Utility maximimization for unusual Leontief utility function

Question

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 $x<y$, I can't find the sub-region where $x+y < 4 \sqrt x$ or $x+y > 4 \sqrt x$.

Cheers!

Answer 1

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)