.webp)
You
have given the utility function
`U(X_(1)X_2)=X_1^(0.4)X_2^(0.6)`
And budget constraint M=P1X1+P2X2
Where,
M=100, P1=10, P2=20
1. Find the Indirect Utility function.
Solution: `U(X_(1)X_2)=X_1^(0.4)X_2^(0.6)`
M=100, P1=10, P2=20
Set Lagrangian function
` L=X1^{0.4}X2^{0.6}+\lambda(M-P_{1\}X_{1\}-\P_2X_{2\})`
Take partial derivatives with respect to `X_1,X_2and\lambda` and set it equal to zero
`\frac{\partial L}{\partial X_1}=0.4X_1^{0.4-1}X_2^{0.6}-\lambda P_1\=0`
`0.4X_1^{0.4-1}X_2^{0.6}-\lambda P_1\=0`.......................(1)
`\frac{\partial L}{\partial X_2}=0.6X_1^{0.4}X_2^{0.6-1}-\lambda P_2\=0`
`0.6X_1^{0.4}X_2^{0.6-1}-\\lambda P_2\=0`...............................(2)
`\frac{\partial L}{\partial\lamda}=\M-P_{1\}X_{1\}-P_{2\}X_{2\}=\0` ......................(3)
Taking the ratio of equation (1) and (2)
`\frac{0.4X_1^{-0.6}\X_2^{0.6}-\lambda P_1}{0.6\X_1^{0.4\}\X_2^{-0.4}\-\lambda P_2}\=\0`
`\frac{0.4}{0.6}\X_1^{-0.6-0.4}\X_{2\}^{0.6+0.4\}=\frac{P_1}{P_2}`
`\frac{0.4}{0.6}\frac{X_2}{X_1}\=\frac{P_1}{P_2}`
`X_{2=\}\frac{0.6}{0.4}\left(\frac{P_1}{P_2}\right)\X_1`
`X_{2=\}\1.5\left(\frac{P_1}{P_2}\right)\X_1`
`X_{1=\}1.5\frac{P_2}{P_1}\left(X_2\right)`
Put the value of X2 in equation (3)
`M-P_{1\}X_1\-P_2\X_{2\}=\0`
`M-P_{1\}X_1\-P_2\left(1.5\frac{P_1}{P_2}\right)\X_1=\0`
`M-P_{1\}X_1\-1.5P_1X_1=\0`
`M=P_{1\}X_1\+1.5P_1X_1`
`M=2.5P_1X_1`
`X_1^\ast=\frac M{2.5P_1}` .................MDF
Put the value of M and P1 in the above equation.
`X_1^\ast=\frac{100}{2.5\left(10\right)}`
`X_1^\ast=\frac{100}{25}\Rightarrow X_1=4`
Put the value of X1 in equation (3) to get `X_2^\ast`
`M=P_1\left(1.5\frac{P_2}{P_1}X_2\right)+P_2X_2`
`M=1.5P_2X_2+P_2X_2`
`M=2.5P_2X_2`
`X_2^\ast=\frac M{2.5P_2}` .........................MDF
Put the values
of M and `P_2` in `X_2^\ast`.
`X_2^\ast=\frac{100}{2.5\left(20\right)}\Rightarrow\frac{100}{50}\Rightarrow2`
`X_2^\ast\=2`
Put the value
of `X_1^\ast` and `X_2^\ast` in the utility function to get the indirect utility function or we get the indirect utility function by putting
Marshallian demand for `X_1^{}` and `X_2^{}` in the utility function.
`U\left(X_1X_2\right)=X_1^{0.4}X_2^{0.6}`
`U=\left(\frac M{2.5P_1}\right)^{0.4}\left(\frac M{2.5P_2}\right)^{0.6}`
`V=\frac M{2.5P_1^{0.4}P_2^{0.6}}`................... Indirect Utility (.)
Now put the
values of M, `p_1`, and `p_2` in IVF.
`V=\frac{100}{2.5(10)^{0.4}\(20)^{0.6}}`
`V=\frac{100}{2.5(15.157)}`
`V=\frac{100}{37.89}`
`V=2.639`
2. Prove that Indirect U(.) is
homogeneous of degree zero in price and income.
According to
this property of indirect utility (.) when price and income increase by the same
proportion utility is unchanged or unaffected.
So, we have
`V=\frac M{2.5P_1^{0.4}\P_2^{0.6}}`
Multiply M and
P by the constant term θ.
`V=\frac{\theta M}{2.5(\theta P_1)^{0.4}\(\theta P_2)^{0.6}}`
`V=\frac\theta{\theta^{0.4+0.6}}\left[\frac M{2.5P_1^{0.4}P_2^{0.6}}\right]`
`V=\theta^{1-1}\left[\frac M{6.25P_1^{0.4}\;P_2^{0.6}}\right]`
`V=\theta^0\left[\frac M{2.5P_1^{0.4}\;P_2^{0.6}}\right]`
`V=\theta^0V`
Thus we proved
that IVF is homogeneous of degree zero in price and income.
3. Prove that Indirect Utility (.) is
not decreasing in income.
This property
of IVF reveals that whenever income increases utility also increases.So Indirect
Utility (.) is not decreasing in income.
Proof:
We have `V=\frac M{2.5P_1^{0.4}P_2^{0.6}}`
Differentiate
with respect to M.
`\frac{\ dV}{\ dM}=\frac1{2.5P_1^{0.4}P_2^{0.6}}>zero`
We solve it
further. So, multiply and divide by M.
`\frac{\ dV}{\ dM}=\frac1M\left(\frac M{2.5P_1^{0.4\;}P_2^{0.6}}\right)`
`\frac{\ dV}{\ dM}=\frac1M\left(V\right)`
`\frac{\ dV}{\ dM}=\frac VM`
Since V ≥ 0
and M ≥ 0. Therefore, the Indirect Utility function is not decreasing in income.
4. Prove that the Indirect Utility function
is not increasing in price.
According to
this property of Indirect Utility function. Indirect Utility is non-increasing
in price which means that when the price increases utility is not increasing rather it
decreases so, IVF is non-increasing in price.
Proof:
`V=\frac M{2.5P_1^{0.4}P_2^{0.6}}`
Or `V=\frac{MP_1^{-0.4}P_2^{-0.6}}{2.5}`
Differentiate
with respect to any one of the prices so, we will differentiate it with respect
to `P_1`.
`\frac{\ dV}{\ dP_1}=\frac{-1MP_1^{-1-1}P_2^{-1}}{6.25}`
`\frac{\dV}{\dP_1}=\frac{-1MP_1^{-2}P_2^{-1}}{6.25}`
`\frac{\ dV}{\ dP_1}=\frac{-0.4MP_1^{-0.4-1}P_2^{-0.6}}{2.5}`
`\frac{\ dV}{\ dP_1}=\frac{-0.4MP_1^{-0.4\}P_1^{-1}P_2^{-0.6}}{2.5}`
`\frac{\dV}{\ dP_1}=-0.4\left(\frac M{2.5P_1^{0.4}P_2^{0.6}}\right)P_1^{-1}`
`\frac{\ dV}{\ dP_1}=-0.4\left(V\right)P_1^{-1}`
`\frac{\ dV}{\dP_1}=\frac{-0.4}{P_1}\left(V\right)<0`
Thus we proved
that IVF is not increasing in income. Here the negative sign shows that IVF is
not increasing in price.
5. The Indirect Utility function satisfies
Roy’s Identity.
According to
Roy’s Identity, the ratio of the partial derivatives of the Indirect Utility
function with respect to income and price is equal to the Marshallian demand
function with a negative sign. So, one important thing here is that MDF can be obtained from IVF.
As `\frac{\ dV}{\ dM}=\frac VM` and `\frac{\ dV}{\ dP_1}=\frac{-0.4}{P_1}\left(V\right)`
Taking ratio
`\frac{\ dV\div \ dP_1}{\ dV\div\ dM}=\frac{-0.4V\div P_1}{V\div M}`
`=-\left(\frac{0.4V}{P_1}\right)\left(\frac MV\right)`
`\frac{\ dV\div \ dP_1}{\ dV\div\ dM}=\frac{-0.4M}{P_1}`..............MDF
Put the values
of M and `P_1`
`\frac{\ dV\div \ dP_1}{\ dV\div\ dM}=\frac{-0.4(100)}{10}`
`\frac{\ dV\div \ dP_1}{\ dV\div\ dM}=-4`
6. The Indirect Utility function is quasi-convex to price.
The indirect Utility function is strictly, quasi convex to prices.
That is
following set is strictly convex.
`A(P^\ast)=\{\frac P{P^\ast}\cong P\}\or\A(P)={\frac P{U(P,M)}\leq U^\ast(P,M)\}`
If we consider
two price vectors `P^a` and `P^b` then their linear combination
`U^\ast`(λ`P^a`+(1- λ)`P^b`,M) < λ`U^\ast`(`P^a`,M)+ (1-λ)`U^\ast` (`P^b`,M)
{utility at
the average at single price set}< { average of utility at two different
price set}
When price
fluctuates; it benefits people.
For more questions visit our website:https://easyeconomicsbyhaj.blogspot.com
