A few clarifications on Utility equations and indifference curves?

Question

I have the utility equation $U(a,b) = a^{2}b^{3}$

How can I tell if the indifference curves are convex? I was under the impression that if:

$U_{a} > 0$ and $U_{b} < 0$

then the curve would be convex. In this case, those conditions don't hold, but when I try graphing the equation, the curve appears convex. What am I missing?

Also are these conditions true:

if $U_{a} > 0$ , there is not diminishing marginal utility. if < 0 , there is diminishing marginal utility, and if = 0 , constant.

Are these conditions the same for good $b$ ?

I've been given lots of conflicting info and am now just confused.

Answer 1

From a mathematical point of view, the indifference curve is an equation

$$I_{\bar u}:a^{2}b^{3} = \bar u$$

for every fixed $\bar u$. Put $b$ on the vertical axis and $a$ on the horizontal axis and write this as a function of $a$:

$$b^3 = \bar ua^{-2} \implies b = (\bar u)^{1/3} a^{-2/3}$$

This fully characterizes the indifference curve for $\bar u$. Now you have a one-dimensional function, and for it to be convex, its second derivative with respect to $a$ must be non-negative. In fact it is strictly positive (we assume goods are measured in positive quantities, and we ignore the uninteresting corner solution for $\bar u=0$ which is a single point).

Answer 2

Convexity is defined by the 5th Axiom of Consumer preferences [A5'].

Convexity is defined thus: $$ if \ x^1 \succeq x^2 $$ then $$ tx^1 + (1-t)x^2 \succeq x^1 \ \ \forall \ x \in \ [0,1] $$

This can be made to define strictly convex by changing $\succeq$ to $\succ$.

In English, this means that if you take any two points that lie along the same indifference curve (IC), than any combination $t(\cdot) +(1-t)(\cdot)$ of the two will have a larger utility.

In economics, this is the mathematical basis for our intuition of balanced preferences. Consumers prefer a balanced basket of goods to one containing "extreme" amounts of either good.

From this you derive your relationship to the MRS. The weaker form of would imply that as you move from northwest to southeast along the IC, the slope (MRS) is either constant or decreasing. Strict convexity would force the above to be strictly decreasing.

Answer 3

You need to check for a diminishing Marginal Rate of Substitution ($MRS$) in order to have convexity. The $MRS$ is defined as: $$-\frac{db}{da}=\frac{MU_a}{MU_b}$$

With $U(a,b)=a^2b^3$ you have that:

• $MU_a=\frac{\partial U(a,b)}{\partial a}=2ab^3$
• $MU_b=\frac{\partial U(a,b)}{\partial b}= 3a^2b^2$

This means that in this case:

$$MRS_{ab}=\frac{2ab^3}{3a^2b^2}=\frac{2}{3} \frac{b}{a}$$

Now you need to interpret the $MRS$, which is quite simple. Suppose that you have $a=1$ and $b=6$, then $MRS_{ab}=4$ which means that you are willing to give up four units of $b$ for one additional unit of $a$. Now suppose you have $a \to +\infty $, then your Marginal Rate of Substitution would be equal to zero. This means that you are not willing to give up an additional unit of $b$ for an additional unit of $a$. The MRS is decreasing if $a$ is increasing thus you have convexity. A diminishing Marginal Rate of Substitution tells us that the individual we are examining prefers balanced bundles of goods over unbalanced ones (5th axiom of Consumer Preference).