这是一份nottingham诺丁汉大学ECON2010作业代写的成功案例

If demand for both varieties is non-negative, ${ }^{9}$ the quantity demanded of variety 1 satisfies
$$
b\left(1-\theta^{2}\right) q_{1}=\left(a_{1}-c_{1}\right)-\theta\left(a_{2}-c_{2}\right)+\theta\left(p_{2}-c_{2}\right)-\left(p_{1}-c_{1}\right)
$$
with an analogous expression for variety 2 . Firm 1’s profit then satisfies
$$
b\left(1-\theta^{2}\right) \pi_{1}=\left(p_{1}-c_{1}\right)\left[\left(a_{1}-c_{1}\right)-\theta\left(a_{2}-c_{2}\right)+\theta\left(p_{2}-c_{2}\right)-\left(p_{1}-c_{1}\right)\right]
$$
The first-order condition for maximizing $\pi_{1}$ with respect to $p_{1}$ gives the equation of firm 1’s price reaction function
$$
2\left(p_{1}-c_{1}\right)=\left(a_{1}-c_{1}\right)-\theta\left(a_{2}-c_{2}\right)+\theta\left(p_{2}-c_{2}\right)
$$
Solving the equations of the two reaction functions gives static Nash equilibrium prices, which satisfy
$$
\begin{aligned}
&\left(4-\theta^{2}\right)\left(p_{1}^{N}-c_{1}\right)=\left(2+\theta-\theta^{2}\right)\left(a_{1}-c_{1}\right)-\theta\left(a_{2}-c_{2}\right) \
&\left(4-\theta^{2}\right)\left(p_{2}^{N}-c_{2}\right)=\left(2+\theta-\theta^{2}\right)\left(a_{2}-c_{2}\right)-\theta\left(a_{1}-c_{1}\right)
\end{aligned}
$$

ECON2010 COURSE NOTES ：

$(U, U)$ : The market boundary for the two firms is given by the location $\vec{x}$ of the consumer who is indifferent between buying from either firm: $p_{1}+t \bar{x}=p_{2}+t(1-\bar{x})$, from which it immediately follows that $\bar{x}=\left(p_{2}-p_{1}+t\right) / 2 t$. As consumers are distributed with a unit density, profits of firm 1 are given by $\pi_{1}=p_{1} \bar{x}$ and those of firm 2 by $\pi_{2}=\left(p_{2}-c\right)(1-\bar{x})$. The unique pair of equilibrium prices is obtained from the first-order conditions as
$$
\left(t+\frac{c}{3}, t+\frac{2 c}{3}\right),
$$
yielding market areas
$$
\left(\frac{1}{2}+\frac{c}{6 t}, \frac{1}{2}-\frac{c}{6 t}\right)
$$
and equilibrium profits
$$
\left(\frac{1}{2 t}\left(t+\frac{c}{3}\right)^{2}, \frac{1}{2 t}\left(t-\frac{c}{3}\right)^{2}\right) .
$$