<p>First of all, I do not really understand how</p>
<p><span class="math-container">$$v=-c\frac{p^{(-β+1)}}{(-\beta+1)}+\frac{y^{(-\gamma+1)}}{(-\gamma+1)},$$</span></p>
<p>can be an indirect utility function since it is a function of <span class="math-container">$c$</span> and the indirect utility function by definition is a function of income and prices.</p>
<p>Nevertheless, if it really is the case that</p>
<p><span class="math-container">$$y = c + pq$$</span></p>
<p>and</p>
<p><span class="math-container">$$q=cp^{-\beta}y^\gamma$$</span></p>
<p>then it follows that</p>
<p><span class="math-container">$$y = c + pcp^{-\beta}y^\gamma,$$</span></p>
<p>hence</p>
<p><span class="math-container">$$y/(1+p^{1-\beta}y^\gamma) = c$$</span></p>
<p>and inserting this into <span class="math-container">$v$</span> you get</p>
<p><span class="math-container">$$v=-y/(1+p^{1-\beta}y^\gamma)\frac{p^{(-β+1)}}{(-\beta+1)}+\frac{y^{(-\gamma+1)}}{(-\gamma+1)},$$</span></p>
<p>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</p>
<p><span class="math-container">$$q(p,y) = - \frac{\frac{\partial v}{\partial p}}{\frac{\partial v}{\partial y}},$$</span></p>
<p>when you have <span class="math-container">$q(p,y)$</span> it then follows from budget constraint that <span class="math-container">$$c(p,y) = y - p q(p,y).$$</span></p>
<p>But it is a little unclear to me whether this is what you want.</p>

CC BY-SA 4.0
2022-02-14T19:57:41.767Z
2022-02-14T19:57:41.767Z
Jesper Hybel
<p>I have this indirect utility function:</p>
<p><span class="math-container">$$v=-c\frac{p^{(-β+1)}}{(-\beta+1)}+\frac{y^{(-\gamma+1)}}{(-\gamma+1)}$$</span></p>
<p>with constraint Y = c + pq</p>
<p>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: <a href="https://economics.stackexchange.com/questions/50229/how-do-i-get-to-this-demand-function-in-the-monocentric-city-model/50231?noredirect=1#comment80920_50231">How do I get to this demand function in the monocentric city model?</a></p>
<p>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?</p>
<p>I thought of solving for c in the resulting demand function:</p>
<p><span class="math-container">$$q={cp^{-\beta}}{y^\gamma}$$</span></p>
<p>Or</p>
<p><span class="math-container">$$ln⁡q= lnc-\beta ln⁡p+ \gamma ln⁡y$$</span></p>
<p>To get</p>
<p><span class="math-container">$$c=\frac{{p^{-\beta}}{y^\gamma}}{q}$$</span></p>
<p>And then substituting that into v and solving for q and p, but that doesn't seem right.</p>
<p>Any help with</p>
<ul>
<li>how to solve for c from the two first</li>
<li>how to get demand function for q as function of income and utility (without c)</li>
<li>how to get the direct utility function</li>
</ul>
<p>Would be greatly appreciated.</p>

<utility><definition><demand><demand-system-estimation><simultaneous-equation-model>
Is it possible to get a demand function as function of income and utility from this log linear indirect utility?
Victor Nielsen
# Is it possible to get a demand function as function of income and utility from this log linear indirect utility?

(Asked by: username_0 on 2022-02-14)

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?](https://economics.stackexchange.com/questions/50229/how-do-i-get-to-this-demand-function-in-the-monocentric-city-model/50231?noredirect=1#comment80920_50231)

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.

**Tags:** utility, definition, demand, demand-system-estimation, simultaneous-equation-model


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## 1 answers

### Answer by username_1 on 2022-02-14. Score: 4 | Accepted answer

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.


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