# Quiz 4: Utility Maximization and Choice

a) P's utility function is His budget constraint is: Setting the Lagrange Yields the first order conditions ...... (1) ...... (2) Dividing equation (1) by equation (2), we get Thus, ...... (3) Substitution of this value of s into the budget constraint We get, Solving for t yields Substituting the above value of t in equation (3), As shown above, P should buy 5 twinkies and 2 cups of soda. b) P's utility when the price of twinkies was $0.10 each is found by substituting the optimal values t = 5 and s = 2 into the utility function. Now, we will maximize P's utility When the price of twinkies is $0.40 The new budget constraint is Where $ x is the increased money his mother gives him so that his utility remains at the initial level of . Setting the Lagrange Yields the first order condition ...... (4) ...... (5) Taking the ratio of the first two terms shows that Thus, ...... (6) Substitution of this value of s into the budget constraint gives: Solving for t yields Substituting the above value of t in equation (6) we get The utility level when t is 1.25 x and s is 2 x Equating this utility level with his initial utility level of We get, As his initial allowance was $1, his mother will have to increase his allowance by $1.

a. Utility function is a mathematical function which gives the level of utility achieved by an individual at different amounts of commodities. Individual's utility is represented by The Budget constraint is given by Setting the Lagrange The first order conditions for the Lagrange equation are given below: …… (1) …… (2) Dividing equation (1) by equation (2), the value of can be calculated as follows: Thus, Putting this value into budget constraint, the value of can be calculated as follows: Substituting the above value of in equation: To maximize her utility, she should purchase bottles of French Bordeaux and bottles of California wine. b. The utility function of the individual isgiven below: When the of price falls to $20, the new budget constraint is given below: The Lagrange equation is given below: The first order condition for the Lagrange equations are given below: …… (3) …… (4) Dividing equation (3) by equation (4), the value of can be calculated as follows: Thus, Putting this value into budget constraint, the value of can be calculated as follows: Substituting the above value of in equation the value of can be calculated as shown below: To maximize her utility, she should purchase bottles of French Bordeaux and bottles of California wine. c. As price of French Bordeaux decreases from $40 to $20 and individual can buy more of French Bordeaux she is better off in second situation.She is consuming same amount of California wine and more French Bordeaux wine. The monetary value that can be put on this utility increase is the additional money that the individual would have had to spend to purchase the additional 10 bottles of French Bordeaux at the initial price of $40. The monetary value of utility increase is calculated below: Therefore, the monetary value of increase in utility is .

a) The utility function of the individual is As cost is not object, there does not exist a budget constraint. First order conditions of the utility function is ...... (1) ...... (2) From equation (1) we get c = 10 From equation (2) we get, b = 3 The individual should consume 3 units of b and 10 units of c to maximize utility. b) The utility function of the individual is given by As instructed by the doctor, his constraint is given by Thus, b = 5 - c Substituting this in the utility function, we get The first order condition is Therefore, Putting this back into equation The individual should consume 1 unit of b and 4 units of c to maximize his utility.