Question

Question: Suppose your utility function from a given bundle of goods, (x.y), isWex.y) (y) (a) Suppose y 1. Make a table showing your utility for x 1, x-2x-3' and x (b) Is utility increasing from more units of x? (c) Is marginal utility increasing from more units of x? (d) Derive the equation describing the marginal utility for x, Mu (e) Plot a representative indifference curve that includes 3 bundles of goods of your choosing. Include on your graph the amount of utility obtained (0 What is the marginal rate of substitution, MRSay, at each of the three points? Interpret it. (g) Why does the marginal rate of substitution decline as you get more x? (h) Given this utility function, why do indifference curves have to slope downwards? MacBook Air FS Fo F7 F8 F9 5 0 0

Answer 1

Table 1

x	U(x)	MUx
1	1
2	1.414214	0.414214
3	1.732051	0.317837
4	2	0.267949

a) The utility function is: u = (xy)^0.5 At y = 1: u = x^0.5 Table-1 gives utility for x = 1, 2, 3, 4 b) From the table it is evident that u(x) increases with x. c) The Mu_x is change in u(x) due to one unit change in x. Thus: Mu_x = 2 = u(x = 2) - u(x = 1) and so on. From this table it is evident that Mu_x decreases with more x. d) The marginal utility of x is: Mu_x = partial differential u/partial differential x therefore Mu_x = 0.5x^0.5 - 1 = 0.5/x^0.5 e) The utility function for u = 4 is given as u = (xy)^0.5 = 4 therefore xy = 16 or, yx = 16/x Take three values of x: x = 1, 2, 4 Resulting y's are: y = 16, 8, 4 Then three point on 1c depicting u = 4 is {1, 16}, {2, 8}: {4, 4} points (A, B, C) \r\n f) The marginal rate of substitution at each point

Figure1

f)

The marginal rate of substitution at each point on the indifference curve is given as

$\small MRS_{x,y}=\frac{MU_x}{MU_y}=\frac{0.5\left ( \frac{y}{x} \right )^{0.5}}{0.5\left ( \frac{x}{y} \right )^{0.5}}=\frac{y}{x}$

This implies that at any point on the indifference curve the consumer is willing to substitute (y/x) unit of y for 1 unit of x. The MRS at each point of IC is

A	16
B	4
C	1

g)

At each point, the MRS declines the consumer move along the IC. This is because due to the diminishing marginal utility the utility of good increases with a decreasing rate. That is as the consumer substitute one good for the constant amount of other good, the consumer has to substitute the lesser and lesser amount of other good to keep utility constant. At the point, A consumer has 1 unit of x and 16 unit of y. The utility from x is much greater than from y. The consumer is willing to substitute 16 units of y for 1 unit of x. As the consumer moved further down along the IC, at point B the consumer is willing to substitute 4 units of y for 1 unit of x. Because to keep utility constant the consumer now need less amount of y for the same amount of x.

h)

The indifference curve for this utility function is downward sloping due to a diminishing marginal rate of substitution. To keep utility constant the consumer need to substitute one good for another.