4. Say we have a utility function u(x) = rc;for a e (0, 1). (a) Construct the Ma

Question

4. Say we have a utility function u(x) = rc;for a e (0, 1). (a) Construct the Marginal rate of substitution. (b) Discuss how the Marginal rate of substitute in related to the slope of an indifference curve at a point r >> 0 (i.e., each component of x is strictly positive). 5. Answer the following: (i) Why is a utility function considered to be an "ordinal" concept? (ii) Show that if u(c) represents a consumer's preference relation, any ûn(z) = 10. u() represents that same utility function. (iii) In question 4, of show that if u(x1, x2)=xfx, for a E (0, 1), the MRS between the two goods for û(c) is not impacted by this strictly increasing transformation. (iv) Actually, show if h(Y) : R R is a strictly increasing continuously differentiable transformation, (c) = h(u(c)) represents that same preferences as u(c). (v) Show in part (iv) that the MRS at the same for both a(c) and u(x). (vi) Show in part (v) that the MRS is the same if (vii) let h(y) = lny. Show the MRS is the same for both () and u(z) if u(x1, 12)=x{ a:1-a) for a e (0, 1).

Explanation / Answer

u(x) = x1a x21-a

To calculate marginal rate of substitution , calculate marginal utilities from both both goods:

Because MRS = MUx1 / MUx2

By doing the partial derivative of utility function with respect to x1 and x2, we get the MUx1 and MUx2 respectively.

MUx1 = ax1a-1 x21-a

MUx2 = (1-a)x1a x21-a-1 = (1-a) x1a x2-a

MRS = MUx1 / MUx2

= ax1a-1 x21-a/ (1-a) x1a x2-a

= a x1a-1-a x21-a+a / (1-a)

= a x1-1 x21 / (1-a)

MRS = a x2 / (1-a) x1

(b) Marginal rate of substitution of x1 for x2 diminishes as more of good x1 is substituted for good  x2. Marginal rate of substitution is represented by (change in good x2/ change in good x1). MRS at a point on the indifference curve is equal to the slope of the indifference curve at that point. And it is possible only when the good value is strictly positive . Otherwise MRS will not be equal to slope of the IC.

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