Questions tagged [utility]
Utility, or usefulness, is the (perceived) ability of something to satisfy needs or wants.
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If utility function is homogenous of order 1, then partial derivatives of demand function are equal
Prove that if $U(\alpha x)=\alpha U(x)$, then $$\frac{\partial x_i(p,w)}{\partial p_j}=\frac{\partial x_j(p,w)}{\partial p_i}$$ for any $i$ and $j$.
I've proved that $x(p,\alpha w)=\alpha x(p,w)$, but ...
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Showing UMP and EMP do not exhibit duality if the assumption of local non-satiation is absent
I have been trying to use the contradiction method to prove this, but it does not seem to be working.
Suppose $x^*$ is optimal in both EMP and UMP. Then $u(x^*) \geq u(x')$ for all $x'$ in $B_pw$.
And ...
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Differentiation of a composite utility function
Taking the utility function $U: ln(c_1)+\beta ln(c_2)$ with $c_t=\sqrt{c_{Nt}c_{Tt}}$, t indicating a time period 1,2. Ignoring any b.c.'s.
I am having issues interpreting the marginal utility wrt $c_{...
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Understanding consumption units normalisation by $u^\prime (c) $
It is frequently noted that if utility is given by $u(c)$, then the object
$$ \frac{u(c)}{u^\prime (c)} $$
puts the utility in consumption units via the normalisation by $u^\prime(c)$. What is the ...
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Log_linearization
Can any one please suggest me how to log-linearize the utility maximization of the Ricardian households in a closed economy DSGE model. The equation is bellow :
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MCQ on income and substitution effects
Person A spends their income only on bread and cheese. Given a rise in the price of bread leaving income constant and the price of cheese constant, the consumer consumes less bread and less cheese. ...
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Change in Hicksian Demand of an Inferior Good when changing Utility
How can you rigorously show that Hicksian demand for an inferior good will decrease when utility increases?
Thanks,
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How the saving rates were derived in Azariadis (1996)'s Impatience Trap?
The household maximizes his lifetime utility function according to
$$
\max_{c_1, c_2} v(c_1, c_2) := \frac{1}{\beta(c_1)} \log c_1 + \log c_2 - A(c_1, c_2)
$$
subject to
$$
\beta(c_1) = \begin{cases}
...
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Basics of MRS. Interpretation and constant utility
I am sure this question has been asked before but I didn't know what to search for since its a quite long question.
Say, I have a utility function U = sqrt(x)*sqrt(y) and the MRS = y/x.
Say I have y = ...
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Why is monotonic preference and monotonic utility function non-decreasing?
Obviously a monotonic function can be either nondecreasing or nonincreasing:
However, in Economics, a quick google search gives:
I am interested in the history or the motivation behind the econ ...
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Example to Demonstrate that with Uncountably Infinite Outcomes We do not Have A Representing Utility Function [duplicate]
Suppose we define $\Omega$ = $\mathbb R_+$. The preference relation $\succeq $ is defined as $$(x_1, x_2)\succeq(y_1,y_2)\iff x_1>y_1 \text{ or } [x_1=y_1 \text{ and }x_2\geq y_2]$$
where $x_1, x_2,...
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The sufficient condition for unique interior solution in utility maximization problem
Suppose the utility function is continuous, differentiable, strictly increasing and strictly quasiconcave. Whether the utility maximization problem has unique interior solution? If not, is there any ...
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Quasiconvexity of the indirect utility function for Cobb-Douglas utility
I've recently started Mas-Colell's, Green's and Whinston's Microeconomic Theory. In section 3.D, the authors define the indirect utility for a price vector $p$ and wealth $w$ as the utility derived ...
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Proving Expected Utility Theorem
I am struggling to understand the proof of the second step in the Expected Utility Theorem, particularly the part that deals with preferences over weighted sums of lotteries. The statement I am trying ...
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In Debreu's representation theorem of ordinal utility, is the assumption of "second countability" necessary?
Debreu's representation theorems
Debreu 1959 states that: second countability, continuity, and weak ordering sufficiently implies the existence of real (continuous) utility function. The second and ...
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How to get a Certain Consumption Equivalent using Epstein-Zin preferences?
In many asset pricing models we use CRRA preferences and Epstein-Zin preferences.
Let's say I have an agent that lives $T$ periods with CRRA preferences:
$$ V_0 = \sum_{t=0}^{T} \beta^t \frac{C_t^{1-\...
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How to calculate direct utility from indirect utility in this exercise?
A consumer has an indirect utility function given by
$$v(p_1, p_2, R)=\frac{R}{p_{1}+p_2}$$
Where $p_1$ and $p_2$ denote the prices of the two goods consumed by the individual and $R$ the income.
How ...
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How do I determine the degree of substitution affect the change in per capita consumption of a good?
Suppose there are two people living together. They have the same utility function (CES)
$$u(x,h)=(x^\rho+h^\rho)^\frac{1}{\rho}$$
where $x$ is a type of good, $h$ is housing. The price of good is $p$, ...
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How to approach this problem of utility?
An individual has the following utility function: U(x, y) = 2xy + 3y, where x and y represent the quantities consumed of goods X and Y, respectively. The budget constraint is given by R = px + qy, ...
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How to find the indirect utility function and the expenditure function through this interesting utility function?
The utility function seems innocent: $u(x_1,x_2)=(x_1^2+x_1)x_2$. I want to find the indirect utility function and the expenditure function, but I've encountered some problems. Here's what I've got:
$$...
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How do I use total derivatives of an implicit function to solve this problem?
Suppose a consumer has a utility function $u(x_1,x_2)$, where $u_1>0, u_2>0, u_{11}<0,u_{22}<0,u_{12}=u_{21}>0$. The prices are fixed. The consumer's income comes from working and the ...
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Quasiconvex and quasiconcave utility function
I saw that the model of the quasi-convex utility function is similar to the concave utility function and also the quasi-concave utility function is similar to the convex utility function. How can ...
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Utility Maximization of a quasi-linear utility function
I am dealing with a quasi-linear utility function. For example
$U=(x_1x_2)^{0.5}+cx_3$ with constrain $w\ge x_1+2x_2+px_3$.By taking c, w and p as constant, I function that by using Lagrange ...
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How do I calculate total utility under discounted utility framework
Suppose we are given an instantaneous utility function and discounted rate with budget constraint. Using the exponentially discounted utility function, we can find the optimal consumption stream of a ...
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Why is the indirect utility function state that higher indirect indifference curves has lower utility when it assumes homogenity in prices and income?
the indirect utility function states that higher indirect indifference curves carry lower utility. however it also assumes homogenity of degree zero in price and income. how?
please explain it to me ...
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Applying dynamic programming to constrained utility
I am trying to solve problem that looks like this; there is utility function that takes $x$ and $y$ as inputs, $x$ is produced by production function that depends on labor $l+y=1$. $x, y$ depend on $t$...
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Does this utility function work?
I'm reading over some models and I found a paper in progress that uses this model
$$U_i(c_i, n, \theta_i) = \log(c_i) + \log(n) + \theta_i$$
where $c_i \text{ and } n \geq 0$ , but when you consider ...
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Can von Neumann Morgenstern utility be negative?
I saw this question in stackexchange that a utility function can take negative values.
I did not read any thing from the axioms that utility cannot be negative. It states they must be in the set of ...
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Intertemporal Utility Optimization For Multiple Goods
I'm building an economic simulation game and I'm trying to solve for the values that a person will spend on each good and the amount they will save in the current period, taking into account all ...
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Understanding the notations in Bayesian game definition
I am having trouble understanding the definition of a Bayesian game based on the following definition from class. I would appreciate it if you could explain the notations and overall meaning for point ...
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What are the real world consequences of diminishing marginal utility?
I've just started a microeconomics course and the instructor has talked about diminishing marginal utility. I understand what it is, but I'm struggling to understand why it matters. What are the real ...
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Why is the Hicksian form of the CES demand used in CGE model forms rather than the Marshallian
I am curious to know why the Hicksian form of the CES is used in CGE models rather than the Marshallian form. I have a few hypotheses, but I am not sure which one is correct. If any?
Hypothesis 1: In ...
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labor leisure model, quasilinear preferences
There is a quasilinear utility function $u= (1-t)wl - p(l)$, where $l$ is labor supply. I don't quite understand what happens, if the budget changes (due to $w$ or $t$) since it is quasilinear. Does ...
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Sufficient conditions for connectedness of indifference sets of a preference relation defined on a compact and convex set only
Let $\succsim$ a complete, reflexive and transitive binary relation defined on $X$, a non-degenerated (i.e not identical to a singleton) convex compact subset of $\mathbb{R}^n_{++}$ (the set of n-...
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What does the Arrow-Pratt risk aversion measure means in the deterministic case?
What type of preferences that are not related to risk aversion can the Arrow-Pratt measure of absolute (or relative) risk aversion model?
So far, it seems to me that low RRA/ARA preferences imply that ...
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Find the set of Pareto efficient allocations. $U_1 = -|x_1-2|$ and $U_2 = −|x_2 − 8|$
A professor has 20 hours to allocate between two PhD students. Let x1 and x2 be the time allocated to the two students. The utility of each student is as follows:
$U_1 = −|x_1 − 2|$ and $U_2 = −|x_2 − ...
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How to derive a utility function using indirect utility function?
I am working on this question to solve for utility function.
The indirect utility function is given as follows:
$V(p_1,p_2,w)=(\frac {1}{p_1} + \frac {1}{p_2})*w$
w stands for income, $p_1$ and $p_2$ ...
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Proof of First-Order Stochastic Dominance with Riemann Sums
Let $A = \mathbb{R}$. For $p,q\in \mathcal{L}(A)$, $p$ first-order stochastically dominates (FOSD) $q$ if $F_p(a)-F_q(a) \leq 0, \forall a\in A$.
Show that $p$ first-order stochastically dominates $q$ ...
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Proof of the Lucas' Cost of Business Cycles
I am trying to derive the parameter used by Lucas to measure the cost of business cycles, namely:
derived in the paper "Macroeconomic Priorities".
I already searched in several papers but I ...
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Is there a name for relative diminishing returns, or relative increasing opportunity cost?
Consider the following hypothetical situation:
You are producing something where the total amount produced, $A$, is equal to the product of two factors, $X$ and $Y$. Hence $A=X*Y$.
$X$ and $Y$ both ...
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Calculate influence of absolute risk aversion on consumption decisions
Say I have the following setup:
A consumer chooses between two goods $x$ and $y$ (a numeraire) such that she maximises:
$$V(x,y)=u(x)+y$$
Under the constraint that her revenue $R$ is such that:
$$R\...
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Euler Equation - small open economy
How would I find the Euler equation for:
$$ U=\sqrt{C_1}+\beta{\sqrt{C_2}} $$, where $$\beta=1/(1+r)$$
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Hessian Matrix Test - When does it fail?
When does the hessian matrix test fail. I understand we are testing the definiteness of the Matrix, and i also understand that because it's a symmetric $n•n$ matrix, we have a principal minor ...
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Minimisation problem turned into Maximisation
My course always converts minimisation problems into maximisation. They given the following reason as outlined in the problem below.
$Min\; P_xx + P_yy \; s.t. \; u(x,y) \le x^{\frac{1}{2}} + y$
&...
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Arguments for Concavity or Quasi-concavity
I'm faced with questions that want me to show that a utility or production function is either concave, or if not then quasi-concave so that we can apply the KKT conditions.
For example the production ...
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Perfect Complement Utility Function Maximisation
When we have the function $U(x_1,x_2) = min\{x_1,3x_2\}$
S.t. $p_1x_1 + p_2x_2 = m$
What's the economic, and mathematical intuition for assuming this constraint is binding, i.e. not having to ...
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If kink, is the agent risk-neutral?
If the graph has a kink, at that point would we assume the agent is risk-neutral?
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How does concavity imply that the marginal utility of a gain is less than the difference between utilities?
Considerin turning down a 50-50 gamble of losing \$ 10 and gaining \$ 11 at wealth w.
$1/2*u(w-10)+1/2*u(w+11)≤u(w)$
Concavity implies that $u'(w+11)*11≤u(w+11)-u(w)$ and
$u(w)-u(w-10)≤u'(w-10)*10$
...
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Is there a way to assess land value independently of zoning policy?
I am aware that the methods used to assess land value of properties, used by government agencies for taxation, are affected by zoning restrictions. Restrictions on how a lot of land may be used by it'...
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Linear utility and interest rate
Consider the following demand side of the economy
Why the subjective discount rate is also the interest rate if we have linear utility?
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