<p>Your mistake probably occurred at the final substitution stage. Optimal utility should be <span class="math-container">$2\sqrt{x^*y^*}$</span>, which is not the same as <span class="math-container">$2\sqrt{x^*y_i}$</span>.</p>

<p>In general, maximizing a two-good Cobb-Douglas utility function <span class="math-container">$u(x,y)=Ax^\alpha y^\beta$</span> subject to budget constraint <span class="math-container">$p_xx+p_yy=m$</span> has general solution 
<span class="math-container">\begin{equation}
x^*=\frac{\alpha}{\alpha+\beta}\frac{m}{p_1}
\quad\text{and}\quad 
y^*=\frac{\beta}{\alpha+\beta}\frac{m}{p_2}.
\end{equation}</span></p>

<p>In your case, the parameters <span class="math-container">$A,\alpha,\beta,p_x,p_y,m$</span> take the following values: 
<span class="math-container">\begin{equation}
A=2,\quad\alpha=\beta=\frac12,\quad p_x=1,\quad p_y=p,\quad m=1+py_i.
\end{equation}</span>
Plugging in the values, you get optimal consumption:
<span class="math-container">\begin{equation}
x^*=\frac{1+py_i}{2},\quad y^*=\frac{1+py_i}{2p}.
\end{equation}</span>
Hence optimal utility is 
<span class="math-container">\begin{equation}
u(x^*,y^*)=2\sqrt{\frac{1+py_i}{2}\cdot \frac{1+py_i}{2p}}=\frac{1+py_i}{\sqrt p}
\end{equation}</span></p>

CC BY-SA 4.0
2018-11-17T01:54:05.593Z
2018-11-17T01:54:05.593Z
Herr K.
<p>There are N agents living in an economy with two goods, <span class="math-container">$X$</span> and <span class="math-container">$Y$</span>. Their preferences are described by the following utility function <span class="math-container">$u(X,Y) = 2 \sqrt{XY}$</span>. Each agent is endowed with 1 unit of <span class="math-container">$X$</span> and <span class="math-container">$y_{i}$</span> units of <span class="math-container">$Y$</span>. Each unit of <span class="math-container">$Y$</span> is sells for <span class="math-container">$p$</span> units of <span class="math-container">$X$</span>. </p>

<p>The question is to show that each agent gets a utility of <span class="math-container">$\frac{1 + py_{i}}{\sqrt p}$</span>.</p>

<p>Here is what I have tried:</p>

<p>The agent chooses <span class="math-container">$X,Y$</span> in order to maximise <span class="math-container">$2\sqrt{XY}$</span> subject to <span class="math-container">$X + pY = 1 + p y_{i}$</span> So I have expressed the resource constraint in units of good <span class="math-container">$X$</span>. The LHS is what the agent can buy and the RHS is the endowment, also expressed in units of <span class="math-container">$X$</span>.</p>

<p>I then re-arrange the constraint in terms of <span class="math-container">$Y$</span>, i.e, <span class="math-container">$Y = \frac{1}{p} + y_{i} - \frac{X}{p}$</span> and substitute this in to the utility function, with the FOC with respect to <span class="math-container">$X$</span> and then substitute the optimal value of <span class="math-container">$X$</span> back in to the utility function to get an expression in terms of <span class="math-container">$y_{i}$</span>. However, the expression is not what I am supposed to get.</p>

<p>If anyone wants to check my algebra, here it is:</p>

<p>The utility function becomes <span class="math-container">$2\sqrt {(X/p) + Xy_{i} - X^2/p}$</span> Differentiating this w.r.t <span class="math-container">$X$</span> I find that <span class="math-container">$X^* = (1/2)(1 + y_{i}p)$</span>. I substitute this in the utility function, <span class="math-container">$2\sqrt{X^*y_{i}}$</span> but this does not equal <span class="math-container">$\frac{1 + py_{i}}{\sqrt p}$</span>.</p>

<p>Any ideas on where I am going wrong?</p>

<microeconomics><competitive-equilibrium>
Finding individual utility
MiltonFried
# Finding individual utility

(Asked by: username_0 on 2018-11-16)

There are N agents living in an economy with two goods, $X$ and $Y$. Their preferences are described by the following utility function $u(X,Y) = 2 \sqrt{XY}$. Each agent is endowed with 1 unit of $X$ and $y\_{i}$ units of $Y$. Each unit of $Y$ is sells for $p$ units of $X$.

The question is to show that each agent gets a utility of $\frac{1 + py\_{i}}{\sqrt p}$.

Here is what I have tried:

The agent chooses $X,Y$ in order to maximise $2\sqrt{XY}$ subject to $X + pY = 1 + p y\_{i}$ So I have expressed the resource constraint in units of good $X$. The LHS is what the agent can buy and the RHS is the endowment, also expressed in units of $X$.

I then re-arrange the constraint in terms of $Y$, i.e, $Y = \frac{1}{p} + y\_{i} - \frac{X}{p}$ and substitute this in to the utility function, with the FOC with respect to $X$ and then substitute the optimal value of $X$ back in to the utility function to get an expression in terms of $y\_{i}$. However, the expression is not what I am supposed to get.

If anyone wants to check my algebra, here it is:

The utility function becomes $2\sqrt {(X/p) + Xy\_{i} - X^2/p}$ Differentiating this w.r.t $X$ I find that $X^\* = (1/2)(1 + y\_{i}p)$. I substitute this in the utility function, $2\sqrt{X^\*y\_{i}}$ but this does not equal $\frac{1 + py\_{i}}{\sqrt p}$.

Any ideas on where I am going wrong?

**Tags:** microeconomics, competitive-equilibrium

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## 1 answers

### Answer by username_1 on 2018-11-17. Score: 2

Your mistake probably occurred at the final substitution stage. Optimal utility should be $2\sqrt{x^\*y^\*}$, which is not the same as $2\sqrt{x^\*y\_i}$.

In general, maximizing a two-good Cobb-Douglas utility function $u(x,y)=Ax^\alpha y^\beta$ subject to budget constraint $p\_xx+p\_yy=m$ has general solution
\begin{equation}
x^\*=\frac{\alpha}{\alpha+\beta}\frac{m}{p\_1}
\quad\text{and}\quad
y^\*=\frac{\beta}{\alpha+\beta}\frac{m}{p\_2}.
\end{equation}

In your case, the parameters $A,\alpha,\beta,p\_x,p\_y,m$ take the following values:
\begin{equation}
A=2,\quad\alpha=\beta=\frac12,\quad p\_x=1,\quad p\_y=p,\quad m=1+py\_i.
\end{equation}
Plugging in the values, you get optimal consumption:
\begin{equation}
x^\*=\frac{1+py\_i}{2},\quad y^\*=\frac{1+py\_i}{2p}.
\end{equation}
Hence optimal utility is
\begin{equation}
u(x^\*,y^\*)=2\sqrt{\frac{1+py\_i}{2}\cdot \frac{1+py\_i}{2p}}=\frac{1+py\_i}{\sqrt p}
\end{equation}


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