Marginal Utilities and the MRS: Find the marginal utilty of x1, the marginal uti
ID: 1189059 • Letter: M
Question
Marginal Utilities and the MRS: Find the marginal utilty of x1, the marginal utilty of x2, and the marginal rate of substitution for the following utilty functions: can you please show the work to get the answer
1. U(x1,x2) = 2x1 + 3x2
2. U(x1,x2) = Ax1 + Bx2
3. U(x1,x2) = 2x1 + x2
4. U(x1,x2) = x1x2
5. U(x1,x2) = (x1^a)(x2^b)
6. U(x1,x2) = (x1+2)+(x2+1)
Monotonic Transformations: Assume preferences for x and y are given by some utility function: U(x,y), and that f(U) is a transformation for U(x,y). Show whether or not f(U) is a valid monotonic transformation of U(x,y), can you please show the work to get the answer
1. f(U) = u + 50
2. f(U) = u - 1000
3. f(U) = -u
4. f(U) = u^2
5. f(U) = 1/u^2
6. f(U) = -(1/u)
Explanation / Answer
Working note:
Marginal Rate of Substitution (MRS) = MUx1 / MUx2
(1) U = 2x1 + 3x2
MUx1 = dU / dx1 = 2
MUx2 = dU / dx2 = 3
MRS = 2 / 3 [This is an utility function for perfect substitutes]
(2) Ax1 + Bx2
MUx1 = dU / dx1 = A
MUx2 = dU / dx2 = B
MRS = A / B [This is a generalized utility function for perfect substitutes]
(3) U = 2x1 + x2 = 2x10.5 + x2
MUx1 = dU / dx1 = (2) (0.5) x1 - 0.5
MUx2 = dU / dx2 = 1
MRS = x1 - 0.5 / 1 = x1 - 0.5
(4) U = x1 x2
MUx1 = dU / dx1 = x2
MUx2 = dU / dx2 = x1
MRS = x2 / x1
(5) U = x1a x2b
MUx1 = dU / dx1 = a (x1)a - 1x2b
MUx2 = dU / dx2 = b. x1a(x2)b - 1
MRS = (a / b). (x2 / x1)
(6) U = (x1 + 2) + (x2 + 1) = x1 + x2 + 3
MUx1 = dU / dx1 = 1
MUx2 = dU / dx2 = 1
MRS = 1/1 = 1
NOTE: Out of 2 multi-part questions, the first question is answered in full.