Answer to Question #273380 in Microeconomics for Cooper

Question

 Robinson's preferences between apples (a) and bananas (b) are expressed by the following:

 

U = (a+2)^0.5(b+1)^0.5

 

(a) Show that Robinson's indifference curves are negatively sloped.

(b) Are they convex to the origin? Explain.

Answer 1

Robinson's interference curves

a.) From U = U = (a + 2)^0.5 ∗(b+ 1)^0.5

Totally differentiating the utility function

$du=\frac{\delta}{\delta a}[(a+2)^{0.5}(b+1)^{0.5}da]+\frac{\delta}{\delta b}[(a+2)^{0.5}(b+1)^{0.5}db]$

$du=0.5(a+2)^{-0.5}(b+1)^{0.5}da+0.5(a+2)^{0.5}(b+1)^{-0.5}db$

along an indifference curve change in utility du=0

Also we may apply (a) along horizontal axis hence,

$0.5(a+2)^{-0.5}(b+1)^{0.5}da+0.5(a+2)^{0.5}(b+1)^{-0.5}db=0$

$\frac{da}{db}=\frac{0.5(a+2)^{0.5}(b+1)^{-0.5}}{0.5(a+2)^{-0.5}(b+1)^{0.5}}=\frac{a+2}{b+1}$

$\frac{da}{db}= slope \ of\ indifference$

$=MRS_{a,b }$ [Nominal rate of substitution between a and b]

As, $MRS_{a,b } \lt0$ implies robinsons indifference curve is negatively sloped

b.) Now $|MRS_{a,b }|=\frac{a+2}{b+1}=\frac{a}{b+1}+\frac{2}{b+1}$

Now $\frac{\delta |MRS_{a,b }|}{\delta a}=\frac{1}{b+1}\gt0 \ as\ b\gt0$

hence they are convex to the origin