please only do: if you can teach explain steps of how to solve each part use lengrange A consumer of two goods has utilityu(x₁, x2) = x²x₂.In addition to her wealth w, the consumer already owns k units of good 1, which she cannotsell (she can, however, purchase more). Thus, for example, if k = 2 and she uses her wealthto purchase 3 units of good 1 and 1 unit of good 2, then she receives utility (3+2)².1 = 25.(a) Sketch this consumer's utility-maximization problem in (₁,2)-space, where I repre-sents the total quantity consumed of good i (i.e. including the k units of good 1 that shealready owns).Solution: The sketch is similar to the typical one except that the budget line is shiftedto the right by k (leaving a gap from x₁ = 0 up to x₁ = k.)(b) Find this consumer's Marshallian demand, expressed in terms of quantity purchased (i.e.not including the k units of good 1 that she already owns).Solution: In terms of quantity consumed, the utility maximization problem is the samethe usual one given this utility function except that the wealth is w+kp₁ and the nonneg-ativity constraint on x1 is replaced with ₁ ≥k. Therefore, using the usual Cobb-Douglasdemand formula and subtracting k from ₁ to get the quantity purchased, and then check-ing the constraint x₁ ≥k, we obtainx(p, w) ==2w-kp1 w+kpi3p₁ 3p₂(0, #)P2if 2w > kp₁,otherwise.

Utility Function : U = x12x2 Budget Constraint : p1x1 +p2x2 = w Optimal consumption point is…

A consumer of two goods has utility u(x₁, x2) = x²x₂. In addition to her wealth w, the consumer already owns k units of good 1, which she cannot sell (she can, however, purchase more). Thus, for example, if k = 2 and she uses her wealth to purchase 3 units of good 1 and 1 unit of good 2, then she receives utility (3+2)².1 = 25. (a) Sketch this consumer's utility-maximization problem in (₁,2)-space, where ; repre- sents the total quantity consumed of good i (i.e. including the k units of good 1 that she already owns). Solution: The sketch is similar to the typical one except that the budget line is shifted