Skip to main content
5 votes
Accepted

Challenging question on mathematical economics

What I say below in my answer is what I can say without having read Acemoglu and Autor's article (unfortunately I haven’t it), an answer based on what you report in your question, of course. I ...
BakerStreet's user avatar
  • 3,887
4 votes

Quasi-linear microeconomics problem

Here is the utility maximisation problem: \begin{eqnarray*}\max_{(x_1,x_2)\in\mathbb{R}^2_+} & x_1+x_2^\beta \\ \text{s.t. } & p_1x_1+p_2x_2\leq M\end{eqnarray*} where $p_1>0, p_2>0, M\...
Amit's user avatar
  • 8,966
3 votes

bayesian nash equilibrium cournot game

Given the demand function: \begin{eqnarray*}P(Q) = \max(60-Q,0) \end{eqnarray*} where $Q=q_1+q_2$. Cost function of company 1 is \begin{eqnarray*}c_1(q_1)= \begin{cases} 18q_1 & \text{with prob. } ...
Amit's user avatar
  • 8,966
3 votes

Proof Mean with a Smaller Sample Size is a Mean Preserving Spread

The following fact is key. Since $\bar{Y}_n=n^{-1}\sum_{i=1}^n Y_i$, we can take expectations conditional on $\bar{Y}_n$ and use that $\{Y_i\}$ is an i.i.d. sequence to obtain $$\bar{Y}_n=\mathbb{E}[\...
Joseph Basford's user avatar
3 votes

Derivation of the Ideal Price Index

The price index is given by $c(p_L, p_H,1)$ which is the minimal cost of producing 1 unit of output. It is given by: $$ \min p_L Y_L + p_H Y_H \text{ s.t. } \left[\gamma_L Y_L^{(\varepsilon -1)/\...
tdm's user avatar
  • 12.3k
3 votes
Accepted

Question About Proving $\alpha(\cdot)$ is Continuous in the Proof of Proposition 3.C.1 from MWG

The sequence you proposed $0,1,0,2,0,3,\ldots$ i.e. $x_n=\begin{cases}0 &\text{if $n$ is odd} \\ \frac{n}{2} & \text{if $n$ is even} \end{cases}$ is not a bounded sequence and is therefore, ...
Amit's user avatar
  • 8,966
3 votes
Accepted

Equivalence of two definitions of monotone preference

Let $\succeq$ be a relation on $\mathbb{R}^l_+$ such that $x\gg y$ implies $x\succeq y$ for all $x,y\in\mathbb{R}^l_+$, and such that all upper contour sets are closed. Then $x\geq y$ implies $x\...
Michael Greinecker's user avatar
3 votes

Do standard consumer theory axioms rule out corner solutions?

Here are some examples of preferences that satisfy (1) completeness, (2) transitivity, (3) continuity, (4) non-satiation, and (5) strict convexity of the indifference curves, along with the ...
Amit's user avatar
  • 8,966
3 votes
Accepted

Do standard consumer theory axioms rule out corner solutions?

A textbook example of a utility function with frequent corner solutions is perfect substitution, e.g. $U(x,y) = x + 2y$. This, however is not strictly convex. But if you think about it, the corner ...
Giskard's user avatar
  • 29.3k
2 votes
Accepted

Do multiple equilibrium points make sense?

Nonlinearity per se is not an issue. As long as demand is decreasing and supply is increasing in price, there will be at most one equilibrium point. However, if e.g. supply is "backwards bending&...
VARulle's user avatar
  • 6,964
2 votes

where does the condition of aggregate demand can be written as function of aggregate wealth come from

Consider a change in the distribution of wealth $dw_i$ that leaves aggregate wealth $\sum_i w_i = W$ unchanged: $$ \sum_i dw_i = 0, $$ Let $X(p, w_1, \ldots, w_N)$ be aggregate demand. Now assume this ...
tdm's user avatar
  • 12.3k
2 votes

Defining Intensive Margin and Extensive Margin

"Extensive margin" is "whether you work at all, or don't work". "Intensive margin" is "how many hours you work, given that you are working". To estimate the ...
Michael Gmeiner's user avatar
2 votes
Accepted

Properties of Multivariate Frechet Distributions

1 $\begin{aligned} \operatorname{Pr}\left(A_j Z_j \leq z\right)=\operatorname{Pr}\left(Z_j \leq \frac{z}{A_j}\right)=F\left(\frac{z}{A_j}\right) =\exp \left(-\left(\frac{z}{A_j}\right)^{-\theta}\right)...
Alalalalaki's user avatar
  • 2,474
2 votes

Why is the marginal utility of money assumed to be constant in Marshallian Theory of Consumer Behaviour

Unlike the ordinal analysis or the revealed preference approach, where there is no need for a measuring rod, the cardinal utility analysis requires a measuring rod. Money acts as this measuring rod. ...
Pallak Goyal's user avatar
2 votes
Accepted

Proof: Let $\epsilon>0$ and $x'\in\mathbb{R}^L_+$ be such that $\|x'-x\|\geq\epsilon$. Then $\alpha(x')$ belongs to some $[\alpha_0,\alpha_1]$

We don't need $X$ (domain of the utility) to be equal to the set of its limit points to use the sequential characterisation of continuity. In other words, the sequential characterisation of continuity ...
Amit's user avatar
  • 8,966
2 votes

Is CES production representing the average of inputs?

Yes, all this does mean that the production function represents the average of two inputs $L$ and $K$ for different values of $\alpha$, given that $0<\gamma<1$. The key thing to consider here is ...
Pallak Goyal's user avatar
2 votes

Nicholson & Snyder V.S. Varian for teaching?

Varian's Intermediate Microeconomics [...] seems friendlier but a bit sloppy This is good insight IMO, and the proper book will depend on your students' goals and abilities. If they will never study ...
Giskard's user avatar
  • 29.3k
1 vote
Accepted

Independence Axiom and Expected Utility Theorem Proof

Let us first show that the function $u$ is linear in the sense that for all lotteries $L$ and $L'$ and all $\alpha \in [0,1]$: $u(\alpha L + (1-\alpha) L') = \alpha u(L) + (1-\alpha) u(L')$. Let $L \...
tdm's user avatar
  • 12.3k
1 vote

All-pay auction question problem

To supplement Amit's answer, we can get the same answer using Myerson's trick. This trick is "halfway" between Amit's answer and using the Revenue Equivalence Theorem. First, we may consider ...
Joseph Basford's user avatar
1 vote

All-pay auction question problem

For the Bayesian all-pay-auction game above where valuations are private information, let $b:[0,1]\rightarrow \mathbb{R}_+$ be the symmetric equilibrium strategy which is strictly increasing and ...
Amit's user avatar
  • 8,966
1 vote
Accepted

What are the most common axioms to define a strict preference relation?

A relation $P$ is negative transitive if $\neg(xPy)$ and $\neg(yPz)$ imply $\neg(xPz)$. If $\succeq$ is a transitive and complete relation, then the relation $\succ$ defined by $x\succ y\iff(x\succeq ...
Michael Greinecker's user avatar
1 vote

What prices do firms impose on perfect substitutes?

TLDR Perfect substitutes is a very theoretical utility function class; consumers with such preferences generally go 'all or nothing'. Because of this, your rationale describes a very rare equilibrium, ...
Giskard's user avatar
  • 29.3k
1 vote

Computing the competitive equilibrium from the edgeworth box

The picture below presents the competitive equilibrium in different situations:
Amit's user avatar
  • 8,966
1 vote

Looking for an universal utility function

Here is an example where $x_i$s are necessities and will always be consumed in positive quantities and $y$ and $z$ will be consumed in positive quantity only when income of the individual is above a ...
Amit's user avatar
  • 8,966
1 vote
Accepted

Counter example for strong and weak Pareto optimality

Here are some more examples of pure-exchange economies (with 2 consumers and 2 goods) where preferences are continuous and monotone, but set of strongly Pareto efficient allocations is not equal to ...
Amit's user avatar
  • 8,966
1 vote

Pareto Set with strictly convex preferences

Observe that the preferences of $A$ and $B$ can be represented by the following utility functions: $u_A(x_A,y_A)=(x_A+3)(y_A+12)$ $u_B(x_B,y_B)=(x_B+9)(y_B+8)$ For interior efficient allocations, the ...
Amit's user avatar
  • 8,966

Only top scored, non community-wiki answers of a minimum length are eligible