Consider the Cobb$-$Douglas utility function given $byu=x_1^{\frac{1}{4}}{x}_2^{\frac{3}{4}}$ and the

budget constraint $p_1{x}_1+p_2{x}_2=m.$

$We$ have derived its Marshallian demands $x(p,m),$ indirect utility function

$v(p,m),$ expenditure function $e(p,u),$ and Hicksian demands $h(p,u).$Check that Marshallian demands and Hicksian demands satisfy the $4$ components $of$ Slutsky equation$(two$ goods$$ two prices$)$ A generic Cobb$-$Douglas utility function $isu=x_1^a{x}_2^b,$ with $a,b>0.If$ one

imposes constant returns $to$ scale, $a+b=1,so$ you can write $u=x_1^a{x}_2^{1-a}.$

$We$ now proceed $to$ calculate relevant consumption functions for the

example $u=x_1^{\frac{1}{4}}{x}_2^{\frac{3}{4}}.$ The general problem $is$ maximise $u$ subject $to$

$p_1{x}_1+p_2{x}_2=m.$

Find the Marshallian demands $x(p,m),$ indirect utility function $v(p,m),$

the expenditure function $e(p,u),$ and Hicksian demands $h(p,u).u=x_1^{\frac{1}{4}}{x}_2^{\frac{3}{4}},s.t.,p_1{x}_1+p_2{x}_2=m$

$we$ have

$L=x_1^{\frac{1}{4}}{x}_2^{\frac{3}{4}}-(p_1{x}_1+p_2{x}_2-m)$

then,

$\frac{\text{d}\text{e}\text{l}\text{L}}{del{x}_1}=\frac{1}{4}{x}_1^{-\frac{3}{4}}{x}_2^{\frac{3}{4}}-p_1=0,=>,\frac{1}{4}{x}_1^{-\frac{3}{4}}{x}_2^{\frac{3}{4}}=p_1$

$\frac{\text{d}\text{e}\text{l}\text{L}}{del{x}_2}=\frac{3}{4}{x}_1^{\frac{1}{4}}{x}_2^{-\frac{1}{4}}-p_2=0,=>,\frac{3}{4}{x}_1^{\frac{1}{4}}{x}_2^{-\frac{1}{4}}=p_2$

$\frac{\text{d}\text{e}\text{l}\text{L}}{\text{d}\text{e}\text{l}}=0=>p_1{x}_1+p_2{x}_2=m$ From the first two equations above $we$ have

$\frac{1}{4}{x}_1^{-\frac{3}{4}}{x}_2^{\frac{3}{4}}\frac{1}{p_1}=\frac{3}{4}{x}_1^{\frac{1}{4}}{x}_2^{-\frac{1}{4}}\frac{1}{p_2},=>,p_2{x}_2=3p_1{x}_1$

Considering the budget constraint $p_1{x}_1+p_2{x}_2=m,we$ have a system $of$ two

equations $in$ two unknown, the solution $of$ which are the Marshalian demands

$x_1=\frac{m}{4p_1}$

and

$x_2=\frac{3m}{4p_2}$

The indirect utility function $is$ obtained $by$ substituting these into $u,$

$V=(\frac{m}{4p_1}{)}^{\frac{1}{4}}(\frac{3m}{4p_2}{)}^{\frac{3}{4}}$The expenditure function $e(p,u)$ can $be$ obtained $as$ follows. First $of$ all

$we$ verify that $we$ have a binding constraint $by$ substituting Marshallian

demands $in$ the expenditure, $p_1{x}_1+p_2{x}_2=p_1*\frac{m}{4p_1}+p_2*\frac{3m}{4p_2}=m.$

Secondly, $we$ write the expenditure $in$ terms $of{x}_2,$ with $x_1=\frac{u^4}{x_2^3}$ from

the expression $of$ the utility, and then substitute the Marshallian demand

expression $of{x}_2.We$ aim $at$ a function $of(p,u)$ :

$e(p,u)=p_1{x}_1+p_2{x}_2=p_1*\frac{u^4}{x_2^3}+p_2{x}_2=p_1*\frac{u^4}{(\frac{3m}{4p_2}{)}^3}+p_2*\frac{3m}{4p_2}$

$By$ simplifying out, and considering that $e(p,u)=mwe$ have

$\frac{1}{4}m=(\frac{4}{3}{)}^3{p}_1{p}_2^3\frac{u^4}{m^3}$

and rearranging $we$ obtain

$e(p_1,p_2,u)=m=\frac{4}{3^{\frac{3}{4}}}{p}_1^{\frac{1}{4}}{p}^{\frac{3}{4}}$uThe Hicksian demands are obtained $by$ differentiation

$h_1(p_1,p_2,u)=\frac{\text{d}\text{e}\text{l}\text{e}}{del{p}_1}=\frac{4}{3^{\frac{3}{4}}}\frac{1}{4}{p}_1^{-\frac{3}{4}}{p}_2^{\frac{3}{4}}u=\frac{1}{3^{\frac{3}{4}}}(\frac{p_2}{p_1}{)}^{\frac{3}{4}}u$

$h_2(p_1,p_2,u)=\frac{\text{d}\text{e}\text{l}\text{e}}{del{p}_2}=\frac{4}{3^{\frac{3}{4}}}\frac{3}{4}{p}_1^{\frac{1}{4}}{p}_2^{-\frac{1}{4}}u=3^{\frac{1}{4}}(\frac{p_1}{p_2}{)}^{\frac{1}{4}}u$