Kit Kats and Oreos
- III.C.2: Solve the utility maximization problem for Marshallian demands, the indirect utility function, and the expenditure function, including corner solutions when prompted.
- III.C.3: Solve the expenditure minimization problem for compensated demands, the expenditure function, and the indirect utility function, including corner solutions when prompted.
- III.C.4: Recognize the graphical tangency condition in the utility maximization and expenditure minimization problems and be able to explain indifference curves and iso-expenditure curves.
Utility maximization and expenditure minimization are, in many ways, two sides of the same coin. To show this we will use the following set up:
Panda’s two favorite treats are Kit Kats, good 1, and Oreos, good 2, and he has the following Cobb-Douglas utility function for the two treats $u(x_1,x_2) = x_1x_2$. However, Panda is also constrained by the candy allowance his mom gives him: $g(x_1, x_2) = m - p_1x_1 - p_2x_2 = 0$. Initially, Panda’s allowance is $m = 16$ and the prices for Kit Kats and Oreos are both $4$; i.e., $p_1 = p_2 = 4$.
- Write down the Lagrangian that corresponds to Panda’s constrained utility maximization problem. Derive the first order conditions, and find the tangency condition (i.e., MRS = MRT).
- In variable form, solve for Panda’s Marshallian demands. Then substitute these demand functions into the utility function to get the indirect utility function $V(p_1,p_2,I)$. Conceptually, what does the indirect utility function tell us?
- Finally, plug in the numbers given above and double check your work knowing that Panda’s utility at the optimal bundle is $U = 4$.
Years later, Panda is a grown up working a job of his own and has (effectively) limitless amount of income to buy Kit Kats and Oreos. However, as an adult, Panda has also come to realize that a diet consisting of only Kit Kats and Oreos (contrary to his previously held belief) is neither the most healthy nor delicious.
Aligning his fitness plans and matured taste buds, Panda has set a new goal to only consume as much Kit Kats and Oreos as he previously did, maintaining the same level of utility, no more, no less. Unfortunately for Panda, inflation has also kicked in during those years and now the price of Kit Kat is $p_1^{new} = 5$ and the price of Oreos is $p_2^{new} = 5$.
More calculations for Panda! Can you help him?
- Write down the Lagrangian that corresponds to Panda’s expenditure minimization problem. Derive the first order conditions, and find the tangency condition (i.e., MRS = MRT).
- In variable form, solve for Panda’s Hicksian (a.k.a., compensated) demands. Without the use of concrete numbers and using the previously derived demands, derive the expenditure function $E(p_1^{new},p_2^{new},\bar{U}^{old} )$. What does the expenditure function function tell us?
- Then, plug in numbers into the expenditure function and find out how much money Panda has to spend to achieve the same level of utility as he did before inflation. Just in case Panda changes his mind about his optimal level of Kit Kats and Oreos, also calculate the costs for $\bar{U}^{new} = 3$ and $\bar{U}^{new} = 6$.
- Finally, let’s return to the initial set up using the original prices of $p_1^{old} = 4$ and $p_2^{old} = 4$, taking what we learn from part (a) and (e) to interpret and evaluate the following expression: \(V(p_1^{old},p_2^{old},E(p_1^{old},p_2^{old},\bar{U}^{old}))\) Reminder: $\bar{U}^{old} = 4$. Think about the allowance $m^{old} = 16$ that Panda had, what part of the above expression does this level of income correspond to. What would plugging these values into the indirect utility function give us?
Now report your findings back to Panda. He sure owes you some Kit Kats for that one…unless you like Oreos more?