Consider an agent who knows that the quality requirement ($t$) of its principal for a product follows the density function $f(t)=\lambda e^{-\lambda (t-x)}$ for all $t>x$ and is zero for $t\leq x$. This is basically an exponential distribution that starts at time $t$. The agent now delivers a product of quality $y$ to the principal who declines the product (The principal has no reason to lie about its requirements). The agent can now use this information to update the pdf via the parameter $x$.
Am I correct that the agent's updated belief about the p.d.f. is $f(t|t>y)=\lambda e^{-\lambda (t-y)}$ for all $t>y$ and is $0$ for $t\leq y$? If not, what would it be?
Assume the agent has an "improper" flat prior on $[0,\infty)$ over $x$. If $y$ is rejected the posterior over $x$ is a flat (improper uniform) distribution over $[y,\infty)$. If it is not rejected we write the likelihood function over $x$ as $L(x| y)=\lambda e^{-\lambda (y-x)}$ when $x \leq y$ and $L(x| y)=0$ otherwise. Because $\lambda e^{-\lambda (y-x)}$ is increasing in $x$, this is maximized when $x=y$ as you suggested.