# Utility function and homogenous of degree zero

I've a utility function which is given by

($$x_i$$-$$b_i$$)$$^{c_i}$$ $$\sqrt{x_2}$$

. What values of b and c can I input to ensure Homogenous of degree zero in prices and wealth? I think c will be positive. However, I have never done these types of questions before. What are the steps that will guide me to solve questions like this?

• You should probably begin to obtain the expression of the indirect utility function – Bertrand May 15 at 15:35

Consider the utility maximisation problem: $$\max_{x_1, x_2} u(x_1, x_2) \text{ s.t. } p_1 x_1 + p_2 x_2 \le m.$$ If we multiply all prices and income by the same number $$t > 0$$, we obtain the problem: $$\max_{x_1, x_2} u(x_1, x_2) \text{ s.t.} t p_1 x_1 + t p_2 x_2 \le t m$$ However if we divide the left and right hand side of the constraint by $$t$$ we get the original problem back.
Summarizing, all demand functions are homogeneous of degree zero whatever the utility function because multiplying all prices and income by the same positive number does not change the problem: everything is $$t$$-times more expensive but you also have $$t$$-times more income.
• @Shane Murply. Yes that's one option. Alternatively, you could recognize your utility function of being of the Stone-Geary type. This one has closed form solution $x_1 = b_1 + \frac{c_1}{1/2 + c_1}\frac{y}{p_1}$ and $x_2 = \frac{1/2}{c_1 + 1/2}\frac{y}{p_2}$. – tdm May 16 at 8:52