Suppose the preference is represented by the following utility function: 𝑈(𝑥, 𝑦) = √𝑥 + √𝑦.
1) Can this preference be represented by 𝑉(𝑥, 𝑦) = 𝑥 + 𝑦?
What about 𝑊(𝑥, 𝑦) = 2√𝑥𝑦 + 𝑥 + 𝑦?
2) Set up the consumer’s utility maximization problem. Write out the Lagrangian and first
order conditions.
3) Derive the tangency condition from the first order conditions.
4) Derive the demand functions for x and y.
5) Calculate the income elasticity for x. Is x a normal good, inferior good or neutral good?
6) Suppose 𝐼 = 10, 𝑃𝑥 = 1, 𝑃𝑦 = 4. What is the optimal consumption bundle A? What is
the optimal consumption bundle C when the price of x becomes 𝑃𝑥 ′ = 4
7) Set up the consumer’s expenditure minimization problem. Write out the Lagrangian and
first order conditions.
8) Derive the tangency condition from the first order conditions.
9) Derive the compensated demand functions for x and y.
10) For the example given in 6), find the decomposition bundle B and derive substitution
effect and income effect. Are they in the same or opposite direction? How is this result
related to question 5)?