Problem Set1
1. Consider the consumption decisions of R.B. Turbo, a new student at T University. Ms. Turbo has only on hand $1,000 in monthly income to spend on the food and housing. In terms of food, Ms. Turbo, a vegetarian, has found there is a nice restaurant called Veggies that offers vegetarian meals. She wants to purchase food from this restaurant and is considering their monthly dinner plan, which costs $125.00 per meal served every day in month (she can consume more than one meal a day). In addition, Ms. Turbo is looking for housing. She has found a cheap apartment building in a polluted area of New Jersey where the monthly rental of a one-bedroom apartment is $250.00, a 2-bedroom is $500, 00, a 3-bedroom is $750.00, and a 4-bedroom is $1,000 and so on. Suppose That Turbo has preferences that can be described by the following utility function:
U = X^{10.5}X^{20.5}
Where X_{1 }is the quantity consumed of food per month (number of meals served every day in a month), and X2 is the number of bedrooms in her apartment.
A. Compute the number of bedrooms that Ms. Turbo’s apartment will have (gives a numerical answer and describes how it is obtained, including any derivations).
B. find out the number of meals that Ms. Turbo will consume every day at Veggies (again, provide a numerical answer and specify how it is obtained).
C. Assume that Ms. Turbo suddenly receives a $500 increase in monthly income. Would there be any changes in her consumption of housing and food? (Assume that she has the ability to upgrade to better apartments (with more bedrooms) and to change her monthly meal plan if she wants, so as to increase or decrease the number of meals she eats a day).
D. What is Ms. Turbo’s income elasticity of demand for housing?
E. Assume that Ms. Turbo has an income of $1,000 (as in parts A and B) but that, all of a sudden, Veggies decides to raise the price of its monthly program of meals from $125 per meal to $150. How many dishes will Ms. Turbo consume each day under the new program? Will she keep the same type of apartment (the same number of bedrooms)?
F. What is Turbo’s price elasticity of demand?
2. Assume that Mr. Chauncey Gardener consumes two goods, X_{1 }and X_{2} .His preferences can be described by the following utility function:
U = X^{10.5}X^{20.5}
He faces the following prices in the market: P1 = $5.00 and P2 = $10.00. His income is $1,000. Assume that he spends all his income.
A. How much will he purchase of commodities 1 and 2? describe your answer.
B. Assume that the price of commodity 2 raises by $2.50 (from $10.00 to $12.50); what will be the change in the quantities consumed of goods 1 and 2? describe.
C. In the previous problem (part B), take the change in the quantity consumed of commodity 2 and decompose it into an amount coupled with the substitution effect and another one connected to the income effect (use the Slutsky decomposition discussed in class).
In answering these problems, remember to give numerical answers and to describe where your answers come from.
3. Consider the following utility function:
U = X_{1}X_{2}
Where X_{1} and X_{2} are quantities consumed of two goods. You are considering the actions of a consumer that maximizes utility. He or she has a fixed income m, and faces prices P_{1} and P_{2}. The consumer spends all of its income on the two commodities.
A. Supposing that the consumer maximizes utility, what would be the accurate condition showing the ratio of each good purchased by the consumer?
B. Please derive the demand functions for X_{1} and X_{2}, providing an algebraic expression for each and a diagrammatic representation.
C. What is the price elasticity of demand for commodity 1? What is the price elasticity of demand for commodity 2?