<p><a href="https://i.sstatic.net/8cj8R.jpg" rel="nofollow noreferrer"><img src="https://i.sstatic.net/8cj8R.jpg" alt="enter image description here"></a>
Above is a plot of the indifference curves for <span class="math-container">$U=1,2,...,8$</span> and also for <span class="math-container">$U=8.5$</span> that I made in MATLAB. For <span class="math-container">$P_x=P_y=1$</span> and <span class="math-container">$m=4$</span>, the budget constraint is simply the the line <span class="math-container">$y=4-x$</span> (not pictured here, but easy to visualize). </p>

<p>Notice this budget constraint overlaps the <span class="math-container">$U=4$</span> indifference curve (fourth contour from the origin) on the line segment connecting <span class="math-container">$(1,3)$</span> and <span class="math-container">$(3,1)$</span>. Any other point on the budget line would yield <span class="math-container">$U&lt;4$</span>, hence consuming on the line segment connecting <span class="math-container">$(1,3)$</span> and <span class="math-container">$(3,1)$</span> is optimal. But since all of these bundles correspond to <span class="math-container">$U=4$</span>, the optimal bundle is not unique, but could be any bundle on this line segment. </p>

<p>Here is the MATLAB code that generated the plot: </p>

<pre><code>u=zeros(60,60);
for i=1:60;
    for j=1:60
        u(i,j)=min([(i+j)/10 4*sqrt(i/10) 4*sqrt(j/10)]);
    end
end
figure
contour(u,[1 2 3 4 5 6 7 8 8.5]);
ylabel('Good Y')
xlabel('Good X')
grid on
grid minor
xt = get(gca, 'XTick');                                 % 'XTick' Values
set(gca, 'XTick', xt, 'XTickLabel', xt/10) 
yt = get(gca, 'YTick');                                 % 'XTick' Values
set(gca, 'YTick', yt, 'YTickLabel', yt/10) 
</code></pre>

CC BY-SA 4.0
2019-03-15T01:26:21.390Z
2019-03-16T20:02:34.293Z
2019-03-16T20:02:34.293Z
dlnB
<p>The problem is basic utility maximization subject to a budget constraint with 
<span class="math-container">$$u(x,y) = min\{x+y,4\sqrt{x},4\sqrt{y}\}$$</span>
<span class="math-container">$$p_x = 1, p_y = 1, M = 4$$</span></p>

<p>I will have to first plot the Indifference Curves. </p>

<p>I am able to figure out the following: </p>

<p>If <span class="math-container">$x &lt; y: u(x,y) = min\{x+y, 4\sqrt x\}$</span>, <br> else <span class="math-container">$u(x,y) = min\{x+y, 4\sqrt y\}$</span>.</p>

<p>But I get stuck here. For ex: if I am in the region where <span class="math-container">$x&lt;y$</span>, I can't find the sub-region where <span class="math-container">$x+y &lt; 4 \sqrt x$</span> or <span class="math-container">$x+y &gt; 4 \sqrt x$</span>.</p>

<p>Cheers!</p>

CC BY-SA 4.0
2019-03-11T10:24:58.727Z
Why can't you find the subregion? Seems like the straightforward solution of a second degree polinomial to me. $$x^2 + x(2y -16) +y^2<0$$.
Giskard
CC BY-SA 4.0
2019-03-11T10:59:28.773Z
Giskard, thanks. But you see, I can't plot that on an x-y plane. Or can't solve for x,y either.
Vizag
CC BY-SA 4.0
2019-03-11T11:06:08.360Z
How come you cannot solve it?  Keeping $y$ constant it is a second degree polinomial. Could you solve $$x^2 + x(constant_1) + constant _2$$?
Giskard
<microeconomics><mathematical-economics><utility>
Utility maximimization for unusual Leontief utility function
Vizag
# Utility maximimization for unusual Leontief utility function

(Asked by: username_0 on 2019-03-11)

The problem is basic utility maximization subject to a budget constraint with
$$u(x,y) = min\{x+y,4\sqrt{x},4\sqrt{y}\}$$
$$p\_x = 1, p\_y = 1, M = 4$$

I will have to first plot the Indifference Curves.

I am able to figure out the following:

If $x < y: u(x,y) = min\{x+y, 4\sqrt x\}$,   
 else $u(x,y) = min\{x+y, 4\sqrt y\}$.

But I get stuck here. For ex: if I am in the region where $x, I can't find the sub-region where $x+y < 4 \sqrt x$ or $x+y > 4 \sqrt x$.

Cheers!

**Tags:** microeconomics, mathematical-economics, utility

> **Comments:**
> **username_1**: Why can't you find the subregion? Seems like the straightforward solution of a second degree polinomial to me. $$x^2 + x(2y -16) +y^2<0$$.  
> **username_0**: username_1, thanks. But you see, I can't plot that on an x-y plane. Or can't solve for x,y either.  
> **username_1**: How come you cannot solve it?  Keeping $y$ constant it is a second degree polinomial. Could you solve $$x^2 + x(constant_1) + constant _2$$?  

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## 1 answers

### Answer by username_2 on 2019-03-15. Score: 1 | Accepted answer

Above is a plot of the indifference curves for $U=1,2,...,8$ and also for $U=8.5$ that I made in MATLAB. For $P_x=P_y=1$ and $m=4$, the budget constraint is simply the the line $y=4-x$ (not pictured here, but easy to visualize). 
Notice this budget constraint overlaps the $U=4$ indifference curve (fourth contour from the origin) on the line segment connecting $(1,3)$ and $(3,1)$. Any other point on the budget line would yield $U<4$, hence consuming on the line segment connecting $(1,3)$ and $(3,1)$ is optimal. But since all of these bundles correspond to $U=4$, the optimal bundle is not unique, but could be any bundle on this line segment. 
Here is the MATLAB code that generated the plot: 
u=zeros(60,60);
for i=1:60;
    for j=1:60
        u(i,j)=min([(i+j)/10 4*sqrt(i/10) 4*sqrt(j/10)]);
    end
end
figure
contour(u,[1 2 3 4 5 6 7 8 8.5]);
ylabel('Good Y')
xlabel('Good X')
grid on
grid minor
xt = get(gca, 'XTick');                                 % 'XTick' Values
set(gca, 'XTick', xt, 'XTickLabel', xt/10) 
yt = get(gca, 'YTick');                                 % 'XTick' Values
set(gca, 'YTick', yt, 'YTickLabel', yt/10)


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