Question 1: Utility maximization and price index Tim consumes haircuts (x₁) and sandwiches (x2). His utility function is: u (X1, X2) = (x₁)* (x₂)¹−º 1-0 The price of ₁ is p₁ and the price of x2 is p2. He has a total income of m. 1. Find Tim's demand function for goods 1 (x₁(P₁, P2, m)) and 2 (x2 (P1, P2, m)), using a Lagrangian. 2. Suppose we'd want to compute a price index for the "average" good that Tim consumes. We define it as such that: P* V(p1, P2, m) = m where P is the price index, V (p₁, P2, m) is the indirect utility that Tim can achieve when prices are p₁ and p2 and his income is m. Solve for the price level as a function of the individual prices, P(p1, P2), and show that it is equal to the inverse of the Lagrange multiplier in 1. Interpret. hint: the indirect utility is obtained by plugging the demand you found in (1) into the utility function: V(p1, P2, m) = u((x1 (P1₁, P2, m), 2 (p1, P2, m)) 3. Does the price index satisfy homogeneity of degree 1? How does the im- portance of each good in the price index vary depending on the parameter 0? Interpret. hint: take the log of the price index and see how 0 mediates the impact of changes in p₁ and p2) 4. Suppose that in the US, the price of a haircut is $16, and the price of a sandwich is $4. In India, a haircut costs $9 and a sandwich $1. Suppose that = 0.5, what is the price index of the US relative to the price index of India? 5. Tim lives in the US, and Tim's cousin lives in India. If Tim earns USD 100 per day and Tim's cousin earns USD 10 per day, how much is Tim richer than his cousin in nominal term, and in PPP terms?

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Form Utility

The term utility in general refers to the satisfaction earned from usage of a particular product or service. Utility always plays an important role because it influences various important elements like demand & thereby the price of the good or service as well. Though utility is not easy to measure, various types of economic models can be used indirectly. There can be various types of utilities like economic utility, marginal utility, cardinal utility, form utility.