8

Not really. There are many compact metrizable topologies you can put on this space, but none that relate meaningfully to the structure of the problem. Let's look first at the case $A_1=[0,1]$ and $A_2=\{0,1\}$. Consider the elevation function $e:A_1\times\Sigma\to\Delta(A_2)=[0,1]$ given by $e(a,\eta)=\eta(a)$. If you want the ultimate action choice of ...


5

Here is the sketch of a proof. All we need is that every continuous weak order on each $X_i$ admits a continuous utility representation. One sufficient condition is that each $X_i$ is a connected separable topological space by a theorem of Eilenberg. A proof of Eilenberg's theorem is given in Debreu's book Theory of Value. Debreu assumes the domain there ...


4

I'm not sure what exactly in plain English means, but here is a try: Think about it as consuming water. Look at condition 1. If you have no water, you would die, making you extremely unhappy. If someone came up to you and offered one bottle of water, it would make you infinitely happier than having no water, as you would live! (You would pay any amount of ...


1

I do not have a full answer, but here are my notes when I studied it that hopefully someone can extend to a full answer. Sketch of Proof: Consider the linear space with basis $\cup_{i =1}^N X_i$, and we can identify any $x \in X$ by $\sum_i x_i$. Define the convex cone $D = \{\lambda(x-y): x\succeq y;\lambda > 0\}$ Let $D^{-}$ be the convex hull of $\{...


1

This is because for small values $x$, $$\ln x_{t+1} - \ln x_{t} \approx \frac{x_{t+1}-x_{t}}{x_{t}}.$$ This holds since the growth rate $g$ can be expressed as follows: $$g= \frac{x_{t+1}-x_{t}}{x_{t}} \implies x_{t+1} = (1+g)x_{t}$$ taking logs we get that: $$ \ln x_{t+1} = \ln (1+g)+ \ln x_{t} \implies \\ \ln x_{t+1} -\ln x_{t} = \ln (1+g) $$ Finally, for ...


1

Probably, the author probably is assuming $x_i$ and $y_i$ are independent in addition to the stated assumption on pg. 7 that types "are orthogonal" (which merely means uncorrelated.) (However, since they are also uniformly distributed, we have, $f_{X,Y}(x,y)=f_X(x)f_Y(y)\equiv 1, \forall x,y$, meaning they are independent.) Assuming independence, $$...


1

This is probably an old question. However there is a book on PDE's and game theory. Game Theory and Partial Differential Equations, Pablo Blanc and Julio Daniel Rossi. If you found other books on this matter, I would like to know as well.


1

There is also Felix Muñoz-Garcia: advanced microeconomic theory (this comes with a book of solved practice questions) and Silberberg: The structure of economics (old but written so well where much the needed math is explained along with the economics) For consumer choice - no firm theory - the book by Deaton and Muellbauer: Economics and consumer behavior is ...


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