This is a utility maximzation problem
maximize $x^a + y^b$ subject to $p_1x+p_2y=w$ (utility maximization problem)
Anyone has any idea, there are no restrictions on $a$ and $b$, as far as i can see it. many thanks!!!
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what is $p_1$,$p_2$? – dato datuashvili Dec 08 '13 at 17:15
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constants, prices, strictly positive. – Alex Dec 08 '13 at 17:16
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I am surprized that there is no restrictions on "x" and "y". AQre there illimited resources ? – Claude Leibovici Dec 08 '13 at 17:37
2 Answers
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You could use Lagrange multipliers but I think that the problem could be easy if you extract "y" from the constraint and plug it in the objective function which becomes only dependent on "x". Now, serach for the maximum.
Claude Leibovici
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When consumers maximize utility $u(x,y)=x^a+y^b$ with respect to a budget constraint $p_1x+p_2y=w$, the indifference curve is tangent to the budget line, therefore: $$ MRS_{xy}=p_1/p_2 $$ with $MRS_{xy}=\frac{u_x}{x_y}$ and $$\begin{align} u_x&=\frac{\partial u(x,y)}{\partial x}=ax^{a-1}\\ u_y&=\frac{\partial u(x,y)}{\partial y}=by^{b-1}\\ \end{align}$$ that is $$ \frac{p_1}{p_2}=\frac{ax^{a-1}}{by^{b-1}} $$ So the optimal $\hat x$ and $\hat y$ are solution of $$ \begin{align} w&=p_1\hat{x}+p_2\left(\frac{p_2}{p_1}\frac{a}{b}\hat{x}^{a-1}\right)^{\frac{1}{b-1}}\\ w&=p_1\left(\frac{p_1}{p_2}\frac{b}{a}\hat{y}^{b-1}\right)^{\frac{1}{a-1}}+p_2\hat{y}\\ \end{align} $$
alexjo
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that's about as far as I got. From there I am stuck, I am simply not able to derive a reduced form solution for neither x nor y. – Alex Dec 08 '13 at 17:44
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the solution cannot be solved in elementary form...unless you give other conditions on $a$ and $b$... – alexjo Dec 08 '13 at 17:57
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I understand. So is there a reduced form solution for a,b both element of (0,1).? – Alex Dec 08 '13 at 18:54