Before:

$M=p_1{x}_1+p_2{x}_2$
$U(x_1,x_2)=x_1{x}_2$
$10=x_1+x_2$
$\frac{\frac{d(u(x_1,x_2)}{d(x_1)}}{\frac{d(u(x_1,x_2)}{d(x_2)}}=\frac{x_2}{x_1}=\frac{p_1}{p_2}=1$
so : $x_1=5;x_2=5$

$TU=25$

After:

$M=p_1{x}_1+p_2{x}_2$
$U(x_1,x_2)=x_1{x}_2$
$10=2.5x_1+x_2$
$\frac{\frac{d(u(x_1,x_2)}{d(x_1)}}{\frac{d(u(x_1,x_2)}{d(x_2)}}=\frac{x_2}{x_1}=\frac{p_1}{p_2}=2.5$
so : $x_1^{\prime }=2;x_2^{\prime }=5$
$TU=10$

TE:

$TE=x_1^{\prime }-x_1=-3$

SE:

$U=25=x_1{x}_2$
$\frac{\frac{d(u(x_1,x_2)}{d(x_1)}}{\frac{d(u(x_1,x_2)}{d(x_2)}}=\frac{x_2}{x_1}=\frac{p_1}{p_2}=2.5$
$x_1^{\prime \prime }=\sqrt{10};x_2^{\prime \prime }=2.5\sqrt{10}$
$SE=x_1^{\prime \prime }-x_1=-(5-\sqrt{10})$

EE:

$EE=TE-SE=-(\sqrt{10}-2)$

the budget line

$c_1{p}_1+(\frac{1+\pi }{1+i})c_2{p}_2=m_1+m_2\frac{1}{1+i}$

$c_1+(\frac{1+5\%}{1+5\%})c_2=100+100(\frac{1}{1+5\%})$

$c_1=195.238-c_2$

A

$d(u_A)/d(c_1)=(1/3)c_1^{(-2/3)}{c}_2^{2/3}$

$d(u_A)/d(c_2)=(2/3)c_1^{(1/3)}{c}_2^{-1/3}$

$\frac{d(u_A)/d(c_1)}{d(u_A)/d(c_2)}=(1/2)\frac{c_2}{c_1}=1$

so :
$c_2=2c_1$
$c_1=195.238-c_2$
$c_1=65.079$
$c_2=130.158$
$u_a=103.306$
saver

B

$d(u_B)/d(c_1)=(2/3)c_1^{(-1/3)}{c}_2^{1/3}$

$d(u_B)/d(c_2)=(1/3)c_1^{(2/3)}{c}_2^{-2/3}$

$\frac{d(u_B)/d(c_1)}{d(u_B)/d(c_2)}=2\frac{c_2}{c_1}=1$
so :
$c_2=1/2c_1$
$c_1=195.238-c_2$
$c_1=130.15867$
$c_2=65.079$
$u_b=103.306$
borrower

$save{r}_a$

$save{r}_A=m_1-c_1^{\ast }=m_1-(m_1+m_2\frac{1}{1+i})/(1+2(\frac{1+\pi }{1+i}))=100-(100+100\frac{1}{1+i})/(1+2(\frac{1+0.05}{1+i}))$

$save{r}_b$

$save{r}_B=m_1-c_1^{\ast }=m_1-(m_1+m_2\frac{1}{1+i})/(1+(1/2)(\frac{1+\pi }{1+i}))=100-(100+100\frac{1}{1+i})/(1+(1/2)(\frac{1+0.05}{1+i}))$

$save{r}_a+save{r}_b$

$sav{e}_{all}=save{r}_a+save{r}_b=100-(100+100\frac{1}{1+i})/(1+2(\frac{1+0.05}{1+i}))+100-(100+100\frac{1}{1+i})/(1+(1/2)(\frac{1+0.05}{1+i}))$

interesting rate raise up to 10%

$c_1{p}_1+(\frac{1+\pi }{1+i})c_2{p}_2=m_1+m_2\frac{1}{1+i}$

$c_1+(\frac{1+5\%}{1+10\%})c_2=100+100(\frac{1}{1+10\%})$

$c_1=190.909-0.95454545c_2$

A after raising

$c_1=190.909-0.95454545c_2$
$c_2=2\ast 0.954545c_1$
$c_1=63.64$
$c_2=133.33$
$u_a=104.1986$
better

B after raising

$c_1=190.909-0.95454545c_2$
$c_2=1/2\ast 0.954545c_1$
$c_1=127.27$
$c_2=66.67$
$u_b=102.5956$
worse

A

$M{U}_c=40c^{3}s$
$M{U}_s=10c^4$
$MRS=\frac{M{U}_c}{M{U}_s}=\frac{40c^{3}s}{10c^4}=\frac{4s}{c}=\frac{p_c}{p_s}=32;s=8c$

budget line: $32c+1s=80$

so : $c=2;s=16$

and utlity = 2560

B

$MRS=\frac{M{U}_c}{M{U}_s}=\frac{40c^{3}s}{10c^4}=\frac{4s}{c}=\frac{p_c}{p_s}=16;s=4c$

budget line: $16c+1s=80$

so $c=4;s=16$

and utlity = 40960

c

$MRS=\frac{M{U}_c}{M{U}_s}=\frac{40c^{3}s}{10c^4}=\frac{4s}{c}=\frac{p_c}{p_s}=16;s=4c$

and utlity = 2560

$c=640^{(1/5)};s=4(640^{(1/5)});M_b=16c+1s=72.822568121$

$SE=640^{(1/5)}-2$

$EE=4-640^{(1/5)}$

$\frac{1}{3}3000+\frac{1}{3}2000+\frac{1}{3}1000=2000$
he is willing to pay 2000 for a car.type B

$0\ast 3000+\frac{1}{2}2000+\frac{1}{2}1000=1500$
he is willing to pay 1500 for a car.type C

$0.6\ast 1000000+0.2\ast 1000000+0.1\ast 1000000=900000$