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St. Louis Fed: https://fred.stlouisfed.org/ Has very good US data. I am not aware of a database that systematically aggregates city-level data. I recommend state government sites and, for larger cities, city government sites.

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You can obtain a definite answer for $$\text{sign}\left\{\frac{\mathrm{d} S_{1}^{*} }{\mathrm{d} \alpha }\right\}$$ given the assumptions. From $$S_{1}^{*} - f\left ( g\left ( S^{*}_{1}, \beta \right ), \alpha \right ) = 0 \equiv F$$ and the Implicit Function Theorem $$\frac{\mathrm{d} S_{1}^{*} }{\mathrm{d} \alpha } = - \frac{\partial F/\partial \alpha}{... 3 "Comparative statics" may refer to two closely related but not identical exercises: a) Regarding a specific equilibrium point, we examine how the properties, associated values etc of it change as we vary the value of an exogenous parameter. b) But also, we can put under "comparative statics" the case where the model has multiple equilibria. Inherently, ... 2 I'm not sure there are enough assumptions here to move forward, in particular I would think you would need some conditions on your function h, at the very least. It's pretty easy to show that g(\cdot,\cdot) will have increasing differences, which implies the single-crossing property for g (this can be quickly seen by letting h be identically zero). ... 2 I believe the analysis in E. Silberberg's "The Structure of Economics" (2n ed. 1990), is illuminating, chapter 7. The fundamental comparative-statics result (for constrained and unconstrained problems) is (eq. 7-10 of the book)$$\frac{\partial ^2 f(x^*,t)}{\partial x\partial t} \cdot \frac {\partial x^*}{\partial t} > 0$$The matter was initially ... 1 \require{cancel} Start with the fact that v = v(c, q) and use the triple product rule$$ \left(\frac{\partial v}{\partial c}\right)\left(\frac{\partial c}{\partial q}\right)\left(\frac{\partial q}{\partial v}\right) = -1\tag{1} $$Now use$$ \color{orange}{c = y - tx - pq} ~~\Rightarrow~~~ \frac{\partial c}{\partial q} = -p \tag{2} $$to conclude$$...

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Also check out quandl.com. The data from the Fed and World bank type of organizations is usually free and updated regularly, as with most of the public economic data. There are other more curated types of sources which are "premium" and range in prices. To download data you will need an account but its super easy and free. (just username, email address and ...

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The approach is not correct, because $\mu^{BGP}$ is a result of underlying structural parameters. So to say "when $\mu^{BGP}$ increases..." immediately begs the question why it increases, which underlying parameter(s) has changed to cause such an increase... assume it was $\epsilon$ that decreased. But in this case $g_k$ is not affected at all, so you see ...

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