HMMT 2003 K9

Two differentiable real functions $f(x)$ and $g(x)$ satisfy

$$\frac{f^{\prime}(x)}{g^{\prime}(x)}=e^{f(x)-g(x)}$$

for all $x$, and $f(0)=g(2003)=1$. Find the largest constant $c$ such that $f(2003)>c$ for all such functions $f, g$.