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30

I completely agree with denesp's answer, however I think you can make it even simpler. On a very small scale, it's certainly true that if I gain, somebody else might lose. If I take away my brother's chocolate, then he will lose it, and will most probably not get anything comparable. OK, let's say I prefer chocolate to wine gums and my brother likes ...


18

This is a fundamental question which economics can answer quite well. I'll rephrase your question a little bit- Is economics a zero sum game? The answer is no. Certainly some transactions are, but for the most part, no. It can be proven a little bit more rigorously and denesp has alluded to that by linking the fundamental theorems of welfare economics. I'll ...


15

As a complement to the great answers already here, let me give an even simpler small scale example in which you win and no one else loses: Suppose you have a broken fan at home. Scenario A: you relax on your couch then go to sleep. Scenario B: you take your fan apart, figure it's just a loose screw, tighten it, put it back together again, and it's fixed. ...


14

Following up on the excellent MWG diagram in Amstell's answer, the fundamental observation needed is that holding $p$ fixed, $e$ and $v$ are inverses of each other. $e$ tells us the amount we need to spend to get a certain amount of utility $u$, while $v$ tells us the maximum amount of utility we can get from a certain expenditure $w$. Whenever we want to ...


14

We can say more generally that lexical preferences are not representable using a continuous utility function. Lexical preferences are not continuous. Note the definition of a continuous preference relation. The preference relation $\succeq$ is continuous if for any sequences of consumption bundles $(x_{i})_{i \in \mathbb{N}}$ and $(y_{i})_{i \in \mathbb{N}}$...


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 ...


13

Not sure how much this will help, but the diagram in Mas-Colell p.75 is something I always have in mind when deriving these functions. I'm not sure what books you're using, but Microeconomics by Mas-Colell et al. is the go to graduate resource. But I prefer Microeconomic Analysis by Varian. Much easier to read and still has the important content needed ...


13

Intuitively, a higher price for pears means that I have to give up more apples to be able to afford an extra pear (or, conversely, if I give up one pear then the number of extra apples that I can afford increases). This is going to make me want to reduce my pear consumption and increase my apple consumption (in orther words, to substitute away from pears ...


11

Yes, under some conditions. This is the classic integrability problem: for detailed discussion, see some excellent notes by Kim Border. Several other technical conditions are required, but the most economically substantive condition is that the Slutsky matrix must always be symmetric and negative semidefinite. To be concrete, if we define the $ij$th element ...


11

Utility functions are invariant with respect to positive monotonic transformations (PMT). Take $U(x,y)=x^\alpha y^{1-\alpha}$, and let $V(x,y)=\log(U(x,y))$ be a PMT of $U$. Thus $V$ and $U$ both represent the same preference, and thus demand functions for $x$ and $y$ are the same.


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 ...


10

No, you should not use Lagrange multipliers here, but sound thinking. Suppose $x\neq y$, say for concreteness $x<y$. Let $\epsilon=y-x$. Then $\min\{x,y\}=x=\min\{x,x\}=\min\{x,y-\epsilon\}.$ So the consumer could reduce her consumption of good 2, without being worse off. On the other hand for all $\delta>0$, we would have $\min\{x+\delta,y-\epsilon/2\}...


10

I think what you need is that if $U(x,y)$ is homothetic then $$ \forall \alpha \in \mathbb{R}_{++}, \forall (x,y) : \hskip 6pt \frac{\frac{\partial U(x,y)}{\partial x}}{\frac{\partial U(x,y)}{\partial y}} = \frac{\frac{\partial U(\alpha \cdot x,\alpha \cdot y)}{\partial x}}{\frac{\partial U(\alpha \cdot x,\alpha \cdot y)}{\partial y}} $$ and love.


10

Use Roy's identity to find demand functions: $$\displaystyle x_1=-\frac{\frac{\partial v}{\partial p_1}}{\frac{\partial v}{\partial y}} =-\frac{\alpha yp_1^{\alpha-1} p_2^\beta}{p_1^\alpha p_2^\beta}=-\frac{\alpha y }{p_1} $$ $$\displaystyle x_2=-\frac{\frac{\partial v}{\partial p_2}}{\frac{\partial v}{\partial y}} =-\frac{\beta yp_1^{\alpha} p_2^{\beta-1}}{...


10

This sounds glib, but The Internet is an ongoing example. In economic models where access to information is an explicit factor, we often gloss over the fact that mere exposure to information about the state of the world is not enough to improve welfare - one also needs the cognitive skills to use those facts to develop plans in support of preferences, and ...


10

The “to be announced” or TBA market for agency mortgage-backed securities is a great example of this. The short version is as follows: The TBA market is a market wherein one can purchase or sell for future delivery pools of mortgages that conform to certain characteristics (e.g., and to vastly simplify, 30 year mortgages at a 4.5% coupon) without ...


9

Models at Dynamic Stochastic General Equilibrium level must be able to replicate real economies to an acceptable degree. One of the features of real economies has been a relatively stable growth rate (see also this post), $\dot x/x=\gamma$, where the dot above a variable denots the derivative with respect to time. So one would want a model that admits a ...


9

The name for the amount $56.25 is certainty equivalent. The expected utility for the individual from taking the bet is calculated as follows: $$E[U]=\frac12U(100+125)+\frac12U(100-100)=75$$ Suppose the individual can pay an amount of money $x$ so that she can avoid taking the bet (which leads to expected utility $75$). What's the maximum amount of money $x$ ...


9

Defn: A function $h:\mathbb{R}^2\rightarrow \mathbb{R}$ is homogenous of degree $k$ if for every nonzero $\alpha$, $h(\alpha x, \alpha y)=\alpha^k h(x,y)$. Defn: A function is homothetic if it is a monotonic transformation of a homogenous function. Lemma: If $f$ is homothetic, i.e. $f=g\circ u$ for some strictly increasing $g$ and homogenous $u$ then $$ \...


8

The primary literature concerned with this type of question (at least where classical results break down) is behavioral economics. There's a great general compilation of papers put together by the Russell Sage Foundation called the "Behavioral Economics Reading List" that includes, among other things, a General Introduction section with overview papers by ...


8

A constrained optimization function maximizes or minimizes an objective subject to one or more constraints. As I understand it, the Lagrangian multiplier approach transforms a constrained optimization problem (I) into an unconstrained optimization problem (II) where the optimal control values to problem II are also the optimal control values to problem I. ...


8

The concept of "marginal utility" (and therefore of decreasing such) has meaning only in the context of cardinal utility. Assume we have an ordinal utility index $u()$, on a single good, and three quantities of this good, $q_1<q_2<q_3$, with $q_2-q_1 = q_3-q_2$. Preferences are well behaved and satisfy the benchmark regularity conditions, so $$u(q_1)&...


8

A good cannot be inferior over the entire income range. The paper A Convenient Utility Function with Giffen Behaviour shows that for a person with utility of the form: $$U(x,y) = \alpha_1 \ln(x-\gamma_x)- \alpha_2 \ln(\gamma_y - y) $$ X is inferior if $\gamma_x$ and $\gamma_y$ are positive, $0<\alpha_1<\alpha_2$, and in the domain $x>\gamma_x$ ...


8

A utility function can certainly be negative. The utility function is nothing more than a way to represent a preference relationship. This is an important conceptual point. In several theorems that typically show up in introductory texts, we show that sets of preferences that satisfy certain regularity conditions can be represented as utility function. Also,...


8

No. Cobb-Douglas utility is monotonic and monotonicity implies L.N.S. The issue here is that you're only considering edge cases. You've correctly reasoned that edge points are not more desirable that the origin. However, LNS simply claims that there exists a more desirable bundle within the open epsilon ball of your allocation under consideration (and this ...


8

My approach would be to define thick indifference curves in terms of a stronger form of local non-satiation: Definition (thick indifference curves) Preferences are said to have thick indifference curves if there exists at least one bundle $A\in\mathbb{R}^l$ and an open ball $\mathscr{B}(A)$ around $A$ such that $A'\sim A$ for every $A'\in\mathscr{B}(A)...


8

If you have quadratic preferences then your utility function is: $$ U(W) = W - \lambda W^2$$ this implies your expected utility function looks like: $$ E[U(W)] = E[W - \lambda W^2] = E[W] - \lambda E[W^2]$$ $$ = E[W] - \lambda E[W^2 - E[W]^2 + E[W]^2]$$ $$ = E[W] - \lambda E[W^2 - E[W]^2] - \lambda E[E[W]^2]]$$ $$ = \mu_w - \lambda \sigma_w^2 - \lambda ...


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