 Solution

Write the first endowment as $(a,t)$. The budget constraint at normalized prices is $px+y\le pa+t$. Because the quasilinear utility is increasing in $y$, the budget binds, and its maximization reduces to maximizing $\log x-px+pa+t$ over $x>0$. Its derivative is $1/x-p$ and its second derivative is $-1/x^2<0$, giving the unique demand correspondenceThe first-good demand has no wealth effect. Since the second consumer's endowment and preference relation are fixed, $Z_t=0$ everywhere. Thus the joint condition of part (a) becomes $Z_p\ne 0$ at equilibrium; it is not automatically satisfied.

Here is a complete smooth counterexample, with the second consumer's consumption set specified explicitly. The question does not impose the first consumer's consumption set on the second consumer. Take endowments $(a,t)=(1/10,t)$ and $(b,c)=(2,1)$. Give the second consumer consumption set $X_2=(-\infty ,A)\times \mathbb{R}$, where $A=43/20$, and put $B=31/20$. Its utility function isThis defines complete, transitive, continuous preference relations. It is in fact smooth on $X_2$: writing $r=A-x>0$ and $s=y-B>0$, the positive branch is $\exp (-r^2/(4s)-r/2-s/4)$; at $s=0$ all derivatives tend to zero locally uniformly where $r>0$. The endowment belongs to $X_2$. Allowing negative holdings is explicit here, just as the first consumer's second coordinate is unrestricted in the printed question.

We now derive, rather than posit, the second consumer's demand correspondence. For $v>0$, the upper contour $U_2\ge e^{-v}$ is described by $(r+s{)}^2\le 4vs$, with $r,s>0$. At a fixed $s$, its cheapest first coordinate uses $r=2\sqrt{vs}-s$. Minimizing total expenditure over $s$ therefore means minimizingThe unique minimum occurs at $\sqrt{s}=p\sqrt{v}/(1+p)$, with $r=vp(p+2)/(1+p{)}^2>0$. The expenditure necessary to reach this contour is consequentlySet $\delta =A-b=3/20$ and $\epsilon =B-c=11/20$. Wealth is $pb+c$, so the best affordable contour hasAll affordable positive-utility bundles have $v\ge v_{\ast }$, and the minimum-expenditure bundle on that contour is unique and spends precisely the budget. Zero-utility bundles cannot be optimal because this positive-utility bundle is available. Substitution yields globally smooth unique demands for every $p>0$:The aggregate excess demand is nowFor every $t>0$, the unique equilibrium price is $p=1$, and $Z_p(1,t)=Z_t(1,t)=0$. At this price the allocations are $(1,t-9/10)$ and $(11/10,19/10)$, whose sum equals the aggregate endowment $(21/10,t+1)$. Second-good excess demand is $-pZ$, so it too is smooth and clears. Hence every economy in this smooth parameterized family is singular. This demonstrates why the first consumer's missing wealth effect alone gives no transversality guarantee; the stronger conclusion in part (c) comes from the second consumer's additional demand restriction.