13. Max has a utility function U(x, y) = 2xy + 1. The prices of x and y are both

Question

13. Max has a utility function U(x, y) = 2xy + 1. The prices of x and y are both $1 and Max has an income of $20.


a. How much of each good will he demand?


b. A tax is placed on x so that x now costs Max $2 while his income and the price of y stay the same. How much of good x does he now demand?


c. Would Max be as well off as he was before the tax if when the tax was imposed, his income rose by an amount equal to $1 times the answer to part (b)?

Explanation / Answer

first of all, derivative with respect to x = 2y, and derivative with respect to y = 2x. so dividing dy/dx = 2x/2y = x/y, but as is stated, x,y both cost $1, so this becomes 1/1 = 1. a) you want to maximize 2xy+1 subject to x+y = 20. so, if x+y=20 then y=20-x so plugging in y from above, 2xy+1 = 2x(20-x)+1 = 40x-2x^2+1. You now need to find critical points of this. Taking the derivative w.r.t x gives -4x+40 = 0 ==> -4x = -40 ==> 4x = 40 ==> x = 10. So, you want 10 units of product x. Now, Y = 20-x = 20-10 = 10. To verify that this is a max, compute 2nd derivative and use 2nd derivative test. 2nd derivative w.r.t x = -4 < 0, so we have a max. So, you would buy 10 units of each of X and Y. b) now, x costs $2. So you now wish to maximize 2xy+1 subject to 2x+y= 20 ==> y = 20-2x so, 2xy+1 = 2x(20-2x)+1 = 40x-4x^2 + 1 taking derivative w.r.t x gives -8x+40 = 0 ==> -8x = -40 ==> 8x = 40 ==> x = 40/8 ==> x = 5. Y = 20-2x = 20-2(5) = 20-10 = 10. once agian, 2nd derivative = -8 < 0, and so we know we have found a max by 2nd derivative test. SO, you would buy 5 units of X and 10 units of Y in case (b).

Related Questions

Navigate

• Browse (All)
• Browse 1
• Subjects

---

---

Integrity-first tutoring: explanations and feedback only — we do not complete graded work. Learn more.