Question

Consider the case of a consumer who decides how many cups of coffee (denote by c) and cups of tea (denote by t) to consume every month. Assume the income endowment for caffeine needs is $300; the price of a cup of tea is $2 and the price of a cup of coffee is $3.  

a) Write down a Cobb‐Douglas utility function with exponents α=0.5 and 1‐α=0.5.

b) Write down the budget constraint for this problem.

c) Set up the Lagrangian and find the optimal consumption bundle for c and t (call this bundle A).

d) Now assume that the price of coffee has increased to $5. Find the Substitution Effect by assuming that you were compensated for the difference in income. What is the optimal bundle for c and t in this case (call it bundle B)? What is the change in the quantities c and t due to the substitution effect?  

e) Find the new optimal bundle for c and t that is actually chosen (that is, without assuming any income compensation), which incorporates now both the Income and Substitution effect. Call the result bundle C.

f) Show the effects of the price change using a graph (or multiple graphs). Indicate in the graph points A, B, C, corresponding to the different bundles. Show the changes due to the substitution effect and to the income effect.

Answer 1

We are supposed to do only four subparts to a question. For solution to other parts of the question please post as separate question.

a) Utility function: ule,t)= costos b) Budget Constraints pe c + Pe tam 30 + 2t - 300 c) Lagrangean. costos & & [ 200-34-2t] 22 0.5, -0.5 tos - 3x = 0 o.5cois trois - 2x=0 at 300 – 30 - 2t –– O ad میرا Binide eq 0 by ® we get. 0.5c-oistos 32 0.5 cos no.5 2 E t = 3 c. c using eq qed 36 +2t - 300 36+3 300 (c = 50 t=75 Budle Ai (Lt) = (50, 75) d) Perts Now Charge in income AM = (5-3)x50 AM - 100 M'= M+AM. M'- Buot 100 40