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# Solving additively separable preference with Lagrange

Suppose that the preferences of a consumer are additively separable in goods 1 and 2, i.e., her utility function is of the form u(x1,x2) = v1(x1) + v2(x2), where v''i(·) < 0 for i = 1,2. The prices of the two goods are p1 and p2, and the exogenous income is m.

(a) Formulate the Lagrangian for this consumer’s constrained utility-maximisation problem. Make sure you indicate the three decision variables. Find the three ﬁrst-order necessary conditions. Denote the decisions variables that solve this system as x1*(p1,p2,m), x2*(p1,p2,m), and λ∗(p1,p2,m).

(b)Substitute the optimal solutions from part (a) into the three ﬁrst-order necessary conditions. Then, partially diﬀerentiate each of the three ﬁrst-order necessary conditions with respect to m. Solve for ∂x1*/ ∂m and ∂x2*/∂m using Cramer’s Rule. Are both goods normal?

I understand getting

v1’(x1)-λp1=0

v2’(x2)-λp2=0

p1x1+p2x2=m

as your 3 first order equations but really can't see where you are supposed to