<p>c is composite good with normalised price, q is good with price p. y is income.</p>
<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>And this Marshallian demand function:</p>
<p><span class="math-container">$$q={cp^{-\beta}}{y^\gamma}$$</span></p>
<p>Or</p>
<p><span class="math-container">$$ln⁡q= \alpha-\beta ln⁡p+ \gamma ln⁡y$$</span></p>
<p>Where alpha is ln c</p>
<p>I need the utility function from these. How do I get there?</p>

CC BY-SA 4.0
2022-02-10T13:25:12.617Z
What can you say about the relationship between direct and indirect utility functions?
Bertrand
CC BY-SA 4.0
2022-02-10T14:55:20.453Z
Maybe you can get inspiration here: https://economics.stackexchange.com/questions/16744/how-to-derive-utility-function-from-indirect-utility-in-this-exercise.
Jesper Hybel
CC BY-SA 4.0
2022-02-10T20:42:56.713Z
@Bertrand. When price and income are both positive, the indirect utility function is increasing in income and decreasing in the price.
Victor Nielsen
CC BY-SA 4.0
2022-02-11T10:05:45.800Z
@Victor Nielsen: are you sure that there is no error in the expression of $v$ or $q$ ?
Bertrand
CC BY-SA 4.0
2022-02-11T10:27:43.343Z
@Betrand This is what my professor has written. If there is an error, I should be able to check it if I had the direct utility function.
Victor Nielsen
CC BY-SA 4.0
2022-02-14T12:55:31.047Z
@JesperHybel I tried to follow you advice by looking at that thread. I think where I get stuck trying to apply their steps to mine is that c is included in my indirect utility function and has normalised price and price for good q is p.
Victor Nielsen
CC BY-SA 4.0
2022-02-14T15:58:28.750Z
@VictorNielsen ... ok I have not tried to solve it. So cannot really help you.
Jesper Hybel
CC BY-SA 4.0
2022-02-15T09:05:22.623Z
@JesperHybel No problem! Your help is appreciated regardless.
Victor Nielsen
<utility><optimization><demand>
How to derive utility function from indirect utility and Marshallian demand?
Victor Nielsen
# How to derive utility function from indirect utility and Marshallian demand?

(Asked by: username_0 on 2022-02-10)

c is composite good with normalised price, q is good with price p. y is income.

I have this indirect utility function:

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

And this Marshallian demand function:

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

Or

$$ln⁡q= \alpha-\beta ln⁡p+ \gamma ln⁡y$$

Where alpha is ln c

I need the utility function from these. How do I get there?

**Tags:** utility, optimization, demand
> **Comments:**
> **username_1**: What can you say about the relationship between direct and indirect utility functions?  
> **username_2**: Maybe you can get inspiration here: https://economics.stackexchange.com/questions/16744/how-to-derive-utility-function-from-indirect-utility-in-this-exercise.  
> **username_0**: @Bertrand. When price and income are both positive, the indirect utility function is increasing in income and decreasing in the price.  
> **username_1**: @Victor Nielsen: are you sure that there is no error in the expression of $v$ or $q$ ?  
> **username_0**: @Betrand This is what my professor has written. If there is an error, I should be able to check it if I had the direct utility function.  
> **username_0**: @JesperHybel I tried to follow you advice by looking at that thread. I think where I get stuck trying to apply their steps to mine is that c is included in my indirect utility function and has normalised price and price for good q is p.  
> **username_2**: @VictorNielsen ... ok I have not tried to solve it. So cannot really help you.  
> **username_0**: @JesperHybel No problem! Your help is appreciated regardless.  

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## 0 answers

