Questions tagged [utility]

Utility, or usefulness, is the (perceived) ability of something to satisfy needs or wants.

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Utility Function Challenge

A utility function is given for two goods Good 1 and Good 2 U = f(G1,G2) = 45G1^0.7, G2^0.3 ^ Did I get this right, the marginal utility functions with respect to G1 and G2? Solution dU/dG1 - MU of ...
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What utility function represents an agent with a time-discount factor?

Consider an agent who has a fixed budget, and should decide how to split it between consumption today and consumption tomorrow. For simplicity, suppose there is no interest, and no borrowing/lending, ...
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marginal utility and Marginal rate of substitution

While studying for my microeconomics class, I realized that I could not fully understand the meanings of Marginal utilities and marginal rate of substitution. Can anyone explain to me the difference ...
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is the investors risk-premium 51/8 [closed]

An investor's utility function for his wealth W is given by u(W) = W^1/3. his wealth after 1 year is either 8 with probability 3/4, or 64 with probability 1/4. Then the investor's risk-premium is ...
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how to solve such UMP where utility function is quasi-linear with cobb-douglas function as the non-linear part [closed]

U =$X_1+X_2^aX_3^{1-a}$ $a ∈[0,1]$ $s.t. p·x≤w , x≥0 $ I have tried FOC for x1 x2 x3 and λ, but I cannot get two pairs of equalities separately in order to express two unknowns as a function of ...
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1answer
74 views

Binary choices in consumer theory

Back when I was a teaching assistant, we were teaching utility theory and how it relates to price determination. So we were looking at continuous goods (e.g. "food") so as to get marginal ...
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1answer
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certainty equivalent and lotteries [closed]

suppose an agent has $u(z)=-e^{-bz}$ where $b>0$ as her Bernoulli utility function and faces two gambles: G1: win 1000 dollars with probability $\frac{1}{2}$ and zero with probability $\frac{1}{2}$ ...
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30 views

CES utility maximization two goods two period

In an Arrow-Debreu economy, there are two periods and N identical agents. In each period, the agent consumes two goods $c_{At}$, $c_{Bt}$ where $ t = 0,1 $ and has the endowments $(e_{a0},e_{b0},e_{a1}...
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1answer
55 views

Why is Deadweight Loss Bad for Society?

I have learned that in a perfectly competitive market in the absence of externalities, taxes will impose a deadweight loss upon society, due to reduced market participation by consumers and producers. ...
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quasilinear utility maximization with non-negativity constraints

I have a utility function: max $U = \frac{c_o^{1-\eta}}{1-\eta} + \frac{\beta}{2} c_{1a} + \frac{\beta}{2} c_{1b} $ subject to the budget constraint $p_0c_0 + p_{1a}c_{1a} + p_{1b}c_{1b} = p_0e_0 + p_{...
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comparing two lotteries

Suppose the prize space (in dollars) is $\mathbb{Z}$ = {1, 2, 3, 4, 5, 6, 7, 8} and consider choices by an agent whose preferences (over lotteries) satisfy the von Neumann-Morgenstern axioms. A risk ...
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Lagrangian in Ramsey model

I struggle to write and solve Lagrangian in the Ramsey model. I have the following individual preferences with a budget constraint: and an additional constraint that is a no-ponzi condition: How to ...
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Market clearing price for a single good with two consumers with different utilities

Say I have a single indivisible good that is randomly allocated to one of two consumers. The first consumer's utility for the good is \$5 and the second consumer's utility for it is $10. If the good ...
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1answer
64 views

Can any three of the four vNM axioms (of expected utility theory) be satisfied without satisfying the fourth?

Is it true that any three of the four vNM axioms (of expected utility theory) can be satisfied without satisfying the fourth? Any examples which support such claim? Basically I'd like to prove that ...
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Utility Maximisation With Infinity

Consider the following utility function. $U(w) = \max x_{1}^{0.25}x_{2}^{0.25}(x_{3} +1)^{0.5} $ $s.t. x_{1} \geq 0, x_{2} \geq 0, x_{3} \in \{0,1\}$ and $1 \times x_{1} +1 \times x_{2} + 1000 \times ...
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How do we show mathematically that the lump sum principle does not apply to perfect complements?

I want to show mathematically that the lump sum principle does not apply to perfect complements. I was able to show it applied with a specific Cobb-Douglas utility function, but I am not sure how to ...
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Log Linearising CES demand

I have been trying to log-linearise the demand function that follows from a standard two-good CES-utility maximamalisation problem. That is: Maximise \begin{eqnarray} U(h,c)= \left(G_1^{\rho}+ G_2^{\...
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32 views

Failure of 1st welfare theorem with non-increasing utility function

I want to find an example that utility function is not increasing but still satisfies that $u(x) \geq u(y)$ for all $x \geq y$ and 1st welfare theorem fails. I want to prove this on edgeworth box not ...
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1answer
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How does this imply that a Pareto optimum maximizes a weighted average of utility functions?

I'm reading a passage from Asset Pricing and Portfolio Choice Theory by Kerry Back, and I don't understand some of it. I would appreciate any help anyone could provide me. In the passage, Back is ...
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Risk seeking investors: Do they benefit from return at all?

Starring on the indifference curves for different types of risk preferences I ask myself why the risk seeking investor would be indifferent between an investment on the far left of his indifference ...
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Von Neuman-Morgenstern utility theorem apply only to linear utility functions?

Does the Von Neuman-Morgenstern utility theorem apply only to linear utility functions? If yes, what extension of this theorem of which theorem needs to be applied to use more specific utility ...
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56 views

Calculate hicksian demand with utility function (with restriction)

$U(x_1, x_2) = 1/2 * x_1 $ I am trying to calculate the Hicksian demand when when $U(x_1, x_2) = 2$ and the value of the minimum expenditure when $p_1 = 9$ and $p_2 = 16$ For the hicksian demand I ...
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Calculate income and sustitution effect from utility funcion

Utility function $U(x_1 , x_2) = x_1 + 4 * x_2 $ $ p_1 = 3, p_2 = 8, m =120 $ $p_2$ changes from $8$ to $10 $ How can I calculate the income and substitution effect. I first thought about calculating ...
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CES utility with negative values

We know that the CES utility function approximates a Leontief utility function as in the following: $$u(x_1,x_2) = (x_1^{-r}+x_2^{-r})^{-1/r}\overset{r\rightarrow\infty}{\longrightarrow}\min\{x_1, x_2\...
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Why are utility functions typically assumed to be concave?

Why is it usually required that utility function be concave? Is it because concavity is a necessary (or sufficient?) assumption for a unique equilibrium? Can someone please spell this out for me? ...
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1answer
77 views

How do you measure the MRS for more than two goods?

For two goods, the MRS is defined as the amount of one good you would trade off for the other. Mathematically, $$\text{MRS} = \frac{dy}{dx}$$ where the amount of goods $X$ and $Y$ are denoted by $x$ ...
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Is addiction a case of increasing marginal utility?

I got to know that alcohol addiction is a case of increasing marginal utility. My professor refutes me bluntly stating that once a person starts consuming he no longer remains to be a 'rational ...
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MRS and IC : The language sounds contradictory

The MRS$_{xy}$ is defined as $\left(-\frac{dy}{dx}\right)$ and Nicholson/Snyder (NS) writes it as the amount of $x$ we can trade for $y$ while remaining equally well off. However, the analytical ...
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Question about Marginal Utility [closed]

The following statement is given as true and I cannot make sense why. It is the only information given. "The $MRS$ of flour in 1kg bags for flour in 2kg bags is negative and constant." I ...
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203 views

What is the problem with this opportunity cost example?

The "stand-up economist" Yoram Bauman used the concept of opportunity cost to make the following joke: [S]omebody offers you a choice between a Snickers bar and a package of M&Ms. ...
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39 views

How come a representative consumer with quasilinear utility need not economize?

Suppose a representative consumer has the following quasilinear utility function: U_i(x_1,x_2)=ax_1+ax_2-(1/2)*[(x_1)^2+(x_2)^2] + k where a>0 is a utility parameter, x_1 and x_2 are the goods, and ...
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1answer
67 views

King–Plosser–Rebelo Preferences and Additively Separable

The wiki of King–Plosser–Rebelo preferences says that the utility function has the multiplicatively separable form $$u(C, L)=\frac{1}{1-\sigma_{c}} C^{1-\sigma_{c}} v(L)$$ and "in the limit case ...
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2answers
58 views

Understanding utility function curve and marginal rate of substitution

This example appears in a different question, but there is something I don't understand. Maybe this question is better suitated for algebra stackexchange. John’s utility function for food (f) and ...
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1answer
51 views

Proof that $U(\sum_{n=1}^{N}{p_nL_n})=\sum_{n}^{N}{p_nU(L_n)}$

I understand the expected value of a lottery is $\sum_n^N{p_nL_n}$ where there are $N$ possible outcomes, each with a probability $p_n$ with $n=1,...,N$ and $\sum_{n}p_n=1$ (that's rather trivial I ...
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How are weak preferences different to strict preferences/indifference?

Given a utility function $u(\cdot)$ and two bundles $x$ and $y$. Assuming $u(x)=u(y)$. I am to prove or disprove that $x \succcurlyeq y$. Now I'm confused by this. We say $x$ is strictly preferred to $...
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Fundamental question on marginal utility

I was just thinking back to some introductory economics courses, but now I'm extremely confused on a fundamental concept. How is marginal utility interpreted as the additional "happiness" ...
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Demand for minimum of $4$ different goods

The consumer has the utility function with $4$ goods $$U=\min\left \{ \sqrt{x+y},z+w \right \}$$ The prices are $p=(3,2,2,1)$ with wage $m=1$. Find the demand. So far I have observed that it is ...
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1answer
34 views

Adding scaled CRRA utility

Suppose our utility function is the usual CRRA utility with $\gamma=2$ so that: $$u(C) = \frac{C^{1-\gamma}}{1-\gamma} = -\frac{1}{C}$$ Now suppose there are 2 goods, A and B, available for ...
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Derivative of $x_1^S(p_1, p_2, \overline{x}_1, \overline{x}_2) \equiv x_1(p_1,p_2,p_1\overline{x}_1 + p_2\overline{x}_2)$ to derive Slutsky equation

Why is the partial derivative of $x_1^S(p_1, p_2, \overline{x}_1, \overline{x}_2) \equiv x_1(p_1,p_2,p_1\overline{x}_1+p_2\overline{x}_2)$ for $p_1$ $$ \frac{\partial x_1^S(p_1, p_2, \overline{x}_1, \...
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Economic theory journals for a refinement theorem about utility function representation

I would like to ask which are the (mathematical) economics journals that publish papers about economic theory and that focus mainly on the mathematical aspects of it. Let me be more precise. If I have ...
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Unsolveable Demand/Utility Problem?

A consume has a preference relation on $\mathbb{R}^4_+$ with a utility function defined as $$ U(x_1,x_2,x_3)=(\ln(3x_1+2x_2+x_3))^3$$ Find the demand at prices $p=(1,1,1)$ and wage $4$. Attempt I ...
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1answer
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Concept of Utility in demand systems

I have seen that researchers use different utility function in demand systems estimation such as Stone Geary. What is the role of these utility functions? What are utility function other than stone ...
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88 views

Utility function and homogenous of degree zero

I've a utility function which is given by ($x_i$-$b_i$)$^{c_i}$ $\sqrt{x_2}$ . What values of b and c can I input to ensure Homogenous of degree zero in prices and wealth? I think c will be positive....
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266 views

Are the goods in additively separable utility functions normal goods?

Inspired by this answer. To make it a bit more precise, by normal good I mean demand is (not necessarily strictly) increasing in income, and by additively separable utility function I mean that a ...
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1answer
130 views

Expected utility theory (Lottery notation)

A wheel of fortune has outcomes $S=\left \{ 1000,100,50,20,0 \right \}$ as money prices. A consumer has the preferences $$20\sim \left ( \frac{2}{100}\cdot1000 \oplus \frac{98}{100} \cdot 0 \right )$$ ...
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1answer
179 views

Question about Social Welfare Function and Social Profile

What are the meanings of a social welfare function and social profile? How are they related?
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1answer
57 views

Derive demand function from utility [closed]

Never encountered such a problem as I am new. $$U(x_1,x_2)=(a\ln(x_1)+b\ln(x_2))^n$$ and $a,b,n>0$ with income $w>0$ and prices $p_1,p_2>0$. Find the demand function. Attempt I am thinking ...
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78 views

The Price and Demand Index in Homothetic Kimball Utility

Suppose with Kimball preferences, utility $Q$ from consuming $\left\{q_{\omega}\right\}_{\omega \in \Omega}$ is implicitly given by $$\int_{\omega \in \Omega} Y\left(\frac{q_{\omega}}{Q}\right) d \...
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1answer
96 views

The Intuition of CES Utility

Suppose a (symmetric) CES utility function $$U(\mathbf{x})=\left[\int_{\Omega}\left(x_{\omega}\right)^{\frac{\sigma-1}{\sigma}} d \omega\right]^{\frac{\sigma}{\sigma-1}}, \sigma>1$$ 1 The indirect ...
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136 views

Is it true that for Cobb-Douglas preferences, the demand function is always iso-elastic?

As we know that $Q*P=const.$ for Cobb-Douglas preferences, we can thus conclude that $\frac{dQ/Q}{dP/P}$ is always $-1$: $$ QP=const. \implies 0=d(PQ)=Q\ dP+P\ dQ \implies \frac{dQ}{Q}=-\frac{dP}{P} $$...

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