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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}) \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
\end {document}