<p>This is the maximisation problem that authors have solved for each period <span class="math-container">$t$</span> and for every individual <span class="math-container">$i\in G_j$</span>. For simplicity of the notation, I have removed the expression <span class="math-container">$t$</span> with parenthesis - <span class="math-container">$(t)$</span> from everywhere:</p>
<p><span class="math-container">\begin{eqnarray*} \max_{(c_{i},x_{ij})\in\mathbb{R}^2_+}&amp;  q_j^\gamma c_i^{1-\gamma}x_{ij}^\gamma \\ \text{s.t } &amp; c_i+p_jx_{ij}=y_i\end{eqnarray*}</span>
which is same as solving:
<span class="math-container">\begin{eqnarray*} \max_{(c_{i},x_{ij})\in\mathbb{R}^2_+}&amp;   c_i^{1-\gamma}x_{ij}^\gamma \\ \text{s.t } &amp; c_i+p_jx_{ij}=y_i\end{eqnarray*}</span>
Solution to the above problem satisfy:</p>
<ul>
<li>MRS <span class="math-container">$=\dfrac{(1-\gamma)x_{ij}}{\gamma c_i}=\dfrac{1}{p_j}$</span> and</li>
<li><span class="math-container">$c_i+p_jx_{ij}=y_i$</span></li>
</ul>
<p>Solving them for <span class="math-container">$x_{ij}$</span>, we get the demand as:
<span class="math-container">$x_{ij}=\dfrac{\gamma y_i}{p_j}$</span>.</p>

CC BY-SA 4.0
2024-09-22T09:07:50.940Z
2024-09-22T09:14:37.487Z
2024-09-22T09:14:37.487Z
Amit
<p><a href="https://i.sstatic.net/mLsrd5jD.png" rel="nofollow noreferrer"><img src="https://i.sstatic.net/mLsrd5jD.png" alt="enter image description here" /></a></p>
<p>Hi I am fairly new to these kinds of optimization problems and I am not clear how from equation (1) the authors derive the demand function for a drug j.
I mean I think they used the Lagrangian and as a budget constrain something like y_i(t)+p_j(t) x_{ji} = income of i but I wouldd like to understand the mathematical steps to reach the result. Also is there a general rule for computing the demand starting from preferences?</p>
<p>Thank you</p>

<microeconomics><optimization><demand>
How to solve for demand?
Lusian
# How to solve for demand?

(Asked by: username_0 on 2024-09-21)

[![enter image description here](https://i.sstatic.net/mLsrd5jD.png)](https://i.sstatic.net/mLsrd5jD.png)

Hi I am fairly new to these kinds of optimization problems and I am not clear how from equation (1) the authors derive the demand function for a drug j.
I mean I think they used the Lagrangian and as a budget constrain something like y\_i(t)+p\_j(t) x\_{ji} = income of i but I wouldd like to understand the mathematical steps to reach the result. Also is there a general rule for computing the demand starting from preferences?

Thank you

**Tags:** microeconomics, optimization, demand

---

## 1 answers

### Answer by username_1 on 2024-09-22. Score: 1

This is the maximisation problem that authors have solved for each period $t$ and for every individual $i\in G\_j$. For simplicity of the notation, I have removed the expression $t$ with parenthesis - $(t)$ from everywhere:

\begin{eqnarray\*} \max\_{(c\_{i},x\_{ij})\in\mathbb{R}^2\_+}& q\_j^\gamma c\_i^{1-\gamma}x\_{ij}^\gamma \\ \text{s.t } & c\_i+p\_jx\_{ij}=y\_i\end{eqnarray\*}
which is same as solving:
\begin{eqnarray\*} \max\_{(c\_{i},x\_{ij})\in\mathbb{R}^2\_+}& c\_i^{1-\gamma}x\_{ij}^\gamma \\ \text{s.t } & c\_i+p\_jx\_{ij}=y\_i\end{eqnarray\*}
Solution to the above problem satisfy:

* MRS $=\dfrac{(1-\gamma)x\_{ij}}{\gamma c\_i}=\dfrac{1}{p\_j}$ and
* $c\_i+p\_jx\_{ij}=y\_i$

Solving them for $x\_{ij}$, we get the demand as:
$x\_{ij}=\dfrac{\gamma y\_i}{p\_j}$.


---

