Q 1.54 This concerns preferences which are additively separable. The cross partial derivatives of the utility function are zero. I think this structure of preferences is known as a Gossen map and it has some rather odd characteristics as we are about to find out.
(a) Suppose good 1 displays increasing marginal utility so f1''>0. Using the result on page 511, D3=f1'f1'fj''+fj'fj'f1''<0 so fj''<0 all j.
(b) From the first order conditions, fi'(xi(p,y))=L(p,y).pi where L(p,y) is the Lagrangian multiplier. Differentiate with respect to y using the Chain Rule to get fi''(xi).dxi/dy=dL/dy.pi and so (fi''.dxi/dy)/pi=(fj''.dxj/dy)/pj all i and j(**). Now p>>0 so if f1''>0 and fj''<0 all j not equal to 1, then I. EITHER if dx1/dy>0 (good 1 normal) then dxj/dy<0 (good j inferior) all j not equal to 1 II. OR if dx1/dy<0 (good 1 inferior) then dxj/dy>0 (good j normal) all j not equal to 1.
I cannot think why the second case should be ruled out. It would seem to be the 'usual' case as it says that inferior goods are rare.
(c) If fj''<0 all j then either dxj/dy<0 all j or dxj/dy>0 all j using (**). But p.x=y so p.dx/dy=1 and since p>>0 we cannot have dxj/dy<0 for all j. Hence dxj/dy>0 all j and all goods are normal.
An excellent answer to 1.57 has already been posted. The first step is to write u=v(p,y) and y=e(p,u), re-arrange to get v(p,y)=a*(p)+b*(p)y where a*(p)=-a(p)/p and b*(p)=1/b(p). (Introduce new notation to save mess! I hope my old tutor Terence Gorman would have approved.) Then use Roy's identity to get xi(p,y), then calculate dxi/dy and get that the income elasticity of demand equals bi*(p)y/(ai*(p)+bi*(p)y) where the subscript denotes differentation with respect to pi.