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The solution The answer can be been found on the internet in any number of places. The function $U$ is a Cobb-Douglas utility function. The Cobb-Douglas function is one of the most commonly used utility functions in economics. The demand functions you should get are: $$x(p,I)=\frac{\alpha I}{(\alpha+\beta)p}\qquad y(p,I)=\frac{\beta I}{(\alpha+\beta)q}$$ ...

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This is a Bertrand duopoly situation. As the game is symmetric, there exists a Nash equilibrium in which both players use the same strategy. Now each firm is a profit maximizer. So we have: $$\pi_{i} = (p_{i} - c)q_{i}(p_{i}, p_{-i})$$ Player $i$ wishes to maximize its profit $\pi_{i}$ and can only vary its price. This leads to the first order condition: $... 1 If$I=px+qy$, then$y = (I-px)/q$, so$x^ay^b =x^a((I-px)/q)^b =x^a(I-px)^b/q^b $. Differentiating, we want$\begin{array}\\ 0 &=(x^a(I-px)^b)'\\ &=ax^{a-1}(I-px)^b-x^apb(I-px)^{b-1}\\ &=x^{a-1}(I-px)^{b-1}(a(I-px)-xpb)\\ &=x^{a-1}(I-px)^{b-1}(aI-apx-xpb)\\ &=x^{a-1}(I-px)^{b-1}(aI-xp(a+b))\\ \text{so}\\ x &=\dfrac{aI}{p(a+b)}\\ \...

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You can certainly simplify that system! $$\lambda = \frac{\alpha x^{\alpha - 1}y^\beta} p$$ $$\lambda = \frac{\beta y^{\beta - 1}x^\alpha} q$$ Thus $$\frac{\alpha x^{\alpha - 1}y^\beta} p = \frac{\beta y^{\beta - 1}x^\alpha} q$$ and $$q\left(\alpha x^{\alpha - 1}y^\beta\right) = p\left(\beta y^{\beta - 1}x^\alpha\right)$$ You can reduce powers: $$q\... 1 \newcommand{\E}{\mathbb{E}}$$\newcommand{\P}{\mathbb{P}}$For positive values,$\psi(x)=x^{\alpha}$is monotone (I think even for all$\alpha \in \mathbb{R}$, but certainly for$0 < \alpha < 1$). Therefore for such$\psi$we have $$X > Y \text{ almost surely } \implies \E[\psi(X)]) > \E[\psi(Y)] \quad \text{and} \quad\E[X] > \E[Y]$$ by ... 1 For Part (a), examine each player's payoff for each of their two possible strategies with the other five players' strategies held fixed. So Player 1's payoff is$6-1=5$if she chooses vaccination and$\frac{3}{6}6+\frac{3}{6}0=3$if not. Player 2's payoff is$6-2=4$if she chooses vaccination, and$\frac{3}{6}6+\frac{3}{6}0=3$if not. The analysis is ... 1 Both lines are wrong but the first one happens to end up with the right result nevertheless. The first one is wrong because you drop the average in the first step, whereas you can only drop it in the last step; in the last step you replace$xx^\top$by$C$even though$C$is defined as the average of that quantity. If you take the average of all ... 1 In (c) you are asked to calculate the income and substitution effects for a discrete change in the price of good$1$from$1$to$3\$. Thus you will not be calculating the effects using derivatives. In (a) which demand functions were you asked to find? Just the Marshallian, or the Hicksian (compensated) demand function as well? The Hicksian demand function ...

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