Is it possible to get a demand function as function of income and utility from this log linear indirect utility?

Question

I have this indirect utility function:

$$v=-c\frac{p^{(-β+1)}}{(-\beta+1)}+\frac{y^{(-\gamma+1)}}{(-\gamma+1)}$$

with constraint Y = c + pq

I have posted before about getting the utility function from it, but I was not able to get it. I only needed that as a preliminary step to get price and demand functions similar to here: How do I get to this demand function in the monocentric city model?

Maybe it's possible to get the demand directly as a function of income and utility (but not as a function of c) without having the direct utility function first?

I thought of solving for c in the resulting demand function:

$$q={cp^{-\beta}}{y^\gamma}$$

Or

$$ln⁡q= lnc-\beta ln⁡p+ \gamma ln⁡y$$

To get

$$c=\frac{{p^{-\beta}}{y^\gamma}}{q}$$

And then substituting that into v and solving for q and p, but that doesn't seem right.

Any help with

• how to solve for c from the two first
• how to get demand function for q as function of income and utility (without c)
• how to get the direct utility function

Would be greatly appreciated.

Answer 1

First of all, I do not really understand how

$$v=-c\frac{p^{(-β+1)}}{(-\beta+1)}+\frac{y^{(-\gamma+1)}}{(-\gamma+1)},$$

can be an indirect utility function since it is a function of $c$ and the indirect utility function by definition is a function of income and prices.

Nevertheless, if it really is the case that

$$y = c + pq$$

and

$$q=cp^{-\beta}y^\gamma$$

then it follows that

$$y = c + pcp^{-\beta}y^\gamma,$$

hence

$$y/(1+p^{1-\beta}y^\gamma) = c$$

and inserting this into $v$ you get

$$v=-y/(1+p^{1-\beta}y^\gamma)\frac{p^{(-β+1)}}{(-\beta+1)}+\frac{y^{(-\gamma+1)}}{(-\gamma+1)},$$

which is a function of prices and income and could be the indirect utility function. From here you can find the Marshall demand using Roy's identity

$$q(p,y) = - \frac{\frac{\partial v}{\partial p}}{\frac{\partial v}{\partial y}},$$

when you have $q(p,y)$ it then follows from budget constraint that $$c(p,y) = y - p q(p,y).$$

But it is a little unclear to me whether this is what you want.