 Solution

When both consumers have the same quasilinear utility on $(0,\infty )\times \mathbb{R}$, the same strictly concave one-variable maximization gives $x_1=x_2=1/p$, independently of their wealth. If their first-good endowments are $a,b$, thenThus the joint regular-value condition always holds. With positive total first-good endowment, the equilibrium isBoth first-good markets and second-good markets clear: second-good demands sum to $p^{\ast }(a+b)+t+c-2=t+c$. If a nonstandard endowment choice has $a+b\le 0$, there is no positive-price zero, so the regular value condition is vacuous. In the usual positive-endowment case every parameter has the displayed unique regular exchange economy.