Question

Marge has the utility function                    U(F,H)=20F2H

where F is the quantity of footwear and H is the quantity of hats she consumes. Suppose the price of footwear is $20 and the price of hats is $5, while Marge has an income of $200/week.

1. Calculate Marge's MRS as a function of the quantities F and H. (2 points)
   
   
   
   
   
   
   
   
   
   
   
2. BONUS: Write down the Lagrangian function for Marge's utility maximization problem. (2 points)

3. Solve for Marge's optimal consumption bundle of footwear and hats. (You can either use the Lagrangian from part b or use any other method you like) (3 points)
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   

Use your answer from part 3 and calculate that, at Marge’s optimal consumption bundle, the slope of the indifference curve is indeed equal to the slope of the budget constraint. (1 point)

Answer 1

U=20F²H PE = $20, Me $200. PH = $5 MRS MUF MUE MUH OUCF,42 ƏF d ap. 20X2 FH 40 FH - MUE And MORE Juti,H) ӘН 2. ( 205?H) ƏИ 2oF² (1) MUH 20 F2 MRS = MUF MUR पOFH 20F? MRS = 27 F Bonus; 2 = Maximize utility subject to Budget constraint Mility. (FH) = 20 FH Budget constratur: PF F t Ph H=M 205751 - 200 20F?H tx [200 - 20F-5H L- 40FH - 201=0 ƏF ToFH = 200 dh 2H 20F? 5) = 0 20F 57 - 2 @ @ 40FH 2o4 5 2052 2 H 4 2 H = 4.5 H = 28 substitute H = af into Budget contraint 20F +5H 200 2of +5(27) - 200 2of tlof - 200 30F 200 200 30 20 H+ = 2F* 2 x 20 Go to = 40 2o > (F*, H 20 ) optimal consumption Bundle

slope of the Indifference Curve MRS - MUF MOH 2 H [as calculation Above Put f*, H 40 3 प 2 2.0 240 20 MRS 4 - At optimal Bundle s slope of gc > Budget line Now slope of PE PH 1 -20 5 4 PE PH s slope of At optina Bundle Budget Courant at IC coustralut Hence opinal consumption bundle, the slope of Cire. MRS) 3 udget lie PE PM MRS = PE 4 lu equal to the Мәр