为了使得市民的效用水平在补贴方式改变前后没有差异，s 最少应 为多少?
改为固定收入补贴之后市民选择的出行方式x_1和x_为多少?
哪一种补贴方式对政府的财政负担较小，价格补贴还是固定收入补 贴？差异为多少元?
3.某竞争性厂商有两个工厂，各自的成本函数是 \(c_{1}\left(y_{1}\right)=2 y_{1}^{2}+90, c_{2}\left(y_{2}\right)=6 y_{2}^{2}+40\) 。如 果该厂商生产 32 单位产品，那么:
每间工厂应该生产多少产品?
厂商的总成本函数是多少?求出规模报酬区间。
solution:
市民的效用最大化问题为：
\(\max : U\left(x_{1}, x_{2}\right)=x_{1}^{0.2} x_{2}^{0.8}\) \(st: \quad p_{1} x_{1}+p_{2} x_{2}=m\)
拉格朗日函数为：
\(\mathcal{L}=x_{1}^{0.2} x_{2}^{0.8}+\lambda\left[m-p_{1} x_{1}-p_{2} x_{2}\right]\)
Focs: \(\left\{\begin{array}{l}\frac{\partial L}{\partial x_{1}}=0.2 x_{1}^{-0.8} x_{2}^{0.8}-\lambda p_{1}=0 \\ \frac{\partial L}{\partial x_{2}}=0.8 x_{1}^{0.2} x_{2}^{-0.2}-\lambda p_{2}=0\end{array}\right.\)
解得： \(\left\{\begin{array}{l}x_{1}=\frac{m}{5 p_{1}} \\ x_{2}=\frac{4 m}{6 p_{2}}\end{array}\right.\)
\(V\left(P_{1}, P_{2}, m\right)=\left(\frac{m}{5 P_{1}}\right)^{\frac{1}{5}}\left(\frac{4 m}{5 P_{2}}\right)^{\frac{4}{5}}\)
1)a.价格补贴下\(\left(p_{1}, p_{2}, m\right)=\left(\frac{p_{1}}{2}, p_{2}, m\right)\)
市民的间接效用函数： \(U_{1}=\left(\frac{2 m}{5 P_{1}}\right)^{\frac{1}{5}}\left(\frac{4 m}{5 P_{2}}\right)^{\frac{4}{5}}\)
b.固定补贴下\(\left(p_{1}, p_{2}, m\right)=\left(p_{1}, p_{2}, m+s\right)\)
市民间接效用函数： \(U_{2}=\left(\frac{m+s}{5 p_{1}}\right)^{\frac{1}{5}}\left(\frac{4(m+s)}{5 p_{2}}\right)^{\frac{4}{5}}\)
两者之间无差异，则 \(U_{1}=U_{2}\)
即至少补贴为 \(s^{*}=\left(2^{\frac{1}{5}}-1\right) \cdot m\)
2)固定补贴下 \(\left(p_{1}, p_{2}, m\right)=\left(p_{1}, p_{2}, m+s^{*}\right)\)
则最优选择为： \(\begin{aligned}\left(x_{1}, x_{2}\right) &=\left(\frac{m+5^{*}}{5 p_{1}}, \frac{4\left(m+s^{*}\right)}{5 p_{2}}\right) \\ &=\left(\frac{2^{\frac{1}{5}} m}{5 p_{1}}, \frac{2^{\frac{11}{5}} m}{5 p_{2}}\right) \end{aligned}\)
3)价格补贴下的支出： \(T_{1}=\frac{p_{1}}{2} x_{1}=\frac{1}{5} m\)
固定收入补贴下的支出为\(T_{2}=s=(2^{\frac{1}{5}}-1) \cdot m\)
由于\(\Delta T=T_{1}-T_{2}=\left(1.2-2^{\frac{1}{5}}\right) \mathrm{m} \doteq 0.05 \mathrm{~m}>0\)
故固定收入补贴下政府的补贴复旦更小，差额为\(\Delta T\)
2生产函数为 \(y=f\left(x_{1}, x_{2}, x_{3}\right)=\left[x_{1}^{\rho}+\left(\min \left\{x_{2}, x_{3}\right\}\right)^{\rho}\right]^{\frac{1}{\rho}}\) 三种投入要素的价格为 \(w_{1}, w_{2}, w_{3}\)，求成本函数。
solution：
由于成本函数表示既定产量下的最优要素选择. \(y=\left[x_{1}^{\rho}+min\{x_{2},x_3\}^{\rho}\right]^{\frac{1}{\rho}}\) 则最优情况下，\(x_2=x_3\)
成本最小化问题为：
min: \(C=w_{1} x_{1}+\left(w_{2}+w_{3}\right) x_{2}\) st: \(y=\left[x_{1}^{\rho}+x_{2}^{\rho}\right]^{\frac{1}{\rho}}\)
拉格朗日函数： \(L=w_{1} x_{1}+\left(w_{2}+w_{3}\right) x_{2}+\lambda\left[y-\left(x_{1}^{\rho}+x_{2}^{\rho}\right)^{\frac{1}{\rho}}\right]\)
Focs: \(\frac{\partial L}{\partial x_{1}}=w_{1}-\lambda \frac{1}{\rho}\left(x_{1}^{\rho}+x_{2}^{\rho}\right)^{\frac{1-\rho}{\rho}} \cdot \rho x_{1}^{\rho-1}=0\) \(\frac{\partial L}{\partial x_{2}}=w_{2}+w_{3}-\lambda \frac{1}{\rho}\left(x_{1}^{\rho}+x_{2}^{\rho}\right)^{\frac{1-\rho}{\rho}} \cdot \rho x_{2}^{\rho-1}=0\)
解得： \(\frac{w_{1}}{w_{1}+w_{3}}=\left(\frac{x_{1}}{x_{2}}\right)^{\rho-1}\)
联合生产函数 \(y=\left(x_{1} \rho+x_{2} \rho\right)^{\frac{1}{\rho}}\)得：
\(x_{1}=\frac{w_{1}^{\frac{1}{\rho-1}}}{[w_{1}^{\frac{\rho}{\rho-1}}+\left(w_{2}+w_{3}\right)^{\frac{\rho}{\rho-1}}]^{\frac{1}{\rho}}} \cdot y\)
\(x_{2}=\frac{(w_{2}+w_{3})^{\frac{1}{\rho-1}}}{[w_{1}^{\frac{\rho}{\rho-1}}+\left(w_{2}+w_{3}\right)^{\frac{\rho}{\rho-1}}]^{\frac{1}{\rho}}} \cdot y\)
\(c(y)=[w_1^{\frac{\rho}{\rho-1}}+(w_2+w_3)^{\frac{\rho}{\rho-1}}]^{\frac{\rho-1}{\rho}}y\)
3.某竞争性厂商有两个工厂，各自的成本函数是 \(c_{1}\left(y_{1}\right)=2 y_{1}^{2}+90, c_{2}\left(y_{2}\right)=6 y_{2}^{2}+40\) 。如 果该厂商生产 32 单位产品，那么:
每间工厂应该生产多少产品?
厂商的总成本函数是多少?求出规模报酬区间。
solution:
1)厂商的成本最小化问题为：
\(\begin{aligned} \min : & c_{1}\left(y_{1}\right)+c_{2}\left(y_{2}\right) \\ \text { st: } & y_{1}+y_{2}=y \quad\left(y_{1} \geqslant 0, \quad y_{2} \geq 0\right) \end{aligned}\)
构建拉格朗日函数：
\(L=2 y_{1}^{2}+90+6 y_{2}^{2}+40+\lambda\left(y-y_{1}-y_{2}\right)\)
Focs: \(\left\{\begin{array}{l}\frac{\partial L}{\partial y_{1}}=4 y_{1}-\lambda=0 \\ \frac{\partial L}{\partial y_{2}}=12 y_{2}-\lambda=0\end{array}\right.\)
解得： \(\left\{\begin{array}{l}y_{1}=\frac{3}{4} y \\ y_{2}=\frac{1}{4} y\end{array}\right.\)
由于\(y=32\),即工厂1生产24单位产品，工厂2生产8单位产品。
2)厂商的成本函数为：
\(\begin{aligned} c(y) &=c_{1}\left(\frac{3}{4} y\right)+c_{2}(\frac{1}4 y) \\ &=\frac{3}{2} y^{2}+130 \end{aligned}\)
平均成本 \(AC( y)=\frac{c(y)}{y}=\frac{3}{2} y+\frac{130}{y}\)
令 \(\frac{d AC( y)}{d y}=\frac{3}{2}-\frac{130}{y^{2}}=0\)
得\(y=\sqrt{\frac{260}{3}} \doteq 9-3\)
故：
当 \(y \in(0,9,3)\)时，生产的规模报酬递增
当\(y \in(9.3,+\infty)\)时，生产的规模报酬递减
当\(y=9.3\)时，生产的规模报酬不变