<p>Without any signal (and assuming that sellers have full bargaining power), a buyer is willing to pay the expected value of the product, which is
<span class="math-container">\begin{equation}
0.5(10000)+0.5(7000)=8500
\end{equation}</span>
This is a price that both types of sellers are willing to accept. Thus, without signaling, <span class="math-container">$H$</span>'s profit is
<span class="math-container">\begin{align}
\pi_H(Y_H=0) &amp; = 8500-8000 = 500 \\
%\pi_L &amp; = 8500-5000 = 3500 
\end{align}</span></p>
<p>Next, consider <span class="math-container">$H$</span>'s preferred outcome when warranty is viewed by the buyer as a credible signal for quality. That is, in this equilibrium, only <span class="math-container">$H$</span> would offer a warranty (<span class="math-container">$Y_H&gt;0$</span>) and <span class="math-container">$L$</span> offers none (<span class="math-container">$Y_L=0$</span>); buyer will pay <span class="math-container">$10000$</span> for a product with warranty and <span class="math-container">$7000$</span> for one without warranty. For this to be an equilibrium, we must have
<span class="math-container">\begin{align} 
\text{$H$'s profit after offering warranty $Y_H$} &amp; \ge \text{$H$'s profit without warranty} \\
10000-8000-500Y_H &amp; \ge 500 \\
\end{align}</span>
and
<span class="math-container">\begin{align}
\text{$L$'s profit after offering warranty $Y_H$} &amp; \le \text{$L$'s profit without warranty} \\
10000-5000-1000Y_H &amp;\le 7000-5000
\end{align}</span>
Solving the two inequalities, we get <span class="math-container">$Y_H=3$</span>.</p>

CC BY-SA 4.0
2024-08-28T03:01:35.053Z
2024-08-28T03:01:35.053Z
Herr K.
<p><em>IF</em> the cost of production for <span class="math-container">$H$</span> is <span class="math-container">$\\\$8000$</span> and the cost of production for <span class="math-container">$L$</span> is <span class="math-container">$\\\$5000$</span> (as you seem to be assuming), then
<span class="math-container">$$
\frac{10000-8000}{500}=Y_{H}^{max} = 4
$$</span>
<span class="math-container">$$
\frac{10000-5000}{1000}=Y_{L}^{max} = 5
$$</span></p>
<p>However, I think there must be more to this question than you have included. For instance, what is the valuation of warranty length by the customers? Are those numbers really the cost of production?</p>
<p>(Not sure if this response follows protocol, but I cannot post comments yet).</p>

CC BY-SA 4.0
2023-04-05T22:30:31.563Z
2023-04-05T22:49:48.697Z
2023-04-05T22:49:48.697Z
A. Miller
<blockquote>
<p>Suppose there are two sellers <span class="math-container">$\{H, L\}$</span> such that <span class="math-container">$H$</span> sells high quality products at <span class="math-container">$\\\$ 8000$</span> and <span class="math-container">$L$</span> sells low quality products at <span class="math-container">$\\\$ 5000$</span>. The customers value the products at prices <span class="math-container">$\\\$10000$</span> and <span class="math-container">$\\\$7000$</span> respectively but they don't know who is selling which product (or at least, they don't trust what the sellers say). Each customer has <span class="math-container">$50\%$</span> chance of buying a high quality product and <span class="math-container">$50 \%$</span> chance of buying a low quality one. If warranty costs <span class="math-container">$500Y$</span> for the high-quality product seller and <span class="math-container">$1000Y$</span> for the low-quality product seller where <span class="math-container">$Y =$</span> number of years of warranty, what's the optimal warranty (in years) that <span class="math-container">$H$</span> will set to signal that his quality of product?</p>
</blockquote>
<p>If the customers get the right signal, they'll pay <span class="math-container">$\\\$10000$</span> for the high quality product. <span class="math-container">$H$</span> can provide a max of <span class="math-container">$\frac{10000 - 8000}{500} = 4$</span> years warranty while <span class="math-container">$L$</span> can provide a max of <span class="math-container">$2$</span> years' warranty. I think <span class="math-container">$H$</span> will give <span class="math-container">$2 + \epsilon $</span> (where <span class="math-container">$\epsilon \in (0, 2]$</span> years warranty. Is that correct? Or do I have to consider the expected price the customers will pay somewhere for this?</p>

CC BY-SA 4.0
2023-04-05T23:00:44.847Z
How does the warranty signal higher quality? I wish I could help but it’s difficult for me to understand your question.
Nicolas Torres
<microeconomics>
Sellers and warranty
warranty
# Sellers and warranty

(Asked by: username_0 on 2023-04-04)

> Suppose there are two sellers $\{H, L\}$ such that $H$ sells high quality products at $\\\$ 8000$ and $L$ sells low quality products at $\\\$ 5000$. The customers value the products at prices $\\\$10000$ and $\\\$7000$ respectively but they don't know who is selling which product (or at least, they don't trust what the sellers say). Each customer has $50\%$ chance of buying a high quality product and $50 \%$ chance of buying a low quality one. If warranty costs $500Y$ for the high-quality product seller and $1000Y$ for the low-quality product seller where $Y =$ number of years of warranty, what's the optimal warranty (in years) that $H$ will set to signal that his quality of product?

If the customers get the right signal, they'll pay $\\\$10000$ for the high quality product. $H$ can provide a max of $\frac{10000 - 8000}{500} = 4$ years warranty while $L$ can provide a max of $2$ years' warranty. I think $H$ will give $2 + \epsilon $ (where $\epsilon \in (0, 2]$ years warranty. Is that correct? Or do I have to consider the expected price the customers will pay somewhere for this?

**Tags:** microeconomics
> **Comments:**
> **username_1**: How does the warranty signal higher quality? I wish I could help but it’s difficult for me to understand your question.  

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## 2 answers

### Answer by username_2 on 2024-08-28. Score: 1

Without any signal (and assuming that sellers have full bargaining power), a buyer is willing to pay the expected value of the product, which is
\begin{equation}
0.5(10000)+0.5(7000)=8500
\end{equation}
This is a price that both types of sellers are willing to accept. Thus, without signaling, $H$'s profit is
\begin{align}
\pi\_H(Y\_H=0) & = 8500-8000 = 500 \\
%\pi\_L & = 8500-5000 = 3500
\end{align}

Next, consider $H$'s preferred outcome when warranty is viewed by the buyer as a credible signal for quality. That is, in this equilibrium, only $H$ would offer a warranty ($Y\_H>0$) and $L$ offers none ($Y\_L=0$); buyer will pay $10000$ for a product with warranty and $7000$ for one without warranty. For this to be an equilibrium, we must have
\begin{align}
\text{$H$'s profit after offering warranty $Y\_H$} & \ge \text{$H$'s profit without warranty} \\
10000-8000-500Y\_H & \ge 500 \\
\end{align}
and
\begin{align}
\text{$L$'s profit after offering warranty $Y\_H$} & \le \text{$L$'s profit without warranty} \\
10000-5000-1000Y\_H &\le 7000-5000
\end{align}
Solving the two inequalities, we get $Y\_H=3$.


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### Answer by username_3 on 2023-04-05. Score: -2

*IF* the cost of production for $H$ is $\\\$8000$ and the cost of production for $L$ is $\\\$5000$ (as you seem to be assuming), then
$$
\frac{10000-8000}{500}=Y\_{H}^{max} = 4
$$
$$
\frac{10000-5000}{1000}=Y\_{L}^{max} = 5
$$

However, I think there must be more to this question than you have included. For instance, what is the valuation of warranty length by the customers? Are those numbers really the cost of production?

(Not sure if this response follows protocol, but I cannot post comments yet).


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