5

Yes if you assume that the sub-utility functions are concave. Notice that this is a standard assumption as otherwise, the utility function $u = \sum_i f_i$ is not guaranteed to be concave (nor quasi-concave). Let denote by $u_i = \dfrac{\partial u}{\partial x_i}$ and by $u_{i,j} = \dfrac{\partial^2 u}{\partial x_i \partial x_j}$. By additivity $u_{i,j} = 0$ ...


4

Below is a graph of the price offer curve of good $y$ when income is $48$ units, $p_x = 8$ and the utility function is $$ U(x,y) = \min\left(2x+2y,x+10\right). $$ (Based on "Simple Utility Functions with Giffen Demand" by Sørensen). Good $y$ exhibits Giffen behavior when $0 < p_y < 8$. A gif of the optimal choice changing as $p_y$ changes ...


4

If $X$ and $Y$ are complements, an increase in the quantity demanded of $X$ also implies an increase in the demand for $Y.$ This is why both the price and the quantity of $Y$ will also have to increase.


3

Yes, whether a good is a Giffen or inferior good depends on "the circumstances", or to be more narrow, it can depend on the income and price levels. Claim 1. Being an inferior good is a local attribute. Proof. Suppose a good $x$ is inferior for all income levels $I$ and all price vectors $\mathbf{p}$. Trivially demand for $x$ at $I = 0$ is 0. If $x$ is ...


3

Actually, neither demand for Veblen good nor for Giffen good is strictly increasing in price. In case of Giffen good the demand actually looks as shown below in picture 1. The reason for this is that you can only increase demand for the Giffen good up until you consume your entire budget. Once the price gets higher then that you still get normal downward ...


2

[Edited: I think that Theoretical Economist is right in highlighting that I'm perpetuating the confusion between the mathematical formalism, and one way of getting to the intuitive understanding. It also comes down to language referring to the slope of the curve versus effects that are positive or negative depending on moving up or down that negatively-...


1

Def 1 is wrong. An inferior good is a good for which the demand decreases after a decrease in the agent's income. (not a decrease in the price of the good). Definition 2 is trying to define the same concept, "an inferior good" so it is also wrong. Instead, both definitions would be appropriate if they were describing a Giffen good. Giffen goods are ...


1

There's also goods for which there is inelastic demand, that is, the demand doesn't change regardless of price. These are things that are essential and cannot be replaced with anything similar, for example, medication for which there is no generic alternative. This is one of the reasons why prescription drug pricing is such a contentious topic - just about ...


1

Since the question is asking for clarification on a given answer, I don't think this question necessarily needs to be closed. To address your concerns: You will recall Walras' Law states $p \cdot x = w$. That is, you spend all your money buying things, given some regularity conditions. So just figure out how much was spent in year 1 and that will give you ...


1

A quick response is that if the law of demand is violated, then the standard definition for substitutes and complements may or may not apply. Consider buying fast food burgers as your only choice of meal. When the price of the meal drops, you might want to consume something healthier like a bowl of salads. You consume less fast food and may increase ...


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