Articles of economics

Is the standard definition of vector wrong?

The definition of a vector is usually something like “a quantity that has both a magnitude and a direction”. But, in the context of, say, economics rather than physics, does this definition make sense? Let’s say we are working in 4 dimensional space? Lastly, how does this definition make sense in matrix algebra when referring […]

Finding mixed Nash equilibria in continuous games

I’m taking my first (graduate-level) game theory class. I understand how to find Nash equilibria in simple games, such as those given in finite tables, and can see (usually) how to find the mixed equilibria in those cases, but most of our problems have been discussing continuous games such as auctions, firm pricings with continuous […]

Does one necessarily need an MS in Math before taking a PhD in Math?

I finished bachelor’s in mathematical finance and am nearly finished with master’s in mathematical finance (I am already done with thesis), and I plan to pursue a PhD not in mathematical finance but in pure mathematics particularly stochastic analysis. Is getting into a PhD in pure mathematics possible without a master’s in pure mathematics? If […]

What is the (expected) outcome of this hybrid auction?

A certain hybrid auction can be accurately modelled as follows. There are $n$ risk-neutral, rational participants $i=1,2,\ldots,n$, and a guy called Zerro: $i=0$. Each, except Zerro, has a private value of the good (only known to themselves) $v_i\sim_\mathrm{i.i.d.} \mathrm{UNIF}(0,1)$. An English auction (EA) is held. Zerro is allowed to, and does, open the bid at […]

How can a social welfare function be a linear combination of von Neumann-Morgenstern utility functions?

The von Neumann-Morgenstern axioms were an attempt to characterize rational decision-making in the presence of risk. The von Neumann-Morgenstern utility theorem says that if someone is vNM-rational, i.e. their preferences obey the axioms, then there exists a function $u$ from the set of outcomes to the real numbers, such that they will have to maximize […]

Mistake in Wikipedia article on St Petersburg paradox?

I suspect that there is a mistake in the Wikipedia article on the St Petersburg paradox, and I would like to see if I am right before modifying the article. In the section “Solving the paradox”, the formula for computing of the expected utility of the lottery for a log utility function is given to […]

Show that the least squares estimator of the slope is an unbiased estimator of the `true' slope in the model.

Under the assumptions of the classical simple linear regression model, show that the least squares estimator of the slope is an unbiased estimator of the `true’ slope in the model. Anyone have any ideas for the following questions?

Lower hemicontinuity of the intersection of lower hemicontinuous correspondences

I have been stumped for long by this exercise (3.12(d)) from Stokey and Lucas’s Recursive Methods in Economic Dynamics. Would greatly appreciate any hints. Let $\phi: X \to Y$ and $\psi: X \to Y$ be lower hemicontinuous correspondences (set-valued functions), and suppose that for all $x \in X$ $$\Gamma(x)=\{y \in Y: y \in \phi(x) \cap […]

Can I get a PhD in Stochastic Analysis given this limited background?

General advice on PhD apps welcome Given my limited background in stochastic analysis and other information (below), can I apply for a PhD with stochastic analysis for my dissertation topic? 1/4 I am currently a masteral student of mathematical finance, expecting to graduate sometime this year. I am not particularly interested in mathematical finance anymore […]

Equilibrium existence proof

Problem: Let $J$ be an integer and let $I$ be an integer multiple of $J$. Let ${\cal I}= \lbrace 1,2,\ldots, I\rbrace$ and ${\cal J}= \lbrace 1,2,\ldots, J\rbrace$. The set $X_{I,J}$ of all $\frac{I}{J}$-subsets of $\cal I$ contains exactly $N=\binom{I}{\frac{I}{J}}$ elements. Now, consider the set $Y_{I,J}$ of all permutations of $X_{I,J}$, viewed as columns enumerating the […]