4
votes
Core in a replicated economy
In the economy provided in the question, competitive equilibrium allocations is equal to the set of efficient allocations. This along with the fact that the competitive equilibrium allocations always ...
- 7,513
3
votes
Solow Model in disrectly and continuously
The discrete time law of motion is given by
$k(t+1) = (1-\delta)k(t) + s f(k(t))$
This can be rewritten as:
$$
(k(t+1)-k(t)) = s f(k(t)) - \delta k(t).
$$
Now, take a Taylor expansion of order 1 of $k(...
- 9,757
3
votes
Accepted
Robinson Crusoe with tax
The profit maximisation problems give:
$$
\max_{L_c} p_c \sqrt{L_c} - w L_c,
$$
and
$$
\max_{L_f} p_f \sqrt{L_f} - w L_f
$$
This gives the following first order conditions:
$$
\frac{p_c}{2w} = \sqrt{...
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3
votes
Accepted
Consumer problem
We consider the following problem:
$$
\max_{x} u(x) \text{ s.t. } p'x \le w.
$$
Let us show that if $u$ is strictly concave, then the solution (if it exists) is unique.
We proof by contradiction. ...
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3
votes
CRS assumption in solow model
From an economic point of view, the assumption of Constant Returns to Scale can have several reasons, and they are not specific of the Solow model.
I can quote what Solow himself says about Constant ...
- 3,037
3
votes
Minimal assumption for a “certainty equivalence” exists
Here is an example that shows that certainty equivalents need not exist: Let $f:\mathbb{R}\to (0,1)$ be an increasing bijection. Let $0<\alpha< f(1)-f(0)$ Define $u:\mathbb{R}^\mathbb{N}\to\...
- 11.4k
3
votes
Minimal assumption for a “certainty equivalence” exists
I take it that $u: \mathbb{R} \to \mathbb{R}$ and not $u: \mathbb{R}^N \to \mathbb{R}$ (as in the question). Otherwise $u(c)$ for $c \in \mathbb{R}$ does not make sense.
tldr:
if $u$ is continuous, a ...
- 9,757
2
votes
Accepted
About Theorem 1.1 in Game Theory: Analysis of Conflict by Roger Myerson
Let's first have a look at the left hand side of the equation. Take an outcome $y$ and a state $r$. There are two cases:
If $y$ is not the worst outcome in state $r$ then the lottery gives:
$$
\left(\...
- 9,757
2
votes
Reasons for why slutsky matrix may be non symmetric
The necessary and sufficient conditions on a demand system are homogeneity of degree 0, Slutsky symmetry and Slutsky negativity.
From a theoretical point of view, symmetry of the Slutsky matrix is due ...
- 9,757
2
votes
How do the assumptions $p'+q_ip''<0$ and $p'-c''<0$ ensure the stability of the Nash equilibrium among private firms in basic mixed oligopoly model?
The stability conditions are from Hahn (1962). They ensure that, under a specific adjustment process, the firms' outputs will converge to the Cournot-Nash equilibrium. The assumed adjustment process (...
- 577
2
votes
Accepted
How to calculate direct utility from indirect utility in this exercise?
To get the Hicksian demand function $h$, you can use Shephard's lemma, which says that
$$h_i(p_1,p_2,u)=\frac{\partial e(p_1,p_2,u)}{\partial p_i}$$
where $e$ is the expenditure function.
To get the ...
- 577
2
votes
Accepted
Proving duality of UMP and EMP arguing with continuity of utility
By definition, a function is continuous if arbitrarily small changes in its value can be assured by choosing sufficiently small changes of its argument. Since $u(x')>u(x^*)$ and the utility ...
- 577
2
votes
Accepted
General equilibrium with market power
Given a pure exchange economy:
$u_A(x_A, y_A)=x_Ay_A$, $u_B(x_B, y_B)=x_By_B^2$
with endowments:
$\omega_A=(80,150)$ and $\omega_B=(210,180)$
To find the equilibrium, we first find the price offer ...
- 7,513
2
votes
Quasiconvex and quasiconcave utility function
Every concave (convex) function is quasiconcave (quasiconvex).
Any nondecreasing transformation of a quasiconcave function is quasiconcave (i.e. if the function $f$ is quasiconcave and $g$ is a ...
- 577
2
votes
Accepted
what to learn to write a comprehensive strong paper?
This depends on exact subfield or even topic you are interested in. For example, there are people who only work on theory, and there you only need strong math background and nothing else. If you work ...
- 52.9k
1
vote
Paradox of 'more quantity means greater satisfaction' in consumer behavior
It is not a paradox.
First, there is no economic law or theory that states more is always in every situation better, and it is not a requirement of indifference curves either.
For example, if you have ...
- 52.9k
1
vote
Accepted
Edgeworth Box for exchange economy
Not quite. The line you have drawn turns out to be the contract curve, but from this figure it's not clear how you get this line, because the indifference curves for the two utility functions are ...
- 6,355
1
vote
How can I write the unit cost function of this nesting production function
You can set up the cost minimization problem explicitly, but it might be easier to consider the similarity with the utility maximising consumer problem and first look at the dual utility maxmisation ...
- 9,757
1
vote
Accepted
About Proof of Theorem 20.8 in Mathematics for Economists by Simon and Blume
Looks rigorous and complete to me. And it is indeed clearer than the corresponding proof in Simon and Blume.
- 6,355
1
vote
Accepted
Consumption with quantity discount
If our consumer buys more than 2 units of $x$ we can rewrite his budget constraint in the following way.
$$
\begin{align*}
&\underbrace{2 \times 2}_{\text{first 2 units of $x$ at price 2}} + \...
- 9,757
1
vote
Walrasian Demand and Indirect Utility Function
Walrasian demand $(x_1^d,x_2^d)$ solves the following problem:
\begin{eqnarray*} \max_{x_1\geq 0,x_2\geq 0} & \min(2x_1+x_2,x_1+2x_2) \\ \text{s.t.} & p_1x_1+p_2x_2\leq m\end{eqnarray*}
where $...
- 7,513
1
vote
CRS assumption in solow model
One way to conceptualize constant returns to scale is by envisioning multiple plants employing the same technology, where it is feasible to initiate as many plants as desired to produce the desired ...
- 7,513
1
vote
Specific term for disincentivized "doing something first"
I would call this the free-riders problem. wikipedia
- 9,757
1
vote
Accepted
Specific term for disincentivized "doing something first"
If I understand it correctly, you are making these core assumptions:
If nobody innovates, no profits are generated.
Innovating pays off, but is costly.
Copying pays off as well, and is not costly, ...
- 6,355
1
vote
General equilibrium with market power
Not sure where you get lost. $u_B$ is Cobb-Douglas type, thus given a price vector $\textbf{p}$ and initial endowment $\textbf{w}^B$ it is easy to determine $x_1^B(\textbf{p}),x_2^B(\textbf{p})$.
Once ...
- 28.4k
Only top scored, non community-wiki answers of a minimum length are eligible