<p>The equation <span class="math-container">$100\sqrt{x_1} = 8x_1^{3/2} - 5$</span> is cubic in <span class="math-container">$\sqrt{x_1}$</span>.  Unless you have been asked for an analytic solution (which is possible but complex), the approximate method I suggest is first to ignore the minus <span class="math-container">$5$</span>, enabling division by <span class="math-container">$\sqrt{x_1}$</span> yielding:</p>
<p><span class="math-container">$$100 = 8x_1$$</span></p>
<p>The first approximation is then <span class="math-container">$x_1 = 100/8 = 12.5$</span>. Since <span class="math-container">$100\sqrt{12.5} \approx 354$</span> is much greater than <span class="math-container">$5$</span>, it can be inferred that the exact solution will be very close to <span class="math-container">$12.5$</span>.  So try successive approximation, perhaps starting with <span class="math-container">$x_1=12.6$</span>.</p>

CC BY-SA 4.0
2020-09-13T12:16:23.450Z
2020-09-13T12:16:23.450Z
Adam Bailey
<p>A consumer with a utility function given by <span class="math-container">$u(x_1, x_2) = \sqrt{x_1} + x_1x_2$</span> has an income of <span class="math-container">$100$</span>. The unit prices of <span class="math-container">$x_1$</span> and <span class="math-container">$x_2$</span> are <span class="math-container">$4$</span> and <span class="math-container">$5$</span>, respectively.</p>
<p>(a) Compute the utility-maximizing commodity bundle, if consumption must be nonnegative.</p>
<p>The necessary condition for this optimization is that
<span class="math-container">$$\partial L/\partial x_1 = \frac1{2\sqrt{x_1}} + x_2 -4\lambda_1+\lambda_2 = 0$$</span>
<span class="math-container">$$x_1 - 5\lambda_1 + \lambda_3 = 0$$</span>
<span class="math-container">$$100 - 4x_1 - 5x_2 = 0$$</span>
<span class="math-container">$$x_1 \ge 0, \quad x_2 \ge 0$$</span></p>
<p>There are two cases we need to consider <span class="math-container">$\{x_1 &gt;0, x_2 &gt;0\}$</span> and <span class="math-container">$\{x_1 &gt;0, x_2 =0\}$</span>. For the first case, if I solve the above system of equations, I have
<span class="math-container">$$100\sqrt{x_1} = 8x_1^{3/2} - 5.$$</span>
In this case, how can we obtain the bundle <span class="math-container">$(x_1^*, x_2^*)$</span>?</p>

CC BY-SA 4.0
2020-09-12T17:45:37.103Z
Is the question how to solve $100\sqrt{x_1} = 8x_1^{3/2} - 5$ for $x_1$?
Michael Greinecker
CC BY-SA 4.0
2020-09-12T18:13:01.233Z
yes............
shk910
<optimization>
Find the utility maximizing bundle [Sundaram, P.169, Q.7 (Kuhn-Tucker Theorem) ]
shk910
# Find the utility maximizing bundle [Sundaram, P.169, Q.7 (Kuhn-Tucker Theorem) ]

(Asked by: username_0 on 2020-09-12)

A consumer with a utility function given by $u(x\_1, x\_2) = \sqrt{x\_1} + x\_1x\_2$ has an income of $100$. The unit prices of $x\_1$ and $x\_2$ are $4$ and $5$, respectively.

(a) Compute the utility-maximizing commodity bundle, if consumption must be nonnegative.

The necessary condition for this optimization is that
$$\partial L/\partial x\_1 = \frac1{2\sqrt{x\_1}} + x\_2 -4\lambda\_1+\lambda\_2 = 0$$
$$x\_1 - 5\lambda\_1 + \lambda\_3 = 0$$
$$100 - 4x\_1 - 5x\_2 = 0$$
$$x\_1 \ge 0, \quad x\_2 \ge 0$$

There are two cases we need to consider $\{x\_1 >0, x\_2 >0\}$ and $\{x\_1 >0, x\_2 =0\}$. For the first case, if I solve the above system of equations, I have
$$100\sqrt{x\_1} = 8x\_1^{3/2} - 5.$$
In this case, how can we obtain the bundle $(x\_1^\*, x\_2^\*)$?

**Tags:** optimization

> **Comments:**
> **username_1**: Is the question how to solve $100\sqrt{x_1} = 8x_1^{3/2} - 5$ for $x_1$?  
> **username_0**: yes............  

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## 1 answers

### Answer by username_2 on 2020-09-13. Score: 1

The equation $100\sqrt{x\_1} = 8x\_1^{3/2} - 5$ is cubic in $\sqrt{x\_1}$. Unless you have been asked for an analytic solution (which is possible but complex), the approximate method I suggest is first to ignore the minus $5$, enabling division by $\sqrt{x\_1}$ yielding:

$$100 = 8x\_1$$

The first approximation is then $x\_1 = 100/8 = 12.5$. Since $100\sqrt{12.5} \approx 354$ is much greater than $5$, it can be inferred that the exact solution will be very close to $12.5$. So try successive approximation, perhaps starting with $x\_1=12.6$.


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