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9 votes
3 answers
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Quasilinear Utility: Pareto Optimality Implies Total Utility Maximization?

I read that if we have quasilinear utility for all consumers, then any pareto optimal allocation maximizes the sum of utility levels of all consumers. That is: $\textbf{What we know:}$ $$1)\quad u^i(...
DornerA's user avatar
  • 1,568
4 votes
2 answers
175 views

Order relations and preferences using logic

I want to understand order relations using their underlying implication mechanics and what this means for certain results, specifically looking at preference relations. Using the logical rules of ...
CormJack's user avatar
  • 899
4 votes
0 answers
160 views

Open Foundational Problems in Mathematical Economics

What are some open problems in mathematical economics which are treated as if they have been solved in economics papers? As an example, one of my teachers has previously pointed out to me that there ...
ertl's user avatar
  • 141
3 votes
2 answers
388 views

Is Varian's definition of continuity of preference equivalent to standard definitions?

Here are two definitions of continuity of preferences. Denote the (weak) preference relation by ≽. We assume completeness, reflexivity and transitivity. Assume non-satiation or strict monotonicity ...
not tdm's twin's user avatar
3 votes
2 answers
754 views

Under what condition is a cost function strictly concave in prices?

Define the unit cost function as $$ c(w) = \min_{z\geq 0} w\cdot z $$ subject to $f(z)\geq 1$. Where $w$ is a vector of input prices, $z$ is the vector of inputs and $f$ is a production function. We ...
user_lambda's user avatar
3 votes
2 answers
1k views

Basic Solow Growth Model: Stability Proof

I am reading through McCandless "The ABCs of RBCs" this summer to get a preview of what I need to know for the coming Fall semester. It did not take long to find a statement that I can easily accept ...
economicist's user avatar
3 votes
2 answers
271 views

Prove that the profits of the firm weakly decreases with input prices

Prove that the profits of the firm weakly decreases with input prices. More formally, suppose that the firm has a production function f, so that its profit function is π(p, w) = max(x≥0) $pf(x) − w · ...
DH00325's user avatar
  • 33
3 votes
1 answer
5k views

Prove all cost functions are concave in input prices and demand for inputs is downward

I've seen proofs that cost functions are concave of the form $C(\lambda w + (1-\lambda)w',q) \ge \lambda c(w,q) + (1-\lambda)c(w',q)$ although this neither feels convincing nor does it seem like a ...
Frank Shmrank's user avatar
3 votes
1 answer
127 views

Is this proof correct (measure theory)?

I'm trying to prove the following statement: I did the following proof but i donno if it makes sense: Proof of $(\Rightarrow)$ direction: Assume $\mathcal{F}$ is a $\sigma$-algebra. Assume also $\{...
Calogero Sortino's user avatar
3 votes
1 answer
1k views

Equivalence of Definitions of Continuity of Preferences

We have two definitions of the continuity of preferences: Def 1: $\succcurlyeq$ is continuous if for any sequences $\{x^n\} \subset X$ and $\{y^n\} \subset X$, then $n \in \mathbb{N}$ such that, $\...
Kitsune Cavalry's user avatar
  • 6,608
3 votes
0 answers
84 views

How to prove that these level curves intersect

I have two functions: $$F(x,y) = U_1'(c_1)(a_1-b_1)+U_2'(c_2) = 0$$ $$G(x,y) = U_1'(c_1)+U_2'(c_2)(a_2-b_2) = 0$$ where $c_i=a_i(x-y)+yb_i, i=1,2.$ the utility functions $U_1,U_2$ are twice ...
vonenzo's user avatar
  • 41
2 votes
1 answer
96 views

Prove that if a production function is such that f'>0 and f''<0, then f'<Average Product

I was told in class that if we have a production function such that $f'(x)>0$ and $f''(x)<0$, then we have that the marginal product is less than the average product. That is $f'(x)<\frac{f(x)...
Mistah White's user avatar
2 votes
1 answer
300 views

Can I assume utility functions strictly increasing?

I am required to show that if: $f:R^L \rightarrow R$ is a strictly monotonic function and $u:R^L \rightarrow R$ is a utility function representing a preference relation $\succsim$, then the function $...
Kinno's user avatar
  • 155
2 votes
1 answer
116 views

How do you establish uniqueness of a rational preference relation?

Going through a proof in Mas Colell and I am not understanding how (iii) shows uniqueness of the rationalizing preference relation. I understand that well $\beta$ is the power set so it contains all ...
Rumi's user avatar
  • 977
2 votes
1 answer
484 views

Proof of Expected utility theorem with three outcomes

I am trying to prove the expected utility theorem with three outcomes. The expected utility with $n$ outcomes is rather cumbersome and long in the economics textbook Mas-Colell. But I was hoping that ...
OGC's user avatar
  • 285
2 votes
1 answer
2k views

How to prove convexity + quasilinear preferences imply concave utility?

Let $\succsim$ be a strictly convex and quasilinear preference relation. It's defined over, say, $\mathbb{R}^2_{+}$ and is quasilinear on good 1. So, $U(x_{1},x_{2}) = x_{1} + f(x_{2})$. How to prove ...
Raul Guarini Riva's user avatar
2 votes
1 answer
71 views

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 ...
Diogo Ferreira's user avatar
1 vote
1 answer
669 views

Weak preferences and negative transitivity

Let $ \succ $ be a binary relationship on the set $X$ such that, given any $ x, y, z\in X $, if $x\succ y$: (Asymmetry): $\neg(y\succ x)$, (Negative transitivity): $(x\succ z) \vee (z\succ y)$. ...
antonio's user avatar
  • 113
1 vote
2 answers
252 views

Expected Utility with expected value and variance

I'm having trouble with a question from Ariel Rubinstein's book, Lecture Notes in Microeconomic Theory. It's the problem 2 from Problem Set 7. Here's the question: Show that the utility function $u(...
Marcelo Gelati's user avatar
1 vote
1 answer
104 views

Intuition of two Measure theory statements

I'm struggling in getting the intuition of two statements about measure theory: Given a measure space $(X,F,\mu)$, $f \in M^+ $, where $M^+ = M^+(F) $ is the set of non negative F-measurable functions ...
Calogero Sortino's user avatar
1 vote
1 answer
110 views

Does x ≽ y imply x > y or x ~ y in preferences?

Mas Collel Micro Theory question: Suppose that X is a set. Let ≽ be a binary preference on X. And ~ represents indifference defined from ≽. If ≽ satisfy completeness, is it okay to assume that: x ≽ y ...
Wizard74's user avatar
1 vote
1 answer
121 views

Can one prove Pareto efficiency in an n-party system by showing all Pareto improvements between any two given parties are made?

I've made a proof of Pareto efficiency of a funding system that I've developed. There are effectively four types of actors. I've shown all Pareto improvements are made between any two given parties ...
bEPIK's user avatar
  • 125
1 vote
1 answer
45 views

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 ...
Lorena_dok's user avatar
1 vote
1 answer
45 views

Formal proof that IR_L IC_H are binding constraints

Two types of customers, equal proportions: $q_H(p_H)=20-0.5p_H$, $q_L(p_L)=20-p_L$ The firm cant differentiate between them. It wishes to create a menu of two possible taarifs, $(p_H,T_H),(p_L,T_L)$ $...
John Pine's user avatar
1 vote
1 answer
109 views

Help with Monopolistic Competition Proof, Prove Love for Variety

I need some help with a proof. Assume η = 2 and there are just two goods. Verify that the following utility function exhibits Love for Variety tastes, show that: u(λa + [1 − λ]b, λb + [1 − λ]...
colorado's user avatar
1 vote
0 answers
37 views

Existence of best and worst lotteries with finite outcome set and IIA

In the context of expected utility theory, I want to prove that if the set of outcomes is finite and an agent has a rational preference relation over the set of lotteries, and if that preference ...
kmf's user avatar
  • 63
1 vote
0 answers
27 views

How to prove Kelly criterion maximizes wealth compared to other betting strategies, and how to generalize such a proof

The Kelly criterion approach to a betting situation basically involves maximizing the expected logarithm of one's wealth. The assumption is that maximizing the logarithm appears to inadvertently ...
Alien's user avatar
  • 111
0 votes
1 answer
254 views

Solution to maximization not Pareto efficient

In my economics class, we saw a proof that if an allocation $ ((\hat x_h), (\hat y_f)), h\in H$ (the set of households), $f\in F$ (the set of firms) is Pareto optimal/efficient, it must necessarily be ...
Julia B's user avatar
  • 76
0 votes
1 answer
583 views

Prove that $u$ is a utility function for $\succsim$

If X is finite, define this function $u : X \rightarrow \mathbb{R}$ by $u(x) = |\{z\in X:z \prec x \}|$. Prove that $u$ is a utility function for $\succsim$. Is it sufficient to prove that the ...
plastico's user avatar
  • 151
0 votes
0 answers
77 views

About Theorem 3.5 (The Finite-Approximation Theorem) in Game Theory: Analysis of Conflict by Roger Myerson

I am study infinite strategy set using Myerson's Game Theory textbook. The author stated the following theorem (Theorem 3.5) but did not prove it. I tried the proof, and would really appreciate it if ...
Beerus's user avatar
  • 325
0 votes
0 answers
34 views

Where can I find an endogeneity proof for bias by LLN?

It is intuitive that E[xe] is non-zero but the result E[xe]=sigma(x)*sigma(e)*corr(xe) seems odd. Please help point me in the right direction.
Geoffrey Turner's user avatar
-1 votes
1 answer
241 views

Representing a Lexicographic Preference in a Natural X Natural Choice Space With Utility Function

my current thinking is i have to dis/prove two things cardinality continuity but im not sure about how it would apply since the above is a natural X natural choice space I know cardinality of ...
theshadowers's user avatar