This question explores the quasi-linear utility function. Consider Thomas who has preferences over food, QF, and clothing, QC. His preferences are represented by
U(Qf,Qc)=aQf^(1/3)+Qc
where a > 0. Let Pf, PC, and I represent the prices of food, clothing, and Thomas's budget for food and clothing.
i. Using the Lagrangian method, solve mathematically for the demand function for each good. Make sure to show each step of your work.
ii. Determine whether food and clothing are normal or inferior goods for Thomas. Prove your answer mathematically.
iii. Are food and clothing substitutes or complements for Thomas? Prove your answer mathematically.
iv. Now assume that a = 3, and I = $1,224.
1. If pf = $3, and pc = $12 what is Thomas's demand for food and clothing?
2. If the price of food goes down to pf = $4/3, what happens to Thomas's demand for food and clothing?
3. If Thomas’ income falls to $996, what happens to quantity consumed of each good?
4. Graph Thomas's inverse demand curve (i.e price on y-axis, quantity on x-axis) for food when his income is I = $996 and the price of clothing is pc = $12. Be as precise as possible, labeling the exact value of any important points on the graph.