<p>In Lemma 1 they say that the support $\left[\underline{p}_t, \bar{p}_t \right]$ is such that $ c &lt; \underline{p}_t $.
So even though the support is connected, it does not extend to $c$, hence the $p \to c$ problem never arises.</p>

CC BY-SA 3.0
2015-04-23T05:11:25.193Z
Your current argument seems strange to me. The equilibrium strategy is such that that $F(\underline{p}_t) = 0$. This is how $\underline{p}_t$ is determined. The authors state this in Appendix A, Claim 5, page 41. This limits how close $p$ can be to $c$. Unless there is another condition that $\underline{p}_t$ has to fulfill I don't see a contradiction?
Giskard
CC BY-SA 3.0
2015-04-22T21:25:32.813Z
2015-04-22T21:25:32.813Z
Giskard
<p>In their equation (5), <a href="http://scholar.princeton.edu/gkaplan/publications/shopping-externalities-and-self-fulfilling-unemployment-fluctuations" rel="nofollow">Kaplan and Menzio</a> claim that the price distribution in their Burdett-Judd market is given by</p>

<p>$$ F(p, u) = \{u \cdot A_1 \left[1 - \left(1 - B_1(u)\right)\frac{(r-c)p}{(p-c)r}y_u\right] \\
+ (1-u) \cdot A_2 \left[1 - (1 - B_2(u))\frac{(r-c)p}{(p-c)r}*w(u)\right]
 \}/C$$</p>

<p>For positive $A_i$, $B_i$, $C$, where $u$ denotes the unemployment rate and $p$ denotes the price. They continue claiming that</p>

<ul>
<li>$F$ is continuous</li>
<li>has connected support</li>
</ul>

<p>$c$ is the households outside option, $r$ is the reservation price, hence the distribution should only give positive mass to prices between $[c, r]$.</p>

<p>$$ B_1(u) = 2\nu(u)\frac{\psi_u}{1+\psi_u}$$</p>

<p>where $\nu(\sigma(u)) = \frac{s}{b} = \frac{1-u}{1+u(\psi_u - \psi_e)}$. In their calibration: $\psi_e = 0.02$, $\psi_u = 0.27$. Hence</p>

<p>$$ B_1(u) = 2\frac{1-u}{1+0.25u}\frac{0.27}{1+0.27}$$</p>

<h3>The Issue</h3>

<p>For example, at an unemployment rate of $0.05$, we have $B_1(0.05) = .38$. However, for $p = c + \epsilon$ (for small $\epsilon$), the denominator $p-c$ becomes very small small.</p>

<p>This means that the product of $1-B_1(0.05)\cdot (r-c)\cdots$ becomes very  large. One minus that is an very large negative number. The denominator $C$ is positive. A similar phenomenon happens with $B_2(0.05)$.</p>

<p>They call $F(p, u)$ the distribution. I assume this means the pdf. Can a pdf have negative values? Or what am I missing here?</p>

CC BY-SA 3.0
2015-04-23T23:50:07.077Z
On the side $F(p,u)$ is a probability, i.e. a cumulative distribution function (CDF), constrained in $[0,1]$.
Alecos Papadopoulos
<macroeconomics><mathematical-economics><price-theory>
Price distribution: Negative values?
FooBar
# Price distribution: Negative values?

(Asked by: username_0 on 2015-04-22)

In their equation (5), [Kaplan and Menzio](http://scholar.princeton.edu/gkaplan/publications/shopping-externalities-and-self-fulfilling-unemployment-fluctuations) claim that the price distribution in their Burdett-Judd market is given by

$$ F(p, u) = \{u \cdot A\_1 \left[1 - \left(1 - B\_1(u)\right)\frac{(r-c)p}{(p-c)r}y\_u\right] \\
+ (1-u) \cdot A\_2 \left[1 - (1 - B\_2(u))\frac{(r-c)p}{(p-c)r}\*w(u)\right]
\}/C$$

For positive $A\_i$, $B\_i$, $C$, where $u$ denotes the unemployment rate and $p$ denotes the price. They continue claiming that

* $F$ is continuous
* has connected support

$c$ is the households outside option, $r$ is the reservation price, hence the distribution should only give positive mass to prices between $[c, r]$.

$$ B\_1(u) = 2\nu(u)\frac{\psi\_u}{1+\psi\_u}$$

where $\nu(\sigma(u)) = \frac{s}{b} = \frac{1-u}{1+u(\psi\_u - \psi\_e)}$. In their calibration: $\psi\_e = 0.02$, $\psi\_u = 0.27$. Hence

$$ B\_1(u) = 2\frac{1-u}{1+0.25u}\frac{0.27}{1+0.27}$$

### The Issue

For example, at an unemployment rate of $0.05$, we have $B\_1(0.05) = .38$. However, for $p = c + \epsilon$ (for small $\epsilon$), the denominator $p-c$ becomes very small small.

This means that the product of $1-B\_1(0.05)\cdot (r-c)\cdots$ becomes very large. One minus that is an very large negative number. The denominator $C$ is positive. A similar phenomenon happens with $B\_2(0.05)$.

They call $F(p, u)$ the distribution. I assume this means the pdf. Can a pdf have negative values? Or what am I missing here?

**Tags:** macroeconomics, mathematical-economics, price-theory

> **Comments:**
> **username_1**: On the side $F(p,u)$ is a probability, i.e. a cumulative distribution function (CDF), constrained in $[0,1]$.  

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## 1 answers

### Answer by username_2 on 2015-04-22. Score: 2 | Accepted answer

In Lemma 1 they say that the support $\left[\underline{p}\_t, \bar{p}\_t \right]$ is such that $ c < \underline{p}\_t $.
So even though the support is connected, it does not extend to $c$, hence the $p \to c$ problem never arises.

> **Comments:**
> **username_2**: Your current argument seems strange to me. The equilibrium strategy is such that that $F(\underline{p}_t) = 0$. This is how $\underline{p}_t$ is determined. The authors state this in Appendix A, Claim 5, page 41. This limits how close $p$ can be to $c$. Unless there is another condition that $\underline{p}_t$ has to fulfill I don't see a contradiction?  

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