Most of the preferences we use in class exhibit convexity.
What does it imply about the shape of indifference curves? Explain, in
words, what this implies about consumer preferences, think of an example
where this might be a bad assumption and try to come up with a utility
function that captures your idea in the example.
To understand the relationship between convex preferences and the shape of indifference curves, we should first define these terms.
Convex preferences refer to the idea that individuals generally prefer a mix of two goods over extremes or pure amounts of either good. In other words, people tend to prefer a variety or combination of goods rather than only one good or an excessive amount of one good. This preference can be represented mathematically using indifference curves.
Indifference curves depict different combinations of two goods that a consumer considers equally desirable or preferable (indifferent) to each other. These curves are typically downward sloping, illustrating the negative trade-off between the two goods.
Given that most preferences exhibit convexity, it implies that the indifference curves associated with these preferences will be convex or bowed inwards towards the origin. This is because as the consumer moves along the indifference curve, the marginal rate of substitution (the rate at which the consumer is willing to exchange one good for another while remaining indifferent) varies. This varying marginal rate of substitution leads to the convex shape of the indifference curves.
To understand this concept better, consider an example of someone's food preferences. Let's assume that this individual likes both pizza and burgers. However, they prefer a combination of the two rather than having only one or the other. As they consume or enjoy more of one food item, their willingness to trade it for the other decreases. Hence, their preferences are convex.
Now, let's suppose you have a friend who doesn't exhibit convex preferences. In this case, they have what we call "bad assumption" preferences. For example, they may have a strong aversion to a particular good or may prefer only extremely large or small amounts of a good. Let's say your friend dislikes fish and would rather have any other food instead. Their indifference curves would depict this by being concave (bowed outwards) towards the origin.
To represent this idea with a utility function, we can use the concept of diminishing marginal utility. Let's say your friend's preferences are not convex, and they strongly dislike fish. We can construct a utility function that captures this dislike and diminishing marginal utility:
U(x, y) = x + ln(y) - αln(fish)
In this utility function, x represents the quantity of pizza consumed, y represents the quantity of burgers consumed, and fish represents the quantity of fish consumed. The α term reflects the strength of aversion to fish, and ln() represents the natural logarithm.
By including the negative αln(fish) term, we are capturing your friend's strong dislike for fish, which would result in concave indifference curves in this case.
Remember, this is just one example, and there could be various utility functions or preferences that capture different consumer behaviors.