Instructions-Attempt all QuestionsQuestion One1Consider an individual with the utility function, U(x₁, x₂) = (α₁x₁² +α₂x;³)¯¼ß. The prices of x₁В21and x₂ and his income are p>0, Px₂ > 0, and I > 0.2x2i. Show that this utility function satisfies diminishing marginal rate of substitution.ii. Derive his Marshallian (uncompensated) demand functions for X₁ and X₂iii. Derive his indirect utility function.iv. Determine the Roy's identityv. Without solving his dual problem (i.e., minimizing expenditure subject to a given utility),derive his expenditure function.Question TwoConsider an economy with two consumers A and B. Consumers A and B have utility111functions: u(x₁^,x^)=(x^)¹³(x^)¹² and u(x²,x²)=(x²) ¹³ (x²)¹². They face prices P₁ and P₂3 A23for good 1 and 2 respectively and have income IA and IB.a) Argue that V(x^^,x^)= ln[u(x^,x^)] and V(x³, x²)= ln[u(x²,x²)] represent the samepreference relationship as u(x,x) and u(x²,x²³b) Show that consumer A's demand functions for the two goods are:21A31AAAx₁^ (P₁, P₂,I^) = ;x₁^ (P₁, P₂,I^)=5p₁5p₂c) Calculate the optimal value of the Lagrange multiplier * and explain its meaningd) Determine B's demand functions for the two goodse) We now assume that the consumers have the following initial endowments e^ = (15,1)and e³=(5,10). Determine the Walrasian equilibrium for this economy, that isi. the equilibrium price (P,P₂)2*AABii. the equilibrium allocation (x^(p₁, p₂),x^(p₁, p₂)) and (x³ (P†‚P₂)‚x²³ (P₁, P²))2X2Question TwoConsider an economy with one consumer and two goods x, (clothing) and x₂ (food) withrespective prices P₁ The utility function is u(x₁, x₂) = x/²x₂4 . The consumer's income1124.P₁ andP₂.is denoted I.i. Write formally the economic problem faced by the consumer and derive the demandfunctions x₁ (P₁, P₂, 1), and x₂ (P₁, P₂,1)I)1919ii. Calculate the optimal value of the Lagrange multiplier 2* for the consumer and explainits meaning.iii. Explain Walras' Law. Why is this law important for the general equilibrium theory?

*Hi there , as per our guidelines we can only solve 1 question with maximum 3 sub parts at a time .…

Instructions-Attempt all Questions Question One Consider an individual with the utility function, U(x₁, x₂) = (a₁x₁² + a₂x²³)¯½³. The prices and x₂ and his income are PX₁ >0, Px₂ > 0, and I > 0. i. Show that this utility function satisfies diminishing marginal rate of substitution. and X₂ ii. Derive his Marshallian (uncompensated) demand functions for X₁ iii. Derive his indirect utility function. iv. Determine the Roy's identity