Question

Suppose that Cassie likes to snack on coffee and pretzels. Her utility function is U(C,P)=CP, where C is the number of cups of coffee she drinks, and P is the number of pretzels packages she eats. The price of coffee is $4 and the price of pretzels is $2. Cassie has $8 to spend per day on coffee and pretzels.

a. What is Cassie's objective function?
b. What is Cassie's specific budget constraint?
c. Write a statement of Cassie's constrained optimization problem. (Hint: this is the math set up that matches the words "maximize utility subject to a budget constraint).
d. Solve Cassie's constrained optimization problem using the Lagrangian - that is, determine how many packages of pretzels and cups of coffee she consumes optimally.

Answer 1

Cassie's objective function is U(C,P)=CP. Her budget constraint is 4C + 2P ≤ 8. To solve her constrained optimization problem, we set up the Lagrangian function and find the optimal values for C and P.

Step-by-step explanation:

a. Cassie's objective function is U(C,P)=CP, where C represents the number of cups of coffee she drinks and P represents the number of pretzel packages she eats.

b. Cassie's specific budget constraint can be determined by dividing her total budget by the price of each item. In this case, her budget constraint is 4C + 2P ≤ 8, where C represents the number of cups of coffee and P represents the number of pretzel packages.

c. Cassie's constrained optimization problem is to maximize her utility function (U) = CP, subject to the budget constraint of 4C + 2P ≤ 8.

d. To solve Cassie's constrained optimization problem using the Lagrangian, we need to set up the Lagrangian function as L = CP + λ(8 - 4C - 2P), where λ is the Lagrange multiplier. Taking the partial derivatives of L with respect to C, P, and λ, we can find the optimal values for C and P.