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7

For notational simplicity, let me define the distribution $G(s) = F^{N-1}(s)$ with density $g(s)$. Let $\underline{s} = 0$ (for simplicity). We have $$ b'(s)G(s)+b(s)g(s)=s g(s) $$ Integrating to $x$ gives us $$ \int_0^x b'(s)G(s)+b(s)g(s) ds = \int_0^x s g(s)ds $$ Notice that $\frac{\partial }{\partial s}(b(s)G(s)) = b'(s)G(s)+b(s)g(s)$, so $$ b(x)G(x) = \...


5

At the intersection of differential equations and game theory one can find differential games. Arguably, the most famous applications of differential games are in warfare, e.g., the homicidal chauffeur problem. However, not all differential games are of the pursuit-evasion kind. Whoever wants to learn differential games may wish to learn optimal control ...


2

As the link you provided mentions: "Differential equations can be used to include the dynamic aspects to economics into a mathematical framework which takes into account the volatility present in economics." So the author does not derive the differential equations as the result of a theoretical model -the author just wants to obtain the movement ...


2

The differential equation $$\dot k = \frac{1}{\sigma} k^\alpha - \delta k$$ has the structure of a Bernoulli equation. We solve it by the following transformation steps: 1) Mulitply throughout by $k^{-\alpha}$: $$k^{-\alpha}\dot k = \frac{1}{\sigma} - \delta k^{1-\alpha} \tag{1}$$ 2) Define the variable $$z \equiv k^{1-\alpha} \implies \dot z = (1-\...


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

Many macroeconomists have a somewhat cavalier attitude towards checking the validity of linearization methods, but definitely not all of them. For an example of the approximation issues being taken seriously, see Appendix A.3 on "Log-Linearization and Determinacy of Equilibrium" in the book Interest and Prices by Michael Woodford.


1

This will be long, but the subject is worth it. The gist of the "exchange rate overshooting" model in Dornbusch, R. (1976). Expectations and exchange rate dynamics. The Journal of Political Economy, 1161-1176. can be given using two equations: Uncovered Interest Rate Parity (UIRP) $$i = i^*+ \dot e\tag{1}$$ where $i$ is local interest rate, $i^*$ is ...


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