<p>Let's assume our agent purchases positive quantities of $x$ and $y$. The utility maximization problem (without constraints on purchasing non-negative quantities) is:</p>

<p>\begin{equation}
 \begin{array}{*2{&gt;{\displaystyle}r}}
 \mbox{maximize (over $x, y$)} &amp; u(x,y) \\
 \mbox{subject to} &amp; p_x x + p_y y \leq w
 \end{array}
\end{equation}</p>

<p>The Lagrangian is:</p>

<p>$$ \mathcal{L}(x, y, \lambda) = u(x,y) - \lambda(p_x x + p_yy - w)$$</p>

<p>Assuming $u$ is concave and differentiable, the problem is a convex optimization problem (in the jargon of optimization problems), and Slater's condition holds, hence the first order conditions are necessary and sufficient for an optimum. The first order conditions are:
$$ \frac{\partial \mathcal{L}}{ \partial x} = \frac{\partial u}{\partial x} - \lambda p_x  \quad \quad  \frac{\partial \mathcal{L}}{ \partial y} = \frac{\partial u}{\partial y} - \lambda p_x = 0 $$</p>

<p>Combining both equations (taking out lambda) we have:</p>

<p>$$\frac{\partial u / \partial x}{p_x} = \frac{\partial u / \partial y}{p_y} $$</p>

<p>This gives you the classic microeconomic theory result that <a href="https://en.wikipedia.org/wiki/Marginal_rate_of_substitution" rel="nofollow noreferrer">the marginal rate of substitution is the ratio of prices</a>. From here can you answer the question?</p>

<p>(Note: as you may have noticed, I slipped into your question a few more assumptions: eg. an assumption that the agent actually buys positive quantities of Option  A and Option B.)</p>

CC BY-SA 3.0
2017-11-28T16:11:05.663Z
2017-11-28T16:11:05.663Z
Matthew Gunn
<p>Imagine a scenario where an agent is faced with two purchasing options, each with an individual utility function. Let us say option $A$ offers $m$ units of $P$ per £, and option $B$ offers $n$ units of $Q$ per pound. Additionally, utility is linear with respect to $P$, but not so for $Q$. How do I find out when the marginal utility of $A$ is greater than $B$?</p>

<utility>
How to Calculate Marginal Utility
Rory
# How to Calculate Marginal Utility

(Asked by: username_0 on 2017-11-28)

Imagine a scenario where an agent is faced with two purchasing options, each with an individual utility function. Let us say option $A$ offers $m$ units of $P$ per £, and option $B$ offers $n$ units of $Q$ per pound. Additionally, utility is linear with respect to $P$, but not so for $Q$. How do I find out when the marginal utility of $A$ is greater than $B$?

**Tags:** utility

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## 1 answers

### Answer by username_1 on 2017-11-28. Score: 3

Let's assume our agent purchases positive quantities of $x$ and $y$. The utility maximization problem (without constraints on purchasing non-negative quantities) is:

\begin{equation}
\begin{array}{\*2{>{\displaystyle}r}}
\mbox{maximize (over $x, y$)} & u(x,y) \\
\mbox{subject to} & p\_x x + p\_y y \leq w
\end{array}
\end{equation}

The Lagrangian is:

$$ \mathcal{L}(x, y, \lambda) = u(x,y) - \lambda(p\_x x + p\_yy - w)$$

Assuming $u$ is concave and differentiable, the problem is a convex optimization problem (in the jargon of optimization problems), and Slater's condition holds, hence the first order conditions are necessary and sufficient for an optimum. The first order conditions are:
$$ \frac{\partial \mathcal{L}}{ \partial x} = \frac{\partial u}{\partial x} - \lambda p\_x \quad \quad \frac{\partial \mathcal{L}}{ \partial y} = \frac{\partial u}{\partial y} - \lambda p\_x = 0 $$

Combining both equations (taking out lambda) we have:

$$\frac{\partial u / \partial x}{p\_x} = \frac{\partial u / \partial y}{p\_y} $$

This gives you the classic microeconomic theory result that [the marginal rate of substitution is the ratio of prices](https://en.wikipedia.org/wiki/Marginal_rate_of_substitution). From here can you answer the question?

(Note: as you may have noticed, I slipped into your question a few more assumptions: eg. an assumption that the agent actually buys positive quantities of Option A and Option B.)


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