Sellers and warranty

Question

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?

Answer 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 $Y_H=3$.

Answer 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).