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}

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