Imperfect Substitutes and Utility Funcitions The utility function for perfect substitutes is defined as U(X,Y) = aX + bY. If the two goods X&amp;Y are imperfect substitutes what would be their utility function?
In general, any utility function that produces non-linear and downward sloping indifference curves will feature imperfect substitutability between goods. (By considering only goods, we're implicitly assuming &quot;more is preferred to less&quot;, or positive marginal utility, for each good.) In other words, look at the marginal rate of substitution and see if it depends on the quantity of either good; if it does, then there is imperfect substitutability.
Examples

Quasi-linear utility: $u(x,y)=v(x)+y$, where $v(x)$ is non-linear in $x$ (e.g. $v(x)=\sqrt x$)
Additively separable utility: $u(x,y)=g(x)+h(y)$, where $g,h$ are non-linear functions
Cobb-Douglas: $u(x,y)=x^\alpha y^\beta$, where $\alpha,\beta$ are constants
Leontief (perfect complement): $\style{text-decoration:line-through}{u(x,y)=\min\{\alpha x,\beta y\}}$ [Leontief utility function features nonsubstitutability rather than imperfect substitutability.]
CES utility: $u(x,y)=\left(\alpha x^\rho+\beta y^\rho\right)^{1/\rho}$, where $\rho\ne1$. Note that the class of CES utility functions includes Cobb-Douglas (as $\rho\to0$), Leontief (as $\rho\to-\infty$), and linear utility (as $\rho=1$) as special cases.

This lecture note by Simon Board has examples and indifference curves corresponding to the cases above.