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6

The only utility function that comes to mind is the Stone-Geary utility function. For 2 goods, $x$ and $y$, this takes the form: $$ u(x,y) = (x - a)^\alpha (y- b)^{1- \alpha}. $$ This is a Cobb-Douglas type of utility function where $a$ and $b$ are subsistence levels, i.e. you need to consume at least $a$ from $x$ and $b$ from $y$ to survive. It is the ...


5

Note first that the function $r\mapsto A(r)^{\mu/p}$ is strictly increasing on $\mathbb{R}_+$. When you look at the minimum in the definition of quasi-concavity, you can therefore ignore this part and it suffices to show that the function $$(x_1,x_2,\ldots,x_n)\mapsto a_1x_1^p+a_2x_2^p+...+a_nx_n^p$$ is quasi-concave. Actually, the function is even concave. ...


3

I think you are a bit confused. The game has the following structure. nature draws $c_1, c_2 \sim U[0,1]$, which is only revealed to the sellers and not to the buyer. The buyer proposes a price $p \in [0,1]$ Sellers decide to sell the good at price $p$. The good is sold if at least one seller agrees with the price. You need to solve this using backwards ...


2

$π = φ^{(σ−1)}*\frac{f_E+f_D}{φ^{(σ−1)}*(1+τ^{(1-σ)})}+φ^{(σ−1)}\frac{f_E+f_D}{φ^{(σ−1)}*(1+τ^{(1-σ)})}-f_E-2f_D$ $π = 2\frac{f_E+f_D}{(1+τ^{(1-σ)})}-f_E-2f_D$ $(1+τ^{(1-σ)})π = 2(f_E+f_D)-(1+τ^{(1-σ)})f_E-2(1+τ^{(1-σ)})f_D$ $(1+τ^{(1-σ)})π = 2f_E+2f_D-f_E-τ^{(1-σ)}f_E-2f_D-2τ^{(1-σ)}f_D$ $(1+τ^{(1-σ)})π = f_E-τ^{(1-σ)}f_E-2τ^{(1-σ)}f_D$ $(1+τ^{(1-σ)})π = ...


3

You question can be answered using a revealed preference argument. Let $B = \{q \in \mathbb{R}^n_+| p' q \le m\}$ be some budget set of a consumer (i.e. $B$ gives all possible bundles that the consumer can choose). Let $q^\ast$ be the optimal choice from $B$, i.e. the bundle that optimizes the utility. Then for any other bundle $q \in B$, it must be that $u(...


4

1. The first question is based on a misreading of the text. First of all it is quite correct that under the model assumptions stated: $$(A) \ \ p''(d) = \frac{t(y)H'(y) - t'(y)H(y)}{H(y)^2} y'(d),$$ however there is no problem because the book does not state that $$(B)\ \ p''(d) = \frac{t(y)H'(y) - t'(y)H(y)}{y'(d)}$$ the book instead states that the second ...


4

I obtained the definition of the principle of variable proportion from this site It states: ... means that upto the use of a certain amount of variable factor, marginal product of the factor may increase and after a certain stage it starts diminishing. The following is a technical derivation. See the picture at the end for an illustration of the various ...


2

Although there already is an accepted answer, there is another way to see the global optimality - or rather the same way with a different formulation. By construction, $$\frac{\partial \pi}{\partial b}(b,x) = - G((\beta)^{-1}(b)) + (x-b) \frac{G'((\beta)^{-1}(b))}{(\beta)'((\beta)^{-1}(b))}\Bigg{|}_{b=\beta(x)}= 0,$$ where $\frac{\partial \pi}{\partial b}(b,...


4

In general the demand for a certain good (say from a consumer) can be written as a function of the prices of all available goods and the total amount of money that the consumer has available. Take the setting of two goods, $q_1$ and $q_2$ with prices $p_1$ and $p_2$ and total income $y$. Then the demands can be written as: $$ q_1 = d_1(p_1, p_2, y)\\ q_2 = ...


4

Let's look at the use of monotonic transformations of utility functions (which I guess is the most frequent occurrence of this concept in econ). Let $u: \mathbb{R}^n_+ \to \mathbb{R}$ be a utility function. We say that $g: \mathbb{R}^n_+ \to \mathbb{R}$ is a monotonic transformation of $u$ if for all $x, y \in \mathbb{R}^n_+$: $$ u(x) \ge u(y) \iff g(x) \ge ...


0

In general, a demand function (for a certain good) gives the demanded quantity as a function of the price of the good and a set of other variables. These other variables include things like prices of the other goods, income, weather, etc. If we consider a framework of two goods and if we excluding all other variables, except for the prices this gives two ...


3

The idea is to consider the hyperplane tangent at the indifference curve through $x^t$ as a "linear budget". These linear budgets have to include the set $\overline{y^t, z^t}$. Then making use of these these linear budgets, we can leverage the results of Matzkin & Richter (JET,1991, Testing strictly concave rationality) to obtain a strictly ...


3

Take a dataset $D = (B^t, x^t)_{t \in T}$ such that for all $t$, $x^t \in B^t$. I'll say that $D$ is rationalisable by the utility function $u$ if for all $t$ and all $x \in B$: $u(x^t) \ge u(x)$. If you only impose convexity on $u$, then any dataset $D$ is rationalisable by the constant (convex) utility function $u(x) = k$. So this has not testable ...


5

$$ G(z) (z-x) = \int_x^z G(z) dy $$ and since $G$ is increasing on $[x,z]$, the right hand side is larger than $\int_x^z G(y) dy$.


2

I think that you need to also assume that the utilities are non-decreasing in the goods. That is, if $x'_1>x_1$, then $u(x'_1,x_2)>u(x_1,x_2)$. The utility functions having the "same preferences" means $\forall (x_1,x_2), (x'_1,x'_2): u(x_1,x_2) > u (x'_1,x'_2) \text{ iff } v(x_1,x_2) > v(x'_1,x'_2) $ So let's assume $u(x_1,x_2) > u (...


2

In your maximization problem the resident chooses the level of pollution $u$. If he can really do this then the maximization problem has no solution, since $U$ becomes infinite for $u\rightarrow 0$. I guess actually the resident doesn't choose the level of pollution, which rather seems to be an exogenously given parameter here. But then $(x^*,k^*)$ doesn't ...


1

This is probably an old question. However there is a book on PDE's and game theory. Game Theory and Partial Differential Equations, Pablo Blanc and Julio Daniel Rossi. If you found other books on this matter, I would like to know as well.


0

From the buyer's point of view, the "best" supplier is the one who offers them the highest utility. Using the utility function for scoring suppliers seems quite natural then. (In this paper the utility function is assumed to be quasilinear in money, which is just for simplicity.)


4

Establishments are business locations (source). The real estate industry does not include construction (under either NAICS or SIC). It’s not really possible to tell what exactly would be included without more detail on the specific industry group you are looking at; it could be anything from realtors’ offices to apartment rental buildings, or both+other ...


2

This has nothing to do with any specific model. For any event $A$, let $I_A$ be the indicator function such that $I_A(\omega)=1$ if $\omega\in A$ and $I_A(\omega)=0$ if $\omega\notin A.$ Then $\mathbb{P}(A)=\mathbb{E}[I_A],$ and here the expectation is given in terms of a density function.


2

The short answer is no. Reveal preference only determines which combination of goods is preferred, it does not affect in any way the allocation of goods, which has more to do with the functional form of the utility function, the types of goods and the types of changes made. For example, the 2 budget lines represent different budget sets when there is a ...


0

I like Besanko, because he provides a lot of examples and good practice problems. I have used this textbook and referred it to a lot of the people I tutor. Below is the 5th edition in pdf. https://economics-pr.weebly.com/uploads/4/8/6/0/48608947/microeconomics-david_besanko_ronald_braeutigam-wiley____2013__zzz.pdf


0

In general, what you are referring to is what we call numeraire pricing. This means that the prices, as you mentioned, are relative to one of the goods (you can pick any good as a numeraire, which is how we measure prices through the whole economy). The reason for this is Walras' Law states (basically) that since we have a bunch (say $N$) goods in the ...


3

It is this probability that is consistent with shares trading for their expected value. The expected value is given by: $$E(S_1+S_2)=E(S_1)+E(S_2)=p_1 \times1+p_2 \times1=p_1+p_2$$ where $S_i$ is the payment of share $i$, $p_i$ is the probability of payment of share $i$ (due to the sales figure falling in some range). So if the expected value is to equal the ...


3

A payoff pair $(x,y)$ is Pareto efficient if it is not Pareto dominated. This means that there does not exist another payoff pair $(x', y')$ such that $x < x'$ and $y < y'$. (Depending on the exact definition, sometimes of of the two inequalities can be weak). All payoffs in the payoff matrix except for $(C,\beta), (B, \beta)$ and $(C, \delta)$ are ...


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