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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?

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  • $\begingroup$ You should probably begin to obtain the expression of the indirect utility function $\endgroup$
    – Bertrand
    May 15, 2021 at 15:35

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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.

In other words, the two problems are identical. This also means that the two problems will also give exactly the same solution.

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.

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  • $\begingroup$ Thanks, I understood what you wrote. I've a follow-up question, in the case of the question, how do I solve the maximization problem, should I merely solve just the way the function is defined, without essentially having any values for b and c? $\endgroup$
    – user31619
    May 15, 2021 at 22:47
  • $\begingroup$ @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 $x_2 = \frac{1/2}{c_1 + 1/2}\frac{y}{p_2}$. $\endgroup$
    – tdm
    May 16, 2021 at 8:52
  • $\begingroup$ Thanks. Although my answer is different compared to yours, but given I want to learn the concept mostly, I would rather overlook that and emphasize more on the knowledge of the concept. To move forward, given these possible solution for x1 and x2, what is your opinion for the simplest values for b1 and c1? My guess would be 1, and 1/2 respectively? $\endgroup$
    – user31619
    May 17, 2021 at 2:30

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