Question

Consider the utility function 𝑈(𝑥, 𝑦) = 𝑥𝑦 + 𝑥, with 𝑀𝑈𝑥 = 𝑦 + 1 and 𝑀𝑈𝑦 = 𝑥. (Verify these.) 

 

a. is the assumption of “more is better” (non-satiation) satisfied for both goods x and y? 

b. does the marginal utility of 𝑥 diminish, remain constant, or increase as the consumer buys more 𝑥? Explain. 

c. what is 𝑀𝑅𝑆𝑥,𝑦? 

d. is 𝑀𝑅𝑆𝑥,𝑦 diminishing, constant, or increasing as the consumer substitutes 𝑥 for 𝑦 along an indifference curve? 

e. on a graph with 𝑥 on the horizontal axis and 𝑦 on the vertical axis, draw a typical indifference curve (not necessarily exactly to scale, but should reflect whether 𝑀𝑅𝑆𝑥,𝑦 is diminishing or not). Label the curve as 𝑈1. 

f. on the same graph, draw a second indifference curve 𝑈2, where 𝑈2 > 𝑈1

Answer 1

Given the utility function $U(x, y)=xy+x$

$MU_x=\frac{dU}{dx}=\frac{d}{dx}(xy+x)=y+1\\ MU_x=\frac{dU}{dy}=\frac{d}{dy}(xy+x)=x\\ Hence \space verified.$

a)

 Since the utility function is increasing in both goods X and Y, hence it can be concluded that the assumption "more is better" is satisfied for both goods X and Y.

b)

$MU_x=y+1$, is constant with respect to x.

Hence the marginal utility of x remain constant as the consumer buys more x.

c)

$MRS_{x,y}=\frac{\frac{dU}{dx}}{\frac{dU}{dy}}\\\space \\ MRS_{x,y}=\frac{y+1}{x}$

d)

the MRS is decreasing as consumer substitutes X for Y

e)

f)