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I have a utility function: $U = x + \min\{x,y\}$

I want to draw the indifference curve and find the demand functions. Will it be the case of the usual perfect complements?

Also, what preferences could such a utility function represent?

Will the optimal solution be at the kink?

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I assume you know how does $\min\{x,y\}$ look like? In order to draw utility function of interest, you need to consider cases: $u(x,y)=x+\min\{x,y\}=\begin{cases}2x, \;\; \mathrm{for} \;\; x \leq y \\ x+y, \;\; \mathrm{for} \;\; x > y\end{cases}$

With $x$ on horizontal and $y$ on vertical axis:

enter image description here

Not sure about the "usual" perfect complements. It is more like a combination of substitutes (below $y=x$) and complements (above $y=x$).

Also, take a look here: Identifying utility function and Algebraic approach towards convexity where you can see more graphs.

EDIT.

As for the demand function, Finding demand function given a utility min(x,y) function.

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    $\begingroup$ Lovely answer. What tool or tools did you use to make this figure? $\endgroup$
    – BKay
    Oct 21, 2019 at 0:28
  • $\begingroup$ Thank you! I used Geogebra $\endgroup$
    – bajun65537
    Oct 21, 2019 at 0:29
  • $\begingroup$ Thank you for the answer! Will the optimal bundle always lie at the kink? $\endgroup$
    – user708015
    Oct 21, 2019 at 1:12
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To provide some real world(ish) interpretation, you could consider the following:

Wallace enjoys eating cheese on its own. He doesn't much care for crackers on their own, but he especially loves eating crackers and cheese together, he makes nice little cracker n cheese sandwiches.

In this example, we can think of cheese (x) and crackers (y) as perfect complements, but cheese will be a perfect substitute for a cracker and cheese sandwich.

If a unit of cheese is cheaper than a unit of crackers he'll skip the sandwiches and spend his whole budget on cheese, but if crackers are cheaper he'll buy cheese and crackers in equal proportions.

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