Consider the problem of maximizing the utility function u = f(c) + g(s) subject to the budget constraint pc + s=y where c is consumption and s is savings

Please see the attached file.

| u1,1u1,2 -p1 | | ∂x1/∂p1 | | λ |
| u2,1u2,2 -p1 | | ∂x2/∂p1 | = | 0 |
| -p1-p2 0 | | ∂λ/∂p1 | = | 0 |
where ui,j is ∂2u/∂xj∂xi
If we denote the 3x3 matrix on the left-hand side as H and create a vector X = (x1, x2, λ), then we can rewrite the abovementioned system as H(∂X/∂p1) = (λ, 0 , 0)T.
Then the Slutsky's equation is
(∂X/∂pi)y = (∂X/∂pi)u - xi(∂X/∂y)P
In case of our problem at hand, we have x1 = c, x2 = s, p1 = p. Therefore, it would be
(∂X/∂p)y = (∂X/∂p)u - c(∂X/∂y)P

where ∂X/∂p and ∂X/∂y represents the impact of a change in the price pi and money income y on the vector of quantities demanded and the Lagrangian multiplier. The notation ( )y, ( )u and ( )P indicates that the derivatives inside of the parentheses are with, respectively, money income y held constant, utility (real income) u held constant, and all prices P held constant.

If g''(s) = 0 then it means that U doesn't depend on s, therefore we have all the entries of the form ui,j in the H matrix containing 2 (either i or j) to be zero.