Quiz 4: Utility Maximization and Choice

Business

a) P's utility function is img His budget constraint is: img Setting the Lagrange img Yields the first order conditions img ...... (1) img ...... (2) Dividing equation (1) by equation (2), we get img Thus, img ...... (3) Substitution of this value of s into the budget constraint img We get, img Solving for t yields img Substituting the above value of t in equation (3), img 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. img img img Now, we will maximize P's utility img When the price of twinkies is $0.40 The new budget constraint is img Where $ x is the increased money his mother gives him so that his utility remains at the initial level of img . Setting the Lagrange img Yields the first order condition img ...... (4) img ...... (5) Taking the ratio of the first two terms shows that img Thus, img ...... (6) Substitution of this value of s into the budget constraint gives: img img Solving for t yields img Substituting the above value of t in equation (6) we get img The utility level when t is 1.25 x and s is 2 x img img Equating this utility level with his initial utility level of img img We get, img 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 img The Budget constraint is given by img Setting the Lagrange img The first order conditions for the Lagrange equation are given below: img …… (1) img …… (2) Dividing equation (1) by equation (2), the value of img can be calculated as follows: img Thus, img Putting this value into budget constraint, the value of img can be calculated as follows: img Substituting the above value of img in equation: img To maximize her utility, she should purchase img bottles of French Bordeaux and img bottles of California wine. b. The utility function of the individual isgiven below: img When the of price img falls to $20, the new budget constraint is given below: img The Lagrange equation is given below: img The first order condition for the Lagrange equations are given below: img …… (3) img …… (4) Dividing equation (3) by equation (4), the value of img can be calculated as follows: img Thus, img Putting this value into budget constraint, the value of img can be calculated as follows: img Substituting the above value of img in equation the value of can be img calculated as shown below: img To maximize her utility, she should purchase img bottles of French Bordeaux and img 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: img Therefore, the monetary value of increase in utility is img .

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