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3

I'll do it for $N=3$ expliciting steps. When you write $f(x_1,x_2,x_3)$ think of $x_1,x_2,x_3$ as placeholders for each corresponding coordinate. Saying it is homogeneous of degree $r$ means that when we have $x_1 = t \overline{x}_1, x_2 = t \overline{x}_2, x_3 = t \overline{x}_3$ then it holds $$f(t \overline{x}_1, t \overline{x}_2, t \overline{x}_3) = t^r ...


3

To be honest "incomple information" is not a good terminology. If players do not know all the relevant information, the game is not really well specified. But for historical reasons, the term has stuck. When you want to model a situation of "incomplete information" what you do in practice is to use Harsanyi's trick: you replace the incomplete information ...


2

Dividing through by $L$ gives you output per worker: $$\frac{Y}{L}= \frac{A L^{\alpha}K^{\beta }}{L}= \frac{A L^{\alpha}K^{\beta }}{L^\alpha L^\beta}=A \left( \frac{K}{L}\right)^\beta.$$ We used the first assumption. Now take logs, which gets you $$\ln \frac{Y}{L}= \ln A + \beta \cdot \ln \frac{K}{L}.$$ Calculate the difference from adjusting $K$ and $L$: ...


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Here's a slightly different take using the fact that logarithmic derivatives are percentage growth rates instead of approximation. Put $y = Y/L$ and $k = K/L$. Then after dividing through by $L$ as in Dimitriy Masterov's answer, differentiate: \begin{align*} \frac{d}{dt}\log y &= \frac{d}{dt}\log A + \beta \frac{d}{dt}\log k\\ \frac{y'}{y} &= ...


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The yellow rectangle on the picture is inaccurately drawn: its lower right vertex should be higher, on the pre-tax supply curve. In present form, it appears to be based on the value of $p$ where the post-tax supply curve meets $q=0$, which isn't a relevant quantity here. Here is my version: The tax amount is $t$, which is the amount by which $S_T$ is ...


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The areas of mathematics to study to get from Pre-Algebra to Calculus is: Basic Math Pre-Algebra Algebra Algebra II Trigonometry Calculus (Source for list, a flow chart a community college created to aid students in choosing math classes http://web.clark.edu/math/docs-students/Math_Flow_Chart.pdf) To see individual concepts per area of math, use ...


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With the hint by achille i was able to do this myself. Set up the following portfolio: Sell (i.e. go short): 2x the strike=100 stock Buy (i.e. go long): 1x the strike=110 stock and 1x the strike=90 stock Then at time $t=0$ we have a value of the portfolio of: $V = -2 C(100) + 1 C(110) + 1 C(90) = -2 \frac{C(110)-C(90)}{2} + 1 C(110) + 1 C(90) = 0$ To ...


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We can assume $b$ is negative, as demand decreases as price increases. Intuitively, we have an initial demand if something is free. If we are getting paid to get something, don't we demand more of it? If we are paying for it, wouldn't we demand less of it? Think of $a$ as an initial endowment. Similarly, with $c, d > 0$, a higher price incentivizes the ...



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