Maximization of Utility

The maximisation of utility

The rational consumer desire to purchase a combination of x1 and x2 form which he derives the highest level of satisfaction. His problem is one of maximisation. However, his income is limited, and he is not able to purchase unlimited amount of the commodities. The consumer’s budget constraint can be written as 

\( y=\; p_{1}x_{1}\; +\; p_{2}x_{2}\)

Where y is the income of the consumer which is fixed and p1 and p2 are the prices of x1 and x2 respectively. The amount he spends on the first commodity p1x1 plus the amount he spends on the second p2x2 equals his income y.

The first and second order conditions

Let the utility function be \(U\; =\; f\left( x_{1},x_{2} \right)\) subject to the constraint given above. From the Langrange function

\(V\; =\; f\left( x_{1},x_{2} \right)\; +\; \lambda \; \left( y\; -\; p_{1}x_{1}\; -\; p_{2}x_{2} \right)\)

The first order conditions are obtained by setting the first partial derivatives of the above equation with respect to x1, x2 and \( \lambda \)  equal to zero :

\(\frac{\partial V}{\partial x_{1}}=\; f_{1}\; -\; \lambda p_{1} = 0\)

\(\frac{\partial V}{\partial x_{2}}=\; f_{2}\; -\; \lambda p_{2} = 0\)

\(\frac{\partial V}{\partial \lambda }=\; y\; -\; p_{1}x_{1}\; -\; p_{2}x_{2} = 0\)

Transposing the second terms in the first two equations above to the right and diving the first by the second yields :

\(\frac{f_{1}}{f_{2}}=\; \frac{p_{1}}{p_{2}}\)

The ratio of the marginal utilities must equal the ratio of prices for a maximum. Since \(\frac{f_{1}}{f_{2}}\) is the Rate of Commodity Substitution, the first-order condition for a maximum is expressed by the quantity of the rate of commodity substitution and the price ratio.

The second order condition as well as the first order condition must be satisfied to ensure that a maximum is actually reached. Denoting the second direct partial derivatives of the utility function by \(f_{11}\; and\; f_{22}\) and the second cross partial derivatives by \(f_{12}\; and\; f_{21}\) , the second order condition for a constraint maximum requires that the relevant bordered Hessian determinant be positive :

\[
\begin{pmatrix}
f_{11} & f_{12} & -p_{1}\\
f_{21} & f_{22} & -p_{2} \\
-p_{1} & -p_{2} & 0 \\
\end{pmatrix}
> 0
\]

Expanding the above equation :

\(2f_{12}p_{_{1}}p_{2}\; -\; f_{11}p_{2}^{2}\; -\; f_{22}p_{1}^{2}\; >\; 0\)

Now substituting \(p_{1}=\; \frac{f_{1}}{\lambda }\; and\; p_{2}\; =\; \frac{f_{2}}{\lambda }\) , we get ,

\(2f_{12}f_{_{1}}f_{2}\; -\; f_{11}f_{2}^{2}\; -\; f_{22}f_{1}^{2}\; >\; 0\)

The above inequality is satisfied by the assumption of regular strict quasi-concavity. This assumption ensures that the second order condition is satisfied at any point at which the first order condition is satisfied.

 

 

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