less special case: concave demand.



 Let demand be $p(q)$, with $p'<0$ and $p''<0$.    Revenue is then $R= p(q)q$ so

marginal revenue is

  \begin{equation} \label{e105}

MR = \frac{R}{dq}  =  p  + p'q

\end{equation}



and the change in marginal revenue with output is

 \begin{equation} \label{e106}

  \frac{MR}{dq}  =  p'  + p''q + p' = 2p'+ p''q

\end{equation}



 This slopes down, and is continuous and differentiable (if  $p''$ is

differentiable).  The shape is a smooth version of what we see in the kinked

demand curve model, however. In choosing output to equate marginal cost and

marginal revenue, the firm will likely  choose output to be in the steep part of

the marginal revenue curve. If marginal costs change, then output will only vary

slightly.





Thus, the kinked-demand curve model certainly has a place as  a special case in

models of firms with market power.  This ``undisputed'' place, however, has

nothing to do with strategic behavior or oligopoly, and is as likely to be found

in a monopoly as in an industry with several firms.





\bigskip

\noindent

  {\bf 3. Sweezy: A Model that Does not Work}





This is not Sweezy's model, but it is model of the idea he was thinking of.









There are 2 firms, differentiated products.



 1. Firms pick $p_1$ and P2.



 2. If Firm 1 wants to switch P1, he can.



 3. If Firm 2 wants to switch P2, he can.



 4. If neither firm has switched in (2) and (3), go to 5.

 Otehrwise, go back to 2.



 5. Consumers buy at the final prices.



Does  this generate a kinked or concave demand?









\begin{equation} \label{e1}

q_1 = \alpha - \beta p_1 + \gamma p_2

\end{equation}



\begin{equation} \label{e2}

q_2 = \alpha - \beta p_2 + \gamma p_1

\end{equation}



Firm 2 will maximize

\begin{equation} \label{e3}

\begin{array}{ll}

\pi_2 &= q_2 (\hat{p}_2-c) = (\alpha - \beta p_2 + \gamma p_1) (\hat{p}_2-c) \\

  & =  \alpha p_2 - \beta p_2^2 + \gamma p_1p_2   -(\alpha - \beta p_2

+ \gamma p_1 )c\\

\end{array}

\end{equation}



Differentiating yields

\begin{equation} \label{e4}

\frac{d \pi_2}{d p_2} =   \alpha - 2\beta p_2  + \gamma p_1   +  \beta

c =0.

\end{equation}



Solving for $p_2$ yields

\begin{equation} \label{e5}

p_2 = \frac{\alpha + \gamma p_1 -\beta c}{2\beta}

\end{equation}



The Stackelberg demand curve for Firm 1 is thus

\begin{equation} \label{e6}

\begin{array}{ll}

 q_1 &= \alpha - \beta p_1 + \gamma \left( \frac{\alpha + \gamma p_1

-\beta c}{2\beta} \right)\\

&=  \alpha - \beta p_1 + \frac{ \gamma^2    p_1  c}{2\beta} +

  \frac{\alpha\gamma     -\beta \gamma c}{2\beta}\\

\end{array}

 \end{equation}



This is still linear in $ p_1$.



So there is still no kink in the demand curve. Whether Firm 1 reduces its price

or increases, Firm 2 will follow, if not proportionately.



 It is hard to think of why one's rival would match price cuts but not

match price increases. That is a basic fallacy, I think.





Tirole's book has a fancier attempt to model the Sweezy intuition. The model is

too contrived, however, and the fact that it takes such contrivance by a great

master of modelling shows that the intuition is unsound. (here do the Tirole

model.)





   There is something to this intuition as far as explaining sticky prices, but

the kinked demand curve is not the way to analyze it.  The intuition is that if

a  group of firms has settled onto a collusive price, then if one firm deviates

to a lower price it can destabilize the cartel and prices fall for everyone. If

he deviates to a higher price, on the other hand, the other firms may follow a

little, but not much. This will generate a discontinuity in the demand curve

facing the firm: a small price reduction would drive prices all the way down to

marginal cost, but a small price rise would just reduce its sales slightly.



 This idea is easily modelled. Imagine an infinitely repeated game between  $N$

firms, with a low discount rate, in which each firm chooses its price and they

have differentiated products.  One equilibrium has price   slightly above

marginal  cost. Another equilibrium has price at the joint-monopoly, fully

collusive, level, enforced by  a ``grim''  equilibrium strategy in which if any

firm deviates in period $t$ by reducing its price, all the  firms price at some

lower level-- the competitive level, or something higher-- in all future

periods. If  a firm deviates by raising its price, on the other hand, that

deviation is ignored, and the equilibrium calls even for that same firm to

return to the collusive price in the next period.









Let us do this with two firms. Let the discount rate be $r$ and let the  demand

curves be  just as above:



\begin{equation} \label{e1a}

q_1 = \alpha - \beta p_1 + \gamma p_2

\end{equation}

 and

\begin{equation} \label{e2a}

q_2 = \alpha - \beta p_2 + \gamma p_1

\end{equation}



[work out this model]



The one-shot equilibrium is: xxx



Under perfect collusion, prices would be xxx.  Oops-- it looks like prices could

be infinite here. We probably need to set up a model based on primitives.











\bigskip

\noindent

  {\bf 4. A Model that Does Work: Switching Costs}



 We can use a model in the style of Klemperer (1987a, 1987b, 1995), but simpler,

because we do not need a dynamic, overlapping-generations  model with new

customers coming in.





What model of consumer preferences and market structure could give rise to

demand curves (from the firms' perspective) that are continuous, but not smooth?

In particular, can a simple model produce a kinked demand curve?



Consider a unit mass of consumers who have budget 1 that they can allocate to

two goods. One of the goods is produced at zero cost by two rival firms with

price-setting power; no differentiation is assumed. The other is an unmodelled

outside good that has price 1. If a consumer buys $q$ units of the first good at

price $p$, the quantity of the second good will be $1-pq$ by the usual budget

constraint. In the example, each consumer obtains   utility

\begin{equation}\label{util}

  u(\hat{p},q) = q^{\frac{1}{2}} + (1-pq) .

\end{equation}

(The values of the exponent is chosen for simplicity and can be set

differently  so long as it is less than $  1/2$).



We assume that each firm initially sells to half of the consumers; however,

consumers who are assigned to firm $i$ can switch to firm $j$ if they pay a non-

monetary switching cost $s>0$ (which does not figure into the budget constraint)

. Consumers' switching costs are distributed on $[\s,\infty)$

according to cumulative distribution function $F$, which  has  convex support

and no mass

points.



If a consumer with switching cost $s$ decides to buy from his assigned supplier

at price $p$, solving \eqref{util} gives us the following individual demand

function:

\[  q( p ) = \frac{1}{4p^2},  \]

which, by the way, has a constant elasticity of -2.





His indirect utility function is then

\[ h^{stay}( p ) = u( p, q( p )) = \left( \frac{1}{4p^2} \right)^{1/2} + \left(

1-\frac{1}{4p^2} p \right) = 1+ \frac{1}{4p}   \]



If he were to switch to the rival supplier, whose price we denote $\hat{p}$,

then he would buy $q(\hat{p})=\frac{1}{4\p^2}$ units and obtain indirect utility

\[ h^{switch}(\hat{p}) = u(\hat{p},q(\hat{p}))-s = 1+ \frac{1}{4\p}   -s. \]

He switches if $h^{stay}(\hat{p})<h^{switch}(\hat{p})$, or

\[ p > \beta(\hat{p},s) \equiv  \p (1+ 4c). \]



A similar calculation applies for consumers who are currently assigned to the

rival firm. They will switch to the firm we are considering if

\[ p < \gamma(\hat{p},s) \equiv \p (1-4c). \]





Notice that $\beta(\hat{p},0)=\gamma(\hat{p},0)=\p$, that $\beta$ increases in

$c$,

and  that $\gamma$ decreases. Imagine both firms charge the same price: $p=\p$.

If

the firm we're considering changes $p$ in any direction away from $\p$, the

first consumers to switch are those with the lowest possible switching cost,

$\s$. Thus, as long as

\[ p \in [\gamma(\hat{p},\s),\beta(\hat{p},\s) ], \]

changes in $p$ do not change the identity of the firm's consumers, but only how

much each of the existing consumers buys. Put differently, all quantity

adjustments are on the internal margin in the given interval, and the firm's

demand is

\begin{equation}\label{demint}

  Q^0(p, \hat{p}) = \frac{1}{2} \frac{1}{4p^2} = \frac{1}{8p^2}

\end{equation}

(recall that the firm sells to half the market).



If, on the other hand, $p>\beta(\hat{p},\s)$, some current consumers will be

lost to the rival firm. It is easy to show that if

\[ s< m( p ,\hat{p}) \equiv  ccc, \]

then a consumer with cost $s$ will switch from the current firm, charging $p$,

to the

competitor, charging the lower price $\hat{p}$. Letting $F(s)$ denote the c.d.f.

of

the   distribution of the switching cost $s$, the  current firm's

demand quantity for a price $p$  above $\beta(\hat{p},\s)$ is

\begin{equation}\label{demext1}

  Q^-(p,\hat{p}) = \frac{1}{2} \left( 1-F(m(p,\hat{p})) \right) q(\hat{p}) =

 \frac{1}{8p^2}  \left(1-F(m(p,\hat{p})) \right).

\end{equation}

The term $m(p,\hat{p})$ in \eqref{demext1} reflects the fact that some

quantity adjustments are on the external margin now, unlike in \eqref{demint}.



Similarly, if $p<\gamma(\hat{p},s_1)$, some consumers who are currently assigned

to the competitor will switch to the current firm, namely those who satisfy $s <

m(\hat{p},p)

$. Note   that the first argument of $m$ is the low price and the second

argument is the high price, which have now  reversed to be $(\hat{p}, p)$.

The current firm's demand in this region is

\begin{equation}\label{demext2}

  Q^+(p,\hat{p}) = \frac{1}{2} \left(1+F(m(  \hat{p},p)) \right)   q(\hat{p}) =

 \frac{1}{8p^2} \left(1+F(m(  \hat{p},p)) \right).

\end{equation}



Putting it all together then: If the rival's price is currently set at $\p$ and

the market is currently split evenly between the firms, a  firm's demand is

composed of the local demand functions, \eqref{demint}--\eqref{demext2},

\begin{equation}\label{demand}

  Q(p,\hat{p}) = \begin{cases}

              Q^-(p,\hat{p})= \frac{1}{8p^2} \left(1-F(m(p,\hat{p})) \right)

&

\text{ if } p > \beta(\hat{p},\s),\\

              Q^0(p,\hat{p})=    \frac{1}{8p^2}  & \text{ if } \beta(\hat{p},\s)

\geq

p \geq \gamma(\hat{p},\s),\\

              Q^+(p,\hat{p})=  \frac{1}{8p^2} \left(1+F(m(  \hat{p},p))

\right) &

\text{ if } p< \gamma(\hat{p},\s).

            \end{cases}

\end{equation}



If $\s>0$ (that is, there are no consumers with switching costs too close to

zero), then \eqref{demand} will be continuously downward sloping with two kinks;

the same properties hold for the inverse demand curve. (Continuity follows from

$m(\beta(\hat{p},\s),\hat{p})=m(\hat{p},\gamma(\hat{p},\s))=\s$, so that

$F(m(\beta(\hat{p},\s),\hat{p}))=F(m(\hat{p},\gamma(\hat{p},\s))) = 0$.)





The derivatives  of the demand curve are:

\begin{eqnarray}

  \frac{\partial Q^0(   p )}{\partial p} & = & -\frac{1}{4p^3}, \label{der1}\\

  \frac{\partial Q^-(p)}{\partial p} & = & -\frac{1}{4p^3} + \frac{1}{4p^3}

F(m(p,\hat{p})) - \frac{1}{8p^2}f(m(p,\hat{p}))ccc,   \label{der2}\\

  \frac{\partial Q^+(p)}{\partial p} & = & -\frac{1}{4p^3} - \frac{1}{4p^3}

F(m(\hat{p},p)) - \frac{1}{8p^2}f(m(\hat{p},p))ccc,

\label{der3}

\end{eqnarray}

Now evaluate \eqref{der1}, \eqref{der2} at the ``upper'' kink where $p=\beta(\p,

\s)$. The middle term in \eqref{der2} vanishes, and

$$

  \frac{\partial Q^-(\hat{p})}{\partial p} < \frac{\partial Q^0(\hat{p})}

{\partial p} < 0,

 $$



  Similarly, evaluate  \eqref{der3} at the ``lower'' kink where $p=\gamma(\p,\s)

$. The middle term in \eqref{der3} vanishes, and

$$

 \frac{\partial Q^+(\hat{p})}{\partial p} < \frac{\partial Q^0(\hat{p})}

{\partial p} < 0.

$$



Thus, the  slope of the demand curve is gentler above the upper kink and below

the lower kink than it is in between in the region in which the company's price

changes do not induce switching.



 It is interesting to know which  value, $\beta$ or $\gamma$,  is closer to

$\hat{p}$.  In this example,   $\beta$ is closer, but we will have to figure out

whether that is true generally, and figure out an intuition.



Figure 1 illustrates the kinked demand curve in this model. (we should draw the

marginal revenue curve on there too)









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{\bf Figure 1: The Doubly Kinked Demand Curve}





 So far, however, we have not determined the level of $\hat{p}$. What is the

equilibrium of this game?  In a pure-strategy equilibrium, $p = \hat{p}$, so the

demand curve is in the middle range. Thus, the price must be at the monopoly

level for that demand curve. We will also need to check that a deviation to a

price in the lower or upper range is not more profitable. That can be done by

finding the most profitable deviation to the lower range and the most profitable

deviation to the upper range.



The firm charging $p$ will maximize, if its costs are  $C(Q)$,

\begin{equation} \label{e100a}

 p Q(p,\hat{p}) - C(Q(p,\hat{p})),

\end{equation}

 which is from  \eqref{demand}, if $p$ is in the middle range,

\begin{equation} \label{e101}

 p  \left( \frac{1}{8p^2} \right) - C( \left( \frac{1}{8p^2} \right)) =

\frac{1}{8p } - C( \left( \frac{1}{8p^2} \right))

\end{equation}

 This has first order condition

\begin{equation} \label{e102}

-\frac{1}{4p^3 } +    C'( \left( \frac{1}{8p^2} \right))   \left( \frac{1}{4p^3}

\right)=0,

\end{equation}



so



\begin{equation} \label{e103}

   C'  \left( \frac{1}{8p^2} \right)  =1,

\end{equation}



  Thus, in equilibrium both firms will choose price to maximize this. We do

need, however, to ask whether a firm can do better by a big  cut from the price

thus discovered, so as to capture the other half of the market.





  Once that is done, we can  find the effect of a change in the firms'

production cost. Suppose we specify that  $C'(q) = \theta q$, so that marginal

cost is linear and increasing.  Equation \eqref{e103}

becomes\footnote{xxxTilman, I'm writing out lots of steps in algebra for easier

checking-- we can condense later.}

\begin{equation} \label{e104}

   \theta \left( \frac{1}{8p^2} \right)  =1,

\end{equation}



 so



\begin{equation} \label{e120}

   \theta    =8p^2,

\end{equation}



and



 \begin{equation} \label{e121}

   p = \sqrt{  \theta/8}

\end{equation}



A monopolist operates where  demand curve starts to become elastic. So maybe

this is not good news for the idea that a small change in costs will not change

prices much.





\bigskip



\noindent

{\bf 5. Concluding Remarks}



Still to be written.



      \bigskip



\newpage

\noindent

{\bf References}





 Athey, Susan,  Kyle Bagwell \& Chris Sanchirico,  ``Collusion and Price

Rigidity,'' Discussion Paper \#:0102-38 Department of Economics Columbia

University New York, http://www.columbia.edu/cu/economics/discpapr/DP0102-

38.pdf, March 2002.



 Bhaskar, V. , S. Machin \&  G. Reid  ``Testing a Model of the Kinked Demand

Curve,'' {\it  Journal of Industrial Economics,}   39(3): 241-254 (March 1991).



d'Aspremont, Gabszewicz \& Thisse (1979) ``On Hotelling's `Stability in

Competition,''  {\it Econometrica,} 47:   1145 (1979).



  Fudenberg, Drew  \& J. Tirole ( 2000) ``Customer Poaching and Brand Switching,

'' {\it Rand Journal of Economics,}  31 : 634-657 (Spring 2000).



Hall, R.L. and C.J. Hitch (1939) ``Price Theory and Business Behavior,"{\it

Oxford Economic Papers,}  2: 12-45 (May 1939).



Hotelling, Harold (1929) ``Stability in Competition,''  {\it Economic Journal,}

39:    41 (1929)



  Klemperer, Paul (1987a) ``The Competitiveness of Markets with Switching Costs,

''{\it Rand Journal of Economics,}  18 : 138-150 (Spring 1987).



  Klemperer, Paul (1987b) ``Markets with Consumer Switching Costs,''{\it

Quarterly Journal of Economics,}  102 (2): 375-394 (May 1987).



Klemperer, Paul ( 1995)  ``Competition When Consumers Have Switching Costs: An

Overview with Applications to Industrial Organization, Macroeconomics, and

International Trade,''{\it  Review of Economic Studies,}  62 (4): 515-539

(October 1995).



Salop, S.C., (1979)  ``Monopolistic Competition with Outside Goods,'' {\it Bell

Journal of Economics,}  10, 141-156.



Scitovsky, Tibor  (1941)  ``Prices Under Monopoly and Competition,'' {\it

Journal of Political Economy,}    49: 663-685 (October 1941).



Stigler, George J.  (1947) ``The Kinky Oligopoly Demand Curve and Rigid Prices,

''{\it Journal of Political Economy,}  55: 432-449 (October 1947).



Stigler, George J. (1978) ``The Literature of Economics: The Case of the Kinked

Oligopoly Demand Curve,'' {\it Economic

  Inquiry,}  16  (2):  185-204 (April 1978).



Sweezy, Paul M.  (1939) ``Demand Under Conditions of Oligopoly,'' {\it Quarterly

Journal of Economics,}   47:  568-573  (August 1939).



Tirole book.



Wolman, Alexander L. (2000)  ``The Frequency and Costs of Individual Price

Adjustment {\it Federal Reserve Bank of Richmond Economic Quarterly,} 86/4 (Fall

2000) http://www.rich.frb.org/pubs/eq/pdfs/fall2000/wolman.pdf.



\newpage





DON'T USE:



 Craig Freedman's ``The Economist as Mythmaker" (Journal of Economic

Issues, March 1995, pp. 175-209).  Bad article.



Galbraith, John Kenneth. "Monopoly Power and Price

Rigidities,"Quarterly

Journal of Economics, vol. 50 (May 1936), pp. 456-75. Not kinked

demand

curve.





 Reid, Gavin (1981) Kinked Demand Curve Analysis of Oligopoly  Edinburgh Univ

Press.

  ISBN: 0852243901







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