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### 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.
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 $u(... 1answer 137 views ### Solution to maximization not Pareto efficient In my economics class, we saw a proof that if an allocation$ ((\hat x_h), (\hat y_f)), h\in H$(the set of households),$f\in F$(the set of firms) is Pareto optimal/efficient, it must necessarily be ... 1answer 321 views ### Proof of Expected utility theorem with three outcomes I am trying to prove the expected utility theorem with three outcomes. The expected utility with$n$outcomes is rather cumbersome and long in the economics textbook Mas-Colell. But I was hoping that ... 1answer 288 views ### Prove that$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$. Is it sufficient to prove that the ... 0answers 96 views ### Open Foundational Problems in Mathematical Economics What are some open problems in mathematical economics which are treated as if they have been solved in economics papers? As an example, one of my teachers has previously pointed out to me that there ... 1answer 3k views ### Prove all cost functions are concave in input prices and demand for inputs is downward I've seen proofs that cost functions are concave of the form$C(\lambda w + (1-\lambda)w',q) \ge \lambda c(w,q) + (1-\lambda)c(w',q)$although this neither feels convincing nor does it seem like a ... 1answer 103 views ### Help with Monopolistic Competition Proof, Prove Love for Variety I need some help with a proof. Assume η = 2 and there are just two goods. Verify that the following utility function exhibits Love for Variety tastes, show that: u(λa + [1 − λ]b, λb + [1 − λ]... 1answer 114 views ### Can one prove Pareto efficiency in an n-party system by showing all Pareto improvements between any two given parties are made? I've made a proof of Pareto efficiency of a funding system that I've developed. There are effectively four types of actors. I've shown all Pareto improvements are made between any two given parties ... 2answers 944 views ### Basic Solow Growth Model: Stability Proof I am reading through McCandless "The ABCs of RBCs" this summer to get a preview of what I need to know for the coming Fall semester. It did not take long to find a statement that I can easily accept ... 1answer 2k views ### How to prove convexity + quasilinear preferences imply concave utility? Let$\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})$. How to prove ... 1answer 1k views ### Equivalence of Definitions of Continuity of Preferences We have two definitions of the continuity of preferences: Def 1:$\succcurlyeq$is continuous if for any sequences$\{x^n\} \subset X$and$\{y^n\} \subset X$, then$n \in \mathbb{N}$such that,$\...
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(...