Consumer Utility Maximisation under Non-Convex Preferences

Questions & Problems
1

Suppose a consumer has preferences represented by the utility function

u(x,y)=e^{ax^2}e^{by^2}

where x,y\geq 0 and a,b>0. Let prices p_x >0, p_y>0 and income M\geq 0 be given.
Solve the consumer’s utility-maximisation problem:

\begin{align} \max_{(x,y)\in\mathbb{R}^2_+} & e^{ax^2}e^{by^2} \\ \text{s.t.} & \quad p_xx+p_yy\leq M\end{align}
2

Maximize:
u(x,y) = e^{ax^2} e^{by^2}
\text{subject to } p_x x + p_y y \le M, \text{ where } a, b > 0.

Simplyfying the equation:
e^A e^B = e^{A+B}.

Therefore,

u(x,y) = e^{ax^2 + by^2}.

Here we will use monotonic tansformation , as we know exponential function is always increasing,

\therefore \text{ maximizing } e^{ax^2+by^2} \iff \text{ maximizing } ax^2+by^2

Now,
Maximize:
u(x,y) = ax^2+ by^2
\text{subject to } p_x x + p_y y \le M, \text{ where } a, b > 0.

Marginal Utilites:
\text{MU}_x = 2ax
\text{MU}_y = 2by

\text{MRS} = \frac{ax}{by}

As we can see that it’s a circle eqauation and they are convex in nature which means that the maximum utility will be at corners points not on the tangency points.

so,
Spend all income on x.

Then

x = \frac{M}{p_x}, y = 0.

Utility:

U_x = {a\left(\frac{M}{p_x}\right)^2}.

Spend all income on y.

Then

x = 0, y = \frac{M}{p_y}.

Utility:

U_x = {b\left(\frac{M}{p_y}\right)^2}.

Now comparing:

a\left(\frac{M}{p_x}\right)^2 = b\left(\frac{M}{p_y}\right)^2.

Which gives

\frac{a}{p_x^2} and \frac{b}{p_y^2}.

So this leads us to the Marshallian Demand:

(x^d, y^d)(Px,Py,M) = \begin{cases} \left(\frac{M}{p_x}, 0\right), & \text{if } \frac{a}{p_x^2} > \frac{b}{p_y^2} \\ \left(0, \frac{M}{p_y}\right), & \text{if } \frac{a}{p_x^2} < \frac{b}{p_y^2} \\ \left\{\left(\frac{M}{p_x}, 0\right), \left(0, \frac{M}{p_y}\right)\right\}, & \text{if } \frac{a}{p_x^2} = \frac{b}{p_y^2} \end{cases}

4

The consumer’s problem is

\max_{x,y \geq 0} \; u(x,y)=e^{ax^2}e^{by^2}

subject to

p_xx+p_yy\leq M

where a,b,p_x,p_y>0

Using the property of exponents,

e^{A}e^{B}=e^{A+B}

we obtain

u(x,y)=e^{ax^2+by^2}

Since the exponential function is strictly increasing, maximizing

e^{ax^2+by^2}

is equivalent to maximizing

ax^2+by^2

∵ the problem becomes

\max_{x,y \geq 0} \; ax^2+by^2

subject to

p_xx+p_yy\leq M

The marginal utilities are

MU_x=2ax,\qquad MU_y=2by

Since

MU_x>0 \quad\text{and}\quad MU_y>0

utility is increasing in both goods. ∴ the budget constraint binds:

p_xx+p_yy=M

The indifference curves

ax^2+by^2=c

The indifference curves are ellipses and the objective function is strictly convex. ∵ the budget set is convex, the maximum is attained at an extreme (corner) point of the budget set

Case 1: Spend all income on good x

x=\frac{M}{p_x}, \qquad y=0

Utility at this bundle is

V_x=a\left(\frac{M}{p_x}\right)^2

Case 2: Spend all income on good y

x=0, \qquad y=\frac{M}{p_y}

Utility at this bundle is

V_y=b\left(\frac{M}{p_y}\right)^2

Comparing the transformed utilities,

\frac{a}{p_x^2} \quad\text{and}\quad \frac{b}{p_y^2}

the Marshallian demand is

(x^d,y^d)= \begin{cases} \left(\dfrac{M}{p_x},\,0\right), & \text{if } \dfrac{a}{p_x^2}>\dfrac{b}{p_y^2} \\[1.2em] \left(0,\,\dfrac{M}{p_y}\right), & \text{if } \dfrac{a}{p_x^2}<\dfrac{b}{p_y^2} \\[1.2em] \left\{ \left(\dfrac{M}{p_x},\,0\right), \left(0,\,\dfrac{M}{p_y}\right) \right\}, & \text{if } \dfrac{a}{p_x^2}=\dfrac{b}{p_y^2} \end{cases}