Questions tagged [consumer-theory]
the study of consumer choice and its fundamental underpinnings in preferences and constraints.
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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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2
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Marshallian demand for x^2+y^2
My question is regarding a simple marshallian demand calculation. Given a utility function $u(x,y)=x^2+y^2$ and a budget constraint $p_1x+p_2y=m$. What are the Marshallian demand functions for each x ...
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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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Exponential Income Consumption Curve
From the Engel's Law we know that as income increases the share of income spent of foods decreases and the share of income spent on luxury goods increases. I wanted to represent this using Consumer ...
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Estimating the form of a utility function on two or more commodities
I am looking for experiments for estimating the form a utility function of a consumer on two or more commodities. In particular, I would like to know e.g. if the utility function of a consumer is of ...
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Reservation price and demand curve
Q1: There are $25$ consumers with each demanding 1 unit and each consumers' valuation of the product is $10$. How would the demand curve look like.
Q2: Now suppose there's a monopolist with $MC=8$ and ...
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21
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Consumer optimization problems with multiples cases
Does anyone know of any resources where the Lagrangian optimization of the consumer problem with one constraint has two cases for an answer? For example, when income is greater than x, the optimal ...
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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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Cobb Douglas Indirect Utility/Expenditure continuity proofs
How should I go about proving that the general Cobb Douglas indirect utility function and expenditure functions are continuous? There are many ways to prove continuity, but which would be the easiest? ...
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Intertemporal consumption with heterogeneous/multiple goods
I'm currently trying to build a CGE, and I'm stuck at the household's problem which is about intertemporal utility maximisation. The household consumes multiple heterogeneous goods $C_i$ (I'm limiting ...
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Consumer theory with subproblem
Say the agent's problem is
$$\max_{c,\{h\}, N}\{U(c, v(\boldsymbol{h} ; \boldsymbol{\theta}))+\lambda(w N-c)\}$$, subject to $\sum_{i=1}^{I} h_{i}+N \leq 1, \quad N \in \mathcal{N}$.
Assume $U(c, v(\...
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$H$ is a constant? Maximizing: $\int _0^Te^{-t}f(x,u)dt$ st $x_t=g(t,x,u)$ and $g$ is independent of $t$
$\max_{x(t),u(t)}\int _0^Te^{-t}f(x(t),u(t))dt$,
st derivative $x_t=g(t,x(t),u(t))$. Prove that $H$ is constant.
My try2:
consider the Hamiltonian
$$
H(x(t), u(t)) = e^{-t}f(x(t), u(t)) + \lambda(t) g(...
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Give bundles $x,y\in \mathbb R^n$, there must exist a budget $B\supset\{x,y\}$ and a demand $D(B)\in[x,y]$?
For a problem in revealed preference. Give bundles $x,y\in \mathbb R^n$, must there exist a budget $B\supset\{x,y\}$ and a demand $D(B)\in[x,y]$?
Intuitively, this mean that we have two bundles, and ...
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Paradox of 'more quantity means greater satisfaction' in consumer behavior
I am trying to understand consumer behavior in microeconomics. Consider a market basket of food and clothing. Utility / Satisfaction of 200 gram food + 2 shirts is always supposed to be greater than ...
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Consumption with quantity discount
A consumer has the utility function:
u(x,y) = x*y
his initial budget constraint is: 12 = 2x+ 1y. He has a budget of $12 for the whole question, which has to be completely used.
so that he consumes 3 x ...
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Properties of Consumer Preferences - Monotonicity
Was reviewing topics and I came across this question. I am confused because there is no reference to strict or weak monotonicity in this case. I first thought that monotonicity is violated b/c an ...
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Reasons for why slutsky matrix may be non symmetric
In demand system estimation, theoretically we require this matrix to be symmetric. This unfortunately is not the case most of the time.
What are some reasons for why symmetry of the slutsky matrix may ...
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Convexity preferences
What is the difference between convexity and strict convexity preferences? What is the difference between quasi-concavity and quasi-convexity? And is MRS still true in concave preferences?
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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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Competitive Equilibrium how to determine subject to functions
Consider a 1-commodity, 2-consumers, 2-periods economy with
S = 2, J = 1. The asset pays one unit (of the commodity) in state 1 and 2
units in state 2. q denotes the price of the asset at time 0. The ...
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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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Logarithmic Utility function Algebra
Question:
I'm told the following (by an exam mark scheme):
Using $a + b =1$
$a[ln(\frac{am}{p_1})] + b[ln(\frac{bm}{p_2})] = ln(m) - aln(p_1) - bln(p_2)$
I can't get this to hold without the ...
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Does an "optimal" MRS exist?
I was reading a case study in Hal Varian, where the author talks about essentially a surge pricing mechanism for incentivizing households to consume less electricity during peak hours (so as to not ...
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Preference relations based on Varian
I understand that there is no universally agreed terminology for preference relations. However I need to pin down a definitive way to think about them (both for my exam, and my own sanity). Please can ...
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Existence and uniqueness of demand, and symmetry implies equal demands given equal prices
Encountered the following problem during self study:
My take on the problem is that if we can show that the equation of the income expansion path is $x_1=x_2$ for all such $U(x_1,x_2)$ then we have ...
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Conflicting Definitions of Weak Monotnocity (preferences)
Strong Montonicity my sources seem to agree on Strong monotonicity, i state equivalent definitions below. But weak montonicity i keep finding what appear to be conflicting definitions. In the ...
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Homothetic Functions and Monotonic Transformations
Using the following definition of a homotheic function (taken from my Mathematical Economics course pack).
A function $f: \mathbb{R^{n+}} \to \mathbb{R}$ is homothetic if it has the form:
$f(x,y) = q(...
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Homotheic Function Definitions
There are a number of different definitions of Homothetic functions i have come across. I have used each of them to prove that a function $f(x, y) = x^a y^b$ with $a+b > 0$ is homothetic. But i ...
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Slutsy Equation and Income effect
Does the Slutsky equation always assume optimal levels of our variables, hence Marshellian demand = Hiscksian demand $x = h$ as indeed this is how the Slutsky is derived?
I originally thought this ...
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Convex Combination of pairs of points
Is it appropriate/meaningful to write vector/points $(a,b) \le (c,d)$, where i would mean component wise each component is $\le$
Specifically is my example below with reference to concavity ...
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What pricing strategies does Amazon use and how do they affect consumers' purchasing decisions?
As a frequent Amazon customer, I have noticed that the prices of products I am interested in buying often fluctuate over time. These changes could either be an increase or decrease in price, and I ...
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Testing for Concavity - Local Maximum & Global Maximum
My question is under which contexts Negative Definitness (ND) vs Negative Semi-Definitness (ND) is required for classifying a global maximiser. And also Global vs Local.
I also want to understand what ...
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How does a lack of incentive to purchase new stuff affect economics?
How does a lack of incentive to purchase new stuff affect economics?
I've perceived as if a lot of economics is rooted in the idea of continued innovation and consumption, but then I've realized that ...
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CES in Slutsky matrix (weird results)
We have a Slutsky matrix:
\begin{bmatrix}
\partial x_{1}^H/\partial P_1 & \partial x_{1}^H/\partial P_2 & \dots & \partial x_{1}^H/\partial P_n \\
\partial x_{2}^H/\partial P_1 &...
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The formula for expansion path
Is there a way how to precisely compute the expansion path?
I know a consumer's utility function $U(\boldsymbol{x})$, I know the budget constraint $\sum P_i x_i \leq M$, I am able to compute the ...
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Prove strict monotonicity of utility function
I have the following utility function: $$ u(x_1, x_2, x_3) = med(x_1, x_2, x_3) $$ Given that $UMG_{i}$ ≥ 0, the utility function represents a strictly monotonic preference. Does this assertion make ...
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Proving that strict convexity is violated
I am given a utility function $u(x)=x_1^2+x_2^2$ and I am asked to see whether this function satisfies strict convexity. The answer is saying this:
We see that $u(3,0) = 9$, $u(0,3) = 9$, $u(1.5,1.5) =...
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Does local non-satiation hold for this problem?
I am getting some confusing results solving this problem:
$max_{c_0\geq0, c_1\geq0} \bigg\{EU = R(1-c_0) [p t_1 + (1-p) c_1^{-2} t_2] \bigg\} ~ s.t. ~c_0+c_1 \leq 1$
where $p$ is the probability of $...
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FOC greater than 0
I couldn't get my head around this part. Basically, I have to prove that a consumer has to hold a positive amount of assets, i.e. $x > 0$.
A hint suggested to find take the FOC, and then set $x = 0$...
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Composite good and preferences
Usually in economics, we could see some versions of multiplicative utility:
$$U(\boldsymbol{x}) = x*y$$
The thing is that most of the time an additional statement is given that $y$ is some composite ...
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How to create a composite good?
Let's say I would like to create some composite score for multiple of goods...
EDIT: More concise version based on @BrsG comments... I would come up with the following scenario. I have a consumer with ...
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Do Aggregated consumers make sense?
Aggregated consumers as a biased concept (in case of cross-price elasticity)?
I try to approach aggregated consumption data as if it was a new consumer (similarly to approaching average data as if it ...
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1
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Solving Lagrangian FOCs: a few difficulties
I have an optimization problem from microeconomics that yields me the following first-order conditions based on a Lagrangian:
$ p_1 = \lambda \qquad(1)$
$ p_2 - \lambda (x_2^2+x_3^2)^{-1/3}x_2=0 \...
2
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1
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Why do we not stick to utilities in calculating supply and demand?
Common microeconomics models give that MC must equal MR in the optimal position for the consumer, therefore, the marginal utility must equal its price. But this is where a mistake has been made, what ...
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Expenditure min problem
The typical expenditure min. problem wants to minimize expenditure under the constraint $u(x) \ge u^{\ast}$.
Why the solution of this problem is such that $u(x^{\ast})=u^{\ast}$ and not $u(x^{\ast})&...
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Expected utility maximization question
If the utility function of an individual is $u(w) = 10 \sqrt{w}$ and the individual starts with $w = 100$ (where $w$ denotes the wealth available to him). If he buys a lottery that costs him $51$ and ...
user43302
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2
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199
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Equivalent of shephard's lemma in consumer theory
I'm studying micro from the Mas-Colell, and I'm trying to understand the proof 2 of proposition 3.G.1. It is about proving that the derivative of the expenditure function w.r.t. the price of a ...
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2
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217
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Why do we need Complementary Slackness Condition for Karush-Kuhn-Tucker Conditions
Complementary slackness condition (CSC) state that
$\lambda_j[g_j(x) − c_j] = 0 \hspace{5pt} \text{for} \hspace{5pt} j = 1, ..., m.$ Therefore, every constraint either needs to be an equality ...
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Example of consumer preferences that switches from being concave to being convex
Question
Is there an example of consumer preferences over consumption bundles $(x,y)\in \Bbb R^2$ that would be concave when $x$ is abundant relative to $y$ and convex otherwise?
Are there known ...
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Nonlinear budget constraints (for quantity discounts)
I was thinking about quantity discounts and if there is a possibility to model them not as bundles (as is typical for second price discrimination) but rather as prices being some continous functions ...
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