I have this indirect utility function:


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:



$$ln⁡q= lnc-\beta ln⁡p+ \gamma ln⁡y$$

To get


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.


1 Answer 1


First of all, I do not really understand how


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

$$y = c + pq$$



then it follows that

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


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

and inserting this into $v$ you get


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.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.