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27

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 ...

16

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 ...

14

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. ...

13

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 ...

12

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 ...

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 ...

10

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 ...

9

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.

9

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\}... 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

If I take away my brother's chocolate, then he will lose it, and will most probably not get anything comparable. The problem with this example is that there is no economy between you and your brother. You simply stole his chocolate. Conquered it if you will. No trade ever took place. The best way I've ever heard this explained is by calling it the "...

9

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 $(y_{i})_{i \in \mathbb{N}}$...

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 $$\... 8 The primary literature concerned with this type of question (at least where classical results break down) is behavioral economics. There's a great general compilation of papers put together by the Russell Sage Foundation called the "Behavioral Economics Reading List" that includes, among other things, a General Introduction section with overview papers by ... 8 A constrained optimization function maximizes or minimizes an objective subject to one or more constraints. As I understand it, the Lagrangian multiplier approach transforms a constrained optimization problem (I) into an unconstrained optimization problem (II) where the optimal control values to problem II are also the optimal control values to problem I. ... 8 I think what you need is that if U(x,y) is homothetic then$$ \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}} $$and love. 8 My approach would be to define thick indifference curves in terms of a stronger form of local non-satiation: Definition (thick indifference curves) Preferences are said to have thick indifference curves if there exists at least one bundle A\in\mathbb{R}^l and an open ball \mathscr{B}(A) around A such that A'\sim A for every A'\in\mathscr{B}(A)... 8 To the pure ordinalist who believes that preferences are purely ordinal, the concept of marginal utility (MU) has no meaning. (And a fortiori, the concept of diminishing MU also has no meaning.) However, the concept of the marginal rate of substitution (MRS) does have meaning. In the course of our work, we may compute something that we call MU. But to the ... 7 There is a type of protection called a liability rule, where I, A, can take something from B, if I pay the damages c which are court-ordered preemptively. Copyright law is all about liability rules. If the damages are correlated with B's valuation appropriately, then efficiency holds. You are interested in the opposite case. If IP law doesn't get it ... 7 In the Arrow-Debreu model, households are endowed with some commodities, producers can use feasible production plans to transform commodities, and households ultimately consume commodities - which they choose optimally given their preferences subject to budget constraints. These commodities can include (potentially time and state-contingent) labor services ... 7 Models at Dynamic Stochastic General Equilibrium level must be able to replicate real economies to an acceptable degree. One of the features of real economies has been a relatively stable growth rate (see also this post), \dot x/x=\gamma, where the dot above a variable denots the derivative with respect to time. So one would want a model that admits a ... 7 A good cannot be inferior over the entire income range. The paper A Convenient Utility Function with Giffen Behaviour shows that for a person with utility of the form:$$U(x,y) = \alpha_1 \ln(x-\gamma_x)- \alpha_2 \ln(\gamma_y - y) $$X is inferior if \gamma_x and \gamma_y are positive, 0<\alpha_1<\alpha_2, and in the domain x>\gamma_x ... 7 Here is one possible rigorous definition of a utility function: Let X be a set of alternatives. Let \succeq be a preference relation over those alternatives. U: X \to \mathbb{R} is a utility function means that U(x) \geq U(y) \iff x \succeq y. Then if for example X is 'amounts of money you might be given', and x \succeq y only if x \geq y, ... 7 A utility function can certainly be negative. The utility function is nothing more than a way to represent a preference relationship. This is an important conceptual point. In several theorems that typically show up in introductory texts, we show that sets of preferences that satisfy certain regularity conditions can be represented as utility function. Also,... 7 The indifference curves are constructed by viewing the utility function as an equation (for a fixed utility index value per curve). So from$$U = U(x_1,x_2)$$where the left side is just a symbol, we move to$$\bar U = U(x_1,x_2)$$where now the left side is a specific number. Take the total differential on both sides to obtain$$0 = U_1dx_1 + U_2dx_2 ...

7

Yes it is: If direction $$x \succ y \Rightarrow x \not \precsim y \Rightarrow u(x) > u(y).$$ Only if direction: For all $x, y \in X$, $$x \succsim y \iff u(x) \geq u(y)$$ implies $$x \sim y \iff u(x) = u(y).$$ Also $$u(x) > u(y) \Rightarrow u(x) \geq u(y) \Rightarrow x \succsim y ,$$  u(x) > u(y) \Rightarrow u(x) \not = u(y) \...

7

The MRS is a function of $x$ and $y$. The fact that $y$ doesn't appear in the formula simply means that this function is constant in $y$. The linearity does not imply indifference. It means that an additional value of $y$ is valued equally by the agent irrespective of his baseline consumption. In your example, this contrasts with the marginal utility of ...

7

Here a short answer: Homothetic, identical preferences have the modelling advantage that the distribution of income across individuals does not matter for aggregate demand. That is, if you want to study, let's say, monetary policy where you do not expect changes in the distribution of income to affect your policy recommendations, then this is a reasonable ...

7

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 ...

7

Recall the definition: The function $u: X \rightarrow \mathbb{R}$ represents $\succeq$ on $X$ if for any $x,y \in X$, then $x \succeq y \iff u(x) \geq u(y)$ We can show that if $u: X \rightarrow \mathbb{R}$ represents $\succeq$ on $X$, then for any strictly increasing function, $f: \mathbb{R} \rightarrow \mathbb{R}$, the function $v(x) = f(u(x))$ also ...

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