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Let $w_1 \dots w_5$ be the respective wealths of economic classes 1, 2, 3, 4, 5. Let $t_1 \dots t_5$ be the new wealths starting from the top, and $b_1 \dots b_5$ the new wealths starting from the bottom. Assuming that each economic class contains the same number of people, then proceeding top to bottom we have: $t_5 = \frac{w_4 + w_5}{2}$ $t_4 = ... 1 You want$20000=\dfrac{x}{1.1^1}+ \dfrac{x}{1.1^2}+ \dfrac{x}{1.1^3}+ \dfrac{x}{1.1^4}+ \dfrac{x}{1.1^5}$so$x=\dfrac{20000}{1.1^{-1}+1.1^{-2}+1.1^{-3}+1.1^{-4}+1.1^{-5}}$which is$20000\times \dfrac{1.1^6-1.1^5}{1.1^5-1}$. 1 Let´s say that the GDP in the year 2000 is$x$. Then the GDP in the year 2003 is$1.03x$The expenditure for R&D in 2000 is$0.0245\cdot x$and the expenditure for R&D in 2003 is$0.0252\cdot 1.03x$. Thus the growth rate for R&D is$\frac{0.0252\cdot 1.03x- 0.0245\cdot x}{0.0245\cdot x}=0.059425\approx 5.9\%$x is cancelling out. 1 So in general I know that this does not hold, because your expression is exactly the myerson virtual value, and there is a long literature about suitable regularity conditions to make it increasing. If you assume log-concavity of your distribution for example then you are done. But this is too strong in that you can actually assume -.5 concavity. I refer you ... 1 The important thing to note is that at your solution, the active constraints are$x+2y\geq 4$,$7x+6y\geq 20$. In other words, at the solution, $$x+2y=4, ~~ 7x+6y=20, ~~ x>0, ~~y>0.$$ For sufficiently small changes in the constraints, the first two constraints will remain the active set. Thus for any sufficiently small$\epsilon$s, you will be able to ... 1 You can investigate the solution of the dual problem: The dual problem is:$\texttt{max} \ 4u+20vu+7v\leq 142u+6v \leq 20u,v \geq 0$The optimal solution$(u^*, v^*)$can be found graphically as well. Interpretation: If the first constraint would be replaced by$x+2y\geq 4+ \epsilon $, then the optimal soulition would be ... 1 The main idea behind the Cremer-McLean result is the following: When agents values are correlated, after an agent learns her own value is$v$, her posterior on the other agent's values is distinct from the case where it is some over value$v'$. Given that the two values$v$and$v'$induce different posteriors on the other agents' values, we can ... 1 The shadow price is the price, which you are maximal willing to pay for one additional unit. The shadow arice of restriction (i) of the primal problem can be calculated by solving the dual problem. The corresponding optimal value of the dual problem is the shadow price. If the shadow price is$\lambda _i$and you raise the bound ($b_i$) by 1 unit, then the ... 1 The horizontal dotted line is to suggest that those two nodes are in the same information set. That's to say that P1 cannot tell whether P2 chose A or B. To represent it in the normal form, you need to consider what the strategies are. Remember that a strategy is not an action -- rather, a strategy defines an action for each player at each of his information ... 1 I can't imagine that A graduate program in economy would include a course in graph theory as an obligatory class. It may be available as an optional course though. As per the prerequisites it depends a lot on the class, if you want an introductory course then you may only need basic set theory and a working knowledge of how proofs work, specially induction ... 1 The equation for the elasticity is:$e=\frac{dy}{dx}\frac{x}{y}$In your case, we have$ln(y)=5-0.1x$, Now take the derivative of both sides and multiply by x:$\frac{dy}{dx}\frac{x}{y} = -0.1x$Therefore, at$x=10$, the elasticity is$-1\$.