Question

Consider the typical example of shopping at Walmart for pants (x1) and shirts (x2). Your income endowment is $300; the price of shirts is $20, and the price of pants is $30 a) Write down a Cobb-Douglas utility function with exponents a=0.5 and 1-a=0.5. b) Write down the budget constraint for this problem. c) Set up the Lagrange and find the optimal consumption bundle for Xi and x2 (call this bundle A) d) Now assume that the price of pants has increased to $50. Find the Substitution Effect by assuming that you were compensated for the difference in income. What is the optimal bundle for Xi and x2 in this case (call it bundle B)? What is the change in the quantities x, and x2 due to the substitution effect? e) Find the new optimal bundle that is actually chosen, which incorporates now both the Income and Substitution effect. To do this, you just need to repeat what you have done in c), but with the new price of $50. 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 please post as separate question.

P, x, & P₂ X2 a) utility function! (x, ,X2) = x 0.5 x 0.5 b) Budget constraints = 30x, +20 x = 300 c) setting up lagrangean. x,0.5x5 + 1 [300-304, 8-20x₂] 0.5*,-05X 0.5 -Box ax, o da 2x2 Or 5x,0.5 x 0.5-2002 300 - - 20x2 =0 21 30x, Divide cq 0 by ②, , we get xos 30x 0.5*,~0.5 0.5x,0.5 +20.5 Nw = ) میرا Using (3 eq get. 30x + 20 20 x2 =300 Gox - 300 5 ) Bundle Ai 2 х, 15

d) Now P = $50 Change in income a AM ДМ - x, An) AM - 5150-30) AMS 100 New income Ma M FAM M'- quo Now Budget Constraint all be sox+20x 400 Now to compute bundle B equate HRS = 50 x2 = // get. Soxt Using Budget line { (20)x, = 400 100 x = 400 x = 4 [x = 10 Bundle B e) {4, 10) = (x, , x ₂) & Substitution effect a Sponsor 11 for x, = a 4-5 -1 2) for X₂ 10 15 5 2 2 ire due to per increase in price of pants demand for pants decreases by one that far shirts increases by runter. - unit