VTT Indifference curve representation as convex or concave

[fosva]

Lecture slides 72-74 mention a VTT indifference curve. In general, these have a convex shape. It seems to me that this only happens when the commodities are 'positive', so something is gained. In case of 'negative' commodities, such as travel time and cost, I believe the curve should be concave.

So instead of the convex curve

I am expecting something concave like this:

I believe this to make more sense. In the convex curve, eventually TT can get very high as TC goes to 0, which is not what happens in real life. I am curious about your thoughts.

Fos

[sandervancranenburgh]

Hi Fos, Excellent question. Whether the shape is convex or concave entirely depends on the nonlinearity of the utility functions. The indifference curve essentially shows how much cheaper the travel cost (y-axis) must get to compensate for the increasing travel time (x-axis). Therefore, if people are increasingly sensitive to longer travel time, they want to be more than linearly compensated. In that case, we will find a concave indifference curve. In contrast, if people have a diminishing marginal (dis)utility for travel time, they need less and less compensation in terms of travel costs to compensate for the increase in travel time. Hence, the indifference curve will be convex (i.e. as in the PowerPoint slide). To flesh this out further, I have added a little notebook file. Feel free to try different nonlinear utility functions and see how they affect the indifference curves!

[fosva]

Hi Sander, thank you for the response. I still think something is different, when comparing to material of previous courses, and other sources I have been able to find on the internet. In general, indifference curves show the relationship between two commodities, and are always convex. The underlying assumptions however include that an increase in both commodities always gives a higher utility. In other words, the commodity should be 'positive'. So, suppose we have a situation where you wake up early to travel to work, when we would consider 'sleeping time' instead of 'travel time', and 'money I earn' instead of 'travel cost', I believe the curve will look convex.

In this situation we are trading in hours of sleep and money, which are both 'positive'.

Suppose, in a reference alternative, I get 8 hours of sleep and earn 50 euros. If I manage to trade in some sleep but earn more money, I will have the same utility. If I have to get up even earlier, I want more money/bonus in return. On the other hand, if I want to sleep in more, I am willing to give up some of my earnings. Then again, if my boss says I can sleep as long as I want, but it means I earn almost nothing, I will not agree and ask again for more money. These two situation make the curve look convex.

I have not been able to come up with an interpretation of a concave curve, and the sources on internet seem to agree. Now for the given example from the slides, the commodities are 'negative' or subtracted from a reference alternative. That means the curve should flip. As of yet, it is unclear to me if in economics this is allowed, or may have a different name. But as far as I know, an indifference curve represents two commodities with positive marginal utility, and is convex. If the marginal utility is negative, the curve should flip and be concave. If it still follows the economic theory, I'm not sure