6

Seems to me $dU = 0$ is true by definition unless $dU$ is not defined because the function $U$ is not differentiable in some variable. E.g. $$U(x,y) = |x| + y$$ because there are a countably infinite number of dimensions but the partial differentials are not always absolute convergent and hence cannot be summed up. E.g.: $$U(x_1,x_2,x_3,x_4...) = x_1 - ...


6

I think the Investopedia article does not use clear terminology which leads to confusion. In Economics, utility can be split across two categories as ordinal and cardinal utility. In the case of ordinal utility assigning numeric values is meaningless. For example, if A is proffered to B, assigning 200 utils to A and 100 utils to B is meaningless because that ...


5

No, you can not really say that. Here is the reason: Utility is generally used to describe preferences. Let's say my happiness is described by this utility of chocolates: $Y(X) = X$. So, two chocolates will give me two units of happiness. However, by definition of preferences, I am allowed to apply monotone transformations to them. Hence, you can just ...


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

$u = \max(x, y)$ represents the preferences over two substitute goods that cannot be consumed together. For example - tea and coffee. In the event that the consumer gets x quantity of tea and y quantity of coffee, consumer choose to consume only one of the them depending on the quantity. He always choose the one that is offered in larger quantity and throws ...


4

Utility functions as ordinarily used are not a measure of well-being comparable among people, but a representation of preferences. Moreover, preferences could principally be elicited from choice experiments. A utility function assigns real numbers to alternatives so that one alternative is preferred to the other if and only if it is assigned a higher number. ...


4

This is just a follow-up on denesp's comment. In economics, the utility is a tool used to represent choices, and therefore the right way of measuring the individual's utility from listening to the song is to elicit his willingness to pay for it. This can be done by a Becker-DeGroot-Marschak mechanism that eliminates moral hazard. Note that this refers to ex-...


3

It appears to be the Theory of Games and Economic Behavior (1944) by John von Neumann & Oskar Morgenstern. I have the 1953 edition which is counted as "3d", but by reading the included introductions to the 2nd and to the 3d editions, it appears that nothing of substance has been changed in chapter 3, where I locate the issue of equivalence up to linear ...


3

Suppose Alex, Bryan, and Chris run in a race. Alex is the fastest and Chris is the slowest. So far I have only given you ordinal information about where they finished. I guess you'd be okay with me taking that ordinal information and saying "Alex finished first, Bryan finished second, and Chris finished third". But then I just defined a function that ...


3

Certain calculations using utility require utility functions to have some particular properties. The point of bringing up the invariance of preferences represented by different utility functions that are just monotonic transformations of each other is to show that these utility-based calculations are not entirely dependent on the utility functions that you ...


3

Your thinking is correct that, in some ways, $x_1, x_2$ are substitute goods. We define substitute goods which have the following property: $$\left.\frac{\partial x_i}{\partial p_j}\right|_{u=\bar u}>0$$ The case of $U(x_1,x_2)=\max\{x_1,x_2\}$ is that of a boundary solution as the indifference curves are now concave to the origin. So equilibrium solution ...


3

It is possible for an addict to be rational. A famous work on this was done by Becker (1988) Theory of Rational addiction. In order for agent to have rational preferences the preferences have to satisfy the following definition (See MWG Microeconomic Theory pp 6): Definition 1.B.1: The preference relation $\succeq$ is rational if it possesses the following ...


2

This is trivially not true. Consider simple example of utility: $$u(x) = x^{1/2}$$ Expected utility $E(u(x)) = E[x^{1/2}]$ Inverse utility is $u^{-1} \implies x = u^2 $ clearly generally $E(u) \neq u^{-1} $.


2

There are no simple methods for estimating cardinal utility (ordinal utility would be a different matter - you could just observe few of your choices). This is not because cardinal utility would necessary be unmeasurable. Although this is not completely settled question (see Moscati (2018) Measuring Utility: From the Marginal Revolution to Behavioral ...


2

Linear utility functions like the one you have ($U = 2X_1+2X_2$) commonly lead to corner solutions where you only buy one of the goods. You can tell that this will occur here because utility is maximized with an internal solution by setting MRS equal to the price ratio, but MRS is always 1 and the price ratio is always .5, so they'll never be equal. In this ...


2

Utility is constant for all points $(q_B,q_A)$ on an indifference curve. So there is a number $u_1$ such that $$ \forall (q_B,q_A) \in IC_1: \ U(q_B,q_A) = u_1. $$ Similarly there is a number $u_2$, such that $$ \forall (q_B,q_A) \in IC_2: \ U(q_B,q_A) = u_2. $$ If $IC_1 \neq IC_2$ then $u_1 \neq u_2$.


2

Utility is subjective, and as you wrote, "it doesn't matter to us", but only to the person who orders her choices. We are not able to quantify a subjective concept like the utility of another person, but each person can do this for herself. This is the reason why utility is ordinal, which means that only the order of the numbers matters, and not the way we ...


2

Yes. In general not. Let's say the individual has initial wealth $W$ and the gamble $g$ has payouts $0$ and $G$, each with probability $1/2$. As you say, the certainty equivalent $C$ of the gamble is the amount $C$ with $$u(W+C)=(u(W)+u(W+G))/2.$$ Now the same individual would be willing to pay at most $P$ to enter the gamble, where $$u(W)=(u(W-P)+u(W+G-P))/...


1

If you assume that agents' utility functions over pairs $(x,t)$ of consumption bundles $x$ and monetary transfers $t$ are quasilinear in money, $u_i(x,t)=v_i(x)+t$, then $v_i(x)$ measures $i$'s WTP for $x$. In a utilitarian framework the sum of utilities is then a social welfare function satisfying Arrow's axioms (apart from universal domain of course) and ...


1

If the preferences do not have an expected utility representation, then either the preferences are not continuous, or they do not satisfy the axiom of independence. For example in prospect theory, consumers display loss aversion, which means that their preferences are not linear in probabilities. This is a case where preferences do not have the expected ...


1

AA approach appears to follow two stages: first, "nature" provides which event obtains, which results in the given lottery; second, the lottery is resolved, therefore revealing the intrinsic probabilities. AA suggests to resolve by backward induction. Under usual conditions for vNM utility, agent may define a function for her utility. I found a ...


1

To the extent you are constructing marginal decisions from preferences, the general answer is no. Any utility function that preserves preference orderings maps to the same partial or total order ranking. As a consequence, $U(w)=\log(w),w>0$ and $U(w)=\sqrt{w},w>0$ preserve preference orderings with respect to $w$. Issues can exist in the multivariate ...


1

According to Wikipedia and this notes from HKU, the indirect utility function is part of consumer theory, and is defined as the maximum utility that can be attained given a consumers' money income, and goods prices. It reflects both the consumer's preferences and market conditions. It is a model which helps to understand how consumers optimize their ...


1

I would start with your generic $$f(t,s) = t \times s$$ and, instead of add weights or factors to this, I would add other variables related to time and satisfaction like: boarding time as function of luggage management ($TL$) times baggage number ($B$), and time to go to seat row ($T_g$) vs time to actually sit ($T_s$) for the different seat types (Window, ...


1

It sounds like you could simply use the Cost Minimization Problem: $$\underset{z_1,...z_N}{min}\sum_{i=1}^N q_iz_i$$ $$s.t.\quad f(z_1,...,z_N)\geq \bar{y}$$ $$z_1,...z_n\geq0$$ Where $z_i$ and $q_i$ are the quantity and price of input $i$, respectively, $\bar{y}$ is some predetermined level of output, and $f(\frac{}{})$ is the production function. A ...


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