# Tag Info

3

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: $$x(p,I)=\frac{\alpha I}{(\alpha+\beta)p}\qquad y(p,I)=\frac{\beta I}{(\alpha+\beta)q}$$ ...

2

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)}\\ \...

1


Only top voted, non community-wiki answers of a minimum length are eligible