<p>You needn't use the Lagrange here. And in general, this will be true for additively separable, linear utility functions. </p>

<p>Note that the marginal rate of substitution between any two goods $i,j$ is a utility function of the form $u(x_1,x_2,...,x_n)=a_1x_1+...+a_nx_n$ will be constant at $MRS_{i,j}=\frac{a_i}{a_j}$</p>

<p>A consumer will buy none of a good $x_i$ whenever $\frac{u_i}{p_i}$ is below the maximum such ratio of the $n$ goods. A consumer will exhaust her budget on whichever good has the maximum such ratio. And if all such ratios are equal, the consumer is indifferent between any consumption bundle that exhuats wealth $w$. </p>

<p>For your case of two goods, indifference curves are linear. For more than two goods, they are hyperplanes. </p>

CC BY-SA 3.0
2017-04-05T21:16:43.343Z
This assumes B is a constant and not a function.
123
CC BY-SA 3.0
2017-04-05T21:00:54.963Z
2017-04-05T21:00:54.963Z
123
<p>I'm trying to solve a utility maximization problem through the Lagrange method. 
The utility function is something like $u(x,y)=x+B(y-a)$.</p>

<p>However, I'm running into problems as x and y do not come up in the First Order Conditions. I was wondering if anyone had an intuitive explanation as for why, and how to get around it?</p>

CC BY-SA 3.0
2017-04-05T07:05:10.353Z
A suggestion: chose some arbitrary values for $B$ and $a$ and draw the indifference curves (i.e. find some bundles that all gives the consumer the same utility). Then draw a budget line and look where the optimal bundle is. This should help you see why the Lagrangian method isn't working.
Ubiquitous
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2017-04-05T11:58:41.470Z
Is `B` a function or a constant? What are the constraints?
Klas Lindbäck
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2017-04-05T12:01:06.667Z
When `x` and `y` don't appear in the differential then it is easy to find lambda. See this example: https://en.wikipedia.org/wiki/Lagrange_multiplier#Example_1
Klas Lindbäck
CC BY-SA 3.0
2017-04-05T12:05:58.597Z
If `x` or `y` don't appear in the differential of the utility function then the partial derivatives are constant which means `u(x,y)` forms a plane.
Klas Lindbäck
CC BY-SA 3.0
2017-04-06T08:41:23.683Z
B is a constant! x>0, y>0, B>0, a>0
Elizabeth 
<microeconomics><utility><choice-theory>
Troubleshooting Utility Maximization with the Lagrange Method
Elizabeth 
# Troubleshooting Utility Maximization with the Lagrange Method

(Asked by: username_0 on 2017-04-05)

I'm trying to solve a utility maximization problem through the Lagrange method.
The utility function is something like $u(x,y)=x+B(y-a)$.

However, I'm running into problems as x and y do not come up in the First Order Conditions. I was wondering if anyone had an intuitive explanation as for why, and how to get around it?

**Tags:** microeconomics, utility, choice-theory
> **Comments:**
> **username_1**: A suggestion: chose some arbitrary values for $B$ and $a$ and draw the indifference curves (i.e. find some bundles that all gives the consumer the same utility). Then draw a budget line and look where the optimal bundle is. This should help you see why the Lagrangian method isn't working.  
> **username_2**: Is `B` a function or a constant? What are the constraints?  
> **username_2**: When `x` and `y` don't appear in the differential then it is easy to find lambda. See this example: https://en.wikipedia.org/wiki/Lagrange_multiplier#Example_1  
> **username_2**: If `x` or `y` don't appear in the differential of the utility function then the partial derivatives are constant which means `u(x,y)` forms a plane.  
> **username_0**: B is a constant! x>0, y>0, B>0, a>0  

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## 1 answers

### Answer by username_3 on 2017-04-05. Score: 1 | Accepted answer

You needn't use the Lagrange here. And in general, this will be true for additively separable, linear utility functions.

Note that the marginal rate of substitution between any two goods $i,j$ is a utility function of the form $u(x\_1,x\_2,...,x\_n)=a\_1x\_1+...+a\_nx\_n$ will be constant at $MRS\_{i,j}=\frac{a\_i}{a\_j}$

A consumer will buy none of a good $x\_i$ whenever $\frac{u\_i}{p\_i}$ is below the maximum such ratio of the $n$ goods. A consumer will exhaust her budget on whichever good has the maximum such ratio. And if all such ratios are equal, the consumer is indifferent between any consumption bundle that exhuats wealth $w$.

For your case of two goods, indifference curves are linear. For more than two goods, they are hyperplanes.

> **Comments:**
> **username_3**: This assumes B is a constant and not a function.  

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