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?


2 Answers 2


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.


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

  • 1
    $\begingroup$ Lovely answer. What tool or tools did you use to make this figure? $\endgroup$
    – BKay
    Commented Oct 21, 2019 at 0:28
  • $\begingroup$ Thank you for the answer! Will the optimal bundle always lie at the kink? $\endgroup$
    – user708015
    Commented Oct 21, 2019 at 1:12

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.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.