<p>The parameters can be identified from the demand function that results from the utility maximization problem. Suppose the budget constraint is:</p>
<p><span class="math-container">$$M = P_XX + P_YY$$</span></p>
<p>where <span class="math-container">$M$</span> is the amount of money the consumer has and <span class="math-container">$P_X, P_Y$</span> are the prices of goods X and Y. Then the consumer's problem can be stated as:</p>
<p><span class="math-container">$$\max X^\alpha Y^\beta\\ s.t.\\ M = P_XX + P_YY$$</span></p>
<p>The langrangian for this problem is:</p>
<p><span class="math-container">$$\mathcal{L} = X^\alpha Y^\beta + \lambda (M - P_XX - P_YY)$$</span></p>
<p>The first order conditions are:</p>
<p><span class="math-container">$$ 
\mathcal{L}_X = \alpha X^{\alpha-1} Y^\beta - \lambda P_X\\
\mathcal{L}_Y = X^{\alpha} \beta Y^{\beta-1} - \lambda P_Y\\ 
\mathcal{L}_\lambda = M - P_XX - P_YY
$$</span></p>
<p>Setting each of these to zero and then taking the ratio of <span class="math-container">$\mathcal{L}_X$</span> to <span class="math-container">$\mathcal{L}_Y$</span> we get:</p>
<p><span class="math-container">$$
\frac{\alpha Y}{\beta X} = \frac{P_X}{P_Y}\\
Y = \frac{\beta P_X }{\alpha P_Y}X 
$$</span></p>
<p>Plugging this back into the budget constraint gives the demand function for <span class="math-container">$X$</span>, which I will denote as <span class="math-container">$X^*$</span></p>
<p><span class="math-container">$$
M =  P_XX^* + P_Y(\frac{\beta P_X }{\alpha P_Y}X)\\
M = X^*P_X(1 + \frac{\beta}{\alpha})\\
M = X^*P_X(\frac{\alpha+\beta}{\alpha})\\
\frac{M}{P_X (\frac{\alpha+\beta}{\alpha})}  = X^*\\
X^* = \frac{\alpha M}{P_X (\alpha+\beta)}
$$</span></p>
<p>Similarly, we the demand function for Y is given by: <span class="math-container">$Y^* = \frac{\beta M}{P_Y (\alpha+\beta)}$</span>.</p>
<p>Finally, to answer your question, we can estimate <span class="math-container">$\alpha$</span> and <span class="math-container">$\beta$</span> given data on: quantities, income (<span class="math-container">$M$</span>), and prices. One way to make this easier is to restrict  <span class="math-container">$\alpha+\beta = 1$</span>. In that case, the parameters are simply equal to the respective good's budget share of income:</p>
<p><span class="math-container">$$
\alpha = \frac{X^*P_X}{M}\\
\beta = \frac{Y^*P_Y}{M}
$$</span></p>
<p>Note that this restriction isn't necessary. I will leave the other case to you but basically, you will have the same number of unknowns as you do equations so they are exactly identified.</p>

CC BY-SA 4.0
2022-01-20T19:30:13.213Z
2022-01-20T19:30:13.213Z
mathguy101
<p>If you transform $X$ and $Y$ into their natural logarithms, then $a$ will be given by the coefficient on $x$ and $b$ will be given the coefficient on $y$</p>

CC BY-SA 3.0
2016-08-15T03:32:36.583Z
Thank you very much for your answer. Please see my comment above.
Marx
CC BY-SA 3.0
2016-08-15T04:56:44.720Z
In that case, I would look into principal component analysis and/or dynamic factor models.
ts_highbury
CC BY-SA 3.0
2016-08-17T08:09:50.907Z
Thank you very much for your comments. I would like to extend these problem with some specific example. Let's U = (1/a)*(CMGS^bEQ^c)‌​^a, U=utility function, C = consumption, MGS = foreign imported goods, EQ = Environmental quality, and a, b, c = parameters. b,c>0; -infinity<a<1; a(b+c)<1; a(1+b+c)<1. In this case, I would like to know how to estimate the parameter values (a, b, and c). If you have any idea, please let me know. Thank you very much in advance.
Marx
CC BY-SA 3.0
2016-08-14T08:56:42.307Z
2016-08-14T08:56:42.307Z
ts_highbury
<p>Utility function is $U = X^aY^b$.</p>

<p>It is a Cobb-Douglas function, and there are data for $X$ and $Y$.</p>

<p>I would like to know how to estimate $a$ and $b$.</p>

CC BY-SA 3.0
2016-08-14T12:17:42.190Z
What kind of data do you have exactly? If I say I ate two apples and an orange you have no idea about my utility. Yet this was data for X and Y.
Giskard
CC BY-SA 3.0
2016-08-15T03:26:42.283Z
There is no data for U and a and b must be positive. If there is data for U, then I can use the natural logarithm. But in this case, a or b might be a negative value. I would like to know how to estimate the positive values of a  and b without data for U.
Marx
CC BY-SA 3.0
2016-08-16T13:21:42.077Z
In addition to stable preferences, you need to know $P_X$ and $P_Y$ to estimate these parameters. These preferences imply a constant expenditure share of X and Y and you can solve for $a$ and $b$ using those expenditure shares, but only if you know the quantities and prices of the goods consumed.
BKay
CC BY-SA 3.0
2016-08-17T07:15:53.190Z
Thank you very much for your comments.  U = (1/a)*(C*MGS^b*EQ^c)^a
Marx
<utility>
How to estimate parameters in a utility function?
Marx
# How to estimate parameters in a utility function?

(Asked by: username_0 on 2016-08-14)

Utility function is $U = X^aY^b$.

It is a Cobb-Douglas function, and there are data for $X$ and $Y$.

I would like to know how to estimate $a$ and $b$.

**Tags:** utility
> **Comments:**
> **username_1**: What kind of data do you have exactly? If I say I ate two apples and an orange you have no idea about my utility. Yet this was data for X and Y.  
> **username_0**: There is no data for U and a and b must be positive. If there is data for U, then I can use the natural logarithm. But in this case, a or b might be a negative value. I would like to know how to estimate the positive values of a  and b without data for U.  
> **username_2**: In addition to stable preferences, you need to know $P_X$ and $P_Y$ to estimate these parameters. These preferences imply a constant expenditure share of X and Y and you can solve for $a$ and $b$ using those expenditure shares, but only if you know the quantities and prices of the goods consumed.  
> **username_0**: Thank you very much for your comments.  U = (1/a)*(C*MGS^b*EQ^c)^a  

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## 2 answers

### Answer by username_3 on 2022-01-20. Score: 2

The parameters can be identified from the demand function that results from the utility maximization problem. Suppose the budget constraint is:

$$M = P\_XX + P\_YY$$

where $M$ is the amount of money the consumer has and $P\_X, P\_Y$ are the prices of goods X and Y. Then the consumer's problem can be stated as:

$$\max X^\alpha Y^\beta\\ s.t.\\ M = P\_XX + P\_YY$$

The langrangian for this problem is:

$$\mathcal{L} = X^\alpha Y^\beta + \lambda (M - P\_XX - P\_YY)$$

The first order conditions are:

$$
\mathcal{L}\_X = \alpha X^{\alpha-1} Y^\beta - \lambda P\_X\\
\mathcal{L}\_Y = X^{\alpha} \beta Y^{\beta-1} - \lambda P\_Y\\
\mathcal{L}\_\lambda = M - P\_XX - P\_YY
$$

Setting each of these to zero and then taking the ratio of $\mathcal{L}\_X$ to $\mathcal{L}\_Y$ we get:

$$
\frac{\alpha Y}{\beta X} = \frac{P\_X}{P\_Y}\\
Y = \frac{\beta P\_X }{\alpha P\_Y}X
$$

Plugging this back into the budget constraint gives the demand function for $X$, which I will denote as $X^\*$

$$
M = P\_XX^\* + P\_Y(\frac{\beta P\_X }{\alpha P\_Y}X)\\
M = X^\*P\_X(1 + \frac{\beta}{\alpha})\\
M = X^\*P\_X(\frac{\alpha+\beta}{\alpha})\\
\frac{M}{P\_X (\frac{\alpha+\beta}{\alpha})} = X^\*\\
X^\* = \frac{\alpha M}{P\_X (\alpha+\beta)}
$$

Similarly, we the demand function for Y is given by: $Y^\* = \frac{\beta M}{P\_Y (\alpha+\beta)}$.

Finally, to answer your question, we can estimate $\alpha$ and $\beta$ given data on: quantities, income ($M$), and prices. One way to make this easier is to restrict $\alpha+\beta = 1$. In that case, the parameters are simply equal to the respective good's budget share of income:

$$
\alpha = \frac{X^\*P\_X}{M}\\
\beta = \frac{Y^\*P\_Y}{M}
$$

Note that this restriction isn't necessary. I will leave the other case to you but basically, you will have the same number of unknowns as you do equations so they are exactly identified.


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### Answer by username_4 on 2016-08-14. Score: 2

If you transform $X$ and $Y$ into their natural logarithms, then $a$ will be given by the coefficient on $x$ and $b$ will be given the coefficient on $y$

> **Comments:**
> **username_0**: Thank you very much for your answer. Please see my comment above.  
> **username_4**: In that case, I would look into principal component analysis and/or dynamic factor models.  
> **username_0**: Thank you very much for your comments. I would like to extend these problem with some specific example. Let's U = (1/a)*(CMGS^bEQ^c)‌​^a, U=utility function, C = consumption, MGS = foreign imported goods, EQ = Environmental quality, and a, b, c = parameters. b,c>0; -infinity<a<1; a(b+c)<1; a(1+b+c)<1. In this case, I would like to know how to estimate the parameter values (a, b, and c). If you have any idea, please let me know. Thank you very much in advance.  

---

