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14

I disagree with @bbecon. I agree with @bbecon that concave utility functions present nice mathematical properties which help theorists develop analytical models. If the OP's question was why utility functions are concave, the fundamental argument is that utility experiences diminishing returns. Examine the image below, taken form here. Let's say good X is ...


11

For theory you have in order of prestige... (I know subjective) Journal of Economic Theory Theoretical Economics AEJ-Micro (only micro) Mathematics of Operations Research (OR related) Games and economic behaviour (only game theory) Economic theory Journal of Mathematical Economics Social choice and welfare (only social choice) Theory and Decision ...


8

The market price is the current price at which something may be bought or sold. If a good is not sold or bought at a particular price, then that is not the market price. Whether or not any particular individual thinks that price is too high or too low makes no difference, as the market price is by definition the price that someone is willing to pay/receive ...


7

The only utility function that comes to mind is the Stone-Geary utility function. For 2 goods, $x$ and $y$, this takes the form: $$ u(x,y) = (x - a)^\alpha (y- b)^{1- \alpha}. $$ This is a Cobb-Douglas type of utility function where $a$ and $b$ are subsistence levels, i.e. you need to consume at least $a$ from $x$ and $b$ from $y$ to survive. It is the ...


7

You can solve it sequentially by noting the nesting structure of the utility function $U$. So first note that the utility function combines utility functions you are probably already familiar with $U=\min\{u_1,u_2\}$ of complements and $u_1=\sqrt{x+y}$ and $u_2 = z+w$ both of which are perfect substitutes. Where $u_1$ and $u_2$ are nested within $U$. The ...


7

How is marginal utility interpreted as the additional "happiness" gained from consuming one more unit of some good? Not sure what you mean. Utility is not interpreted as some biological measure of happiness. A bundle of goods with high utility is prefered by the consumer to bundles with lower utilities. This is all that utility describes; it is a ...


7

More or less, yes. Making the right assumption on the shape of the utility function allows you to prove existence or uniqueness of the equilibrium. The exact assumption you need depends on what exactly you are trying to prove and how general you want your result to be. In the case of concavity, it also makes the equilibrium easier to find using the first-...


6

It is basically a restatement of the first order condition - at an extrema (maxima or minima) of a well-behaved function its first derivative is equal to zero. If you are at the point of maximization, any deviation should either be of no benefit to you or violate some constraints. By continuity, it means that, unless you are constrained, at the optimal point ...


6

The vast majority of economists subscribe today to the subjective theory of value that was in economics introduced by Jevons, Walras, and Menger. Subjective theory of value posits that value is subjective. A corollary to that is that there is no correct objective price. However, if you talk about market price existing as an objective number that is ...


5

Let's look at the use of monotonic transformations of utility functions (which I guess is the most frequent occurrence of this concept in econ). Let $u: \mathbb{R}^n_+ \to \mathbb{R}$ be a utility function. We say that $g: \mathbb{R}^n_+ \to \mathbb{R}$ is a monotonic transformation of $u$ if for all $x, y \in \mathbb{R}^n_+$: $$ u(x) \ge u(y) \iff g(x) \ge ...


5

Here is the sketch of a proof. All we need is that every continuous weak order on each $X_i$ admits a continuous utility representation. One sufficient condition is that each $X_i$ is a connected separable topological space by a theorem of Eilenberg. A proof of Eilenberg's theorem is given in Debreu's book Theory of Value. Debreu assumes the domain there ...


5

The following is essentialy due to Debreu. The result is formulated in terms of linear orders, but each complete and transitive relation induces a linear order on the indifference classes: Theorem: Let $S$ be a set and $\preceq$ be a linear order on $S$. Then $\preceq$ has a utility representation if and only if there exists a countable set $C\subseteq S$ ...


5

In its most general formulation, a social welfare function is just a utility function representing the preferences of "society as a whole" (or the preferences of a hypothetical "benevolent social planner" who makes decisions for the society). Let $X$ be some space of "social outcomes". (Social outcomes could be anything. But ...


5

$$ \left(\frac{2}{100} \cdot 1000 \oplus \frac{98}{100}\cdot 0\right) $$ is the lottery where you get $1000$ with probability $2/100$ and $0$ with probability $98/100$. The expression $$ 20 \sim \left(\frac{2}{100} \cdot 1000 \oplus \frac{98}{100}\cdot 0\right) $$ usually says that the decision maker is indifferent (in terms of preferences) between taking ...


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 $u_{i,j} = 0$ ...


5

(1) Satisfying completeness, independence, and continuity but not transitivity: Take two outcomes, $\{0,1\}$, and the associated lottery space $[0,1]$. Consider the preference relation $\succsim$ where $x\sim y$ if and only if $x=y$, $x=0$ and $y=1$ or vice versa, $x\succ y$ if and only if $x>y$, except for $x=0$ and $y=1$ or vice versa. Transitivity is ...


4

Your steps look okay, and the solutions are correct. You can rule out corner solutions by evaluating the utility function at the "corners" and compare those values to the optimum you found. You should be able to verify that $U(50,20)>\max\{U(0,120),\,U(60,0)\}$.


4

You can do whatever you want, it's your paper. Will it make it more difficult to publish? Yeah. Referees are fickle and easily annoyed. You would not be the first person to go down this route. For example, the term Maskin monotonicity is common place in implementation theory. However, some authors use another term, I believe Maskin invariance. The claim is ...


4

Decision under uncertainty is sometimes called a "game against chance", and can thus be modeled as a two-player normal form game: the decision-maker vs Nature/Chance. The possible states would form a set of pure strategies for Nature, and Nature commits to a publicly known mixed strategy that randomizes over those pure strategies (assume Nature is ...


4

Converting my comments into an answer: At the bottom of p.268, the authors say: The set of consumption vectors $X_i$ available to individual $i$ $(=1,\cdots,m)$ is a closed convex subset of $R^l$ which is bounded from below. [Emphasis added.] Since the Heine-Borel Theorem establishes that $S\subset R^n$ is compact if and only if $S$ is both closed and ...


4

Real income and utility are not the same thing but utility can be expressed as a function of income because income is what allows you to consume goods and services. For example, let us assume there are only two goods $x_1$ an $x_2$ and consumer is given budget $p_1 x_1 + p_2 x_2 = m$, where $p_i$ are prices for good 1 and 2 respectively and $m$ is an income. ...


4

Income elasticity of demand Let $q(y)$ be the Engel curve for a good, i.e. it gives the demanded quantity for a given level of income $y$ (keeping prices fixed). The income elasticity of demand is then given by: $$ \varepsilon^y_q = \frac{\partial q}{\partial y} \frac{y}{q} $$ It measures the percentage point change in demand $q(y)$, due to a 1$\%$ increase ...


4

Yes. I have to include at least 30 characters in an answer, so let me repeat: Yes.


4

This is more of a calculus question. Recall the total differential of a function $f(z_1,z_2,z_3)$: \begin{equation} \mathrm df(z_1,z_2,z_3)=\frac{\partial f(z_1,z_2,z_3)}{\partial z_1}\mathrm dz_1 + \frac{\partial f(z_1,z_2,z_3)}{\partial z_2}\mathrm dz_2+\frac{\partial f(z_1,z_2,z_3)}{\partial z_3}\mathrm dz_3. \end{equation} Hence, \begin{equation} \frac{\...


4

To provide another answer with less equations: Consider first that the inner utility function $u_1 = \sqrt{x+y}$ and $u_2 = z + w$ are perfect substitutes implying that consumer only buy the cheapest of $x$ and $y$ and similarly only the cheapest of $z$ and $w$. Because prices are given we know that $y$ is cheaper than $x$ and $w$ is cheaper than $z$. Using ...


4

To expand on @1muflon1's answer. The theory of rational addiction assumes that the utility of a consumer at time instance $t$ depends both on current consumption of the addicitve good, say $c_t$, and the consumption of the addictive good in the past. For simplicity say $c_{t-1}$. So at period $t$ the instantaneous utility looks something like: $$ u(c_t, c_{t-...


4

Perhaps I misunderstand the question, because it seems trivial coming from such an established researcher. As @HerrK. points out, utility functions that represent intertemporal discounting are generally of the form $$ U\left((x_i)_{i=1}^T\right) = u(x_1) + \delta_1 u(x_2) + \delta_2^2 u(x_3) + \dots $$ where $\delta_i$ is the discount factor and $x_i$ is the ...


4

No, I don't think so. Let $a_i$ be your choice (e.g. ice tea) in situation $i$ (e.g. summer). Separability over states says that if for some $b_j$, $a_i$ and $c_i$: $$ u(a_i, b_j) \ge u(c_i, b_j), $$ then for all $d_j$ $$ u(a_i, d_j) \ge u(c_i, d_j). $$ Notice that both comparisons keep the choice in some state fixed but this choice changes. So your ...


4

I think the core issue with this question (and the other related one the OP posted Nonseparable utility across states of nature: an intuitive example) is we need to clarify what is meant by "separable". Unfortunately, "separable" is among the most overused adjectives across formal theories, in econ and beyond (including in pure math ...


3

This might just be a misunderstanding. The textbook contains various passages like e.g. "An alternative approach holds real income (or utility) constant while examining reactions to changes in $p_x$" (p. 151). When this terminology is introduced, however, "real" stands in quotation marks. This is in the context of explaining the ...


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