#### 2. Derive consumer's expenditure function by minimizing total expenditure; \(y = {p_1}{x_1} + {p_2}{x_2}\) subject to utility constraint \(\bar u = {q_1}{q_2}\).

Expenditure funtion can be derived if we know either indirect utility function or the compensated demand curve of the two goods. Let us first derive the the compensated demand curve for the two goods.

Here objective is to minimize total expenditure subject to utility constraint. We can write Lagrangian function as follows: \[L = {p_1}{q_1} + {p_2}{q_2} + \lambda (\bar u - {q_1}{q_2})\] First order condition for minimization is: \[\frac{{\partial L}}{{\partial {q_1}}} = \frac{{\partial L}}{{\partial {q_2}}} = \frac{{\partial L}}{{\partial \lambda }} = 0\] Taking partial derivative of the Lagransian function with respect to q_{1}and equating to zero, we get: \[\frac{{\partial L}}{{\partial {q_1}}} = 0\] \[{p_1} - \lambda {q_2} = 0\] \[\lambda = \frac{{{p_1}}}{{{q_2}}}\] Similarly, taking partial derivative of the Lagransian function with respect to q

_{2}and equating to zero, we get: \[\frac{{\partial L}}{{\partial {q_2}}} = 0\] \[{p_2} - \lambda {q_1} = 0\] \[\lambda = \frac{{{p_2}}}{{{q_1}}}\] We can equate the values of \(\lambda \): \[\frac{{{p_1}}}{{{q_2}}} = \frac{{{p_2}}}{{{q_1}}}\] Rearranging the equation, we get: \[{q_2} = \frac{{{p_1}{q_1}}}{{{p_2}}}\] Sustituting this value of \({q_2}\) in the utility constraint, we get: \[\bar u = {q_1} \times \frac{{{p_1}{q_1}}}{{{p_2}}}\] Rearranging: \[\bar u = q_1^2 \times \frac{{{p_1}}}{{{p_2}}}\] \[{q_1} = \sqrt {\bar u\frac{{{p_2}}}{{{p_1}}}} \] This is the compensated demand function for Good - 1. Sustituting this value of \({q_1}\) in the utility constraint, we get: \[\bar u = \sqrt {\bar u\frac{{{p_2}}}{{{p_1}}}} \times {q_2}\] Rearranging: \[{q_2} = \sqrt {\bar u\frac{{{p_1}}}{{{p_2}}}} \] This is the compensated demand function for Good - 2.

Now, we can find expenditure function by substituting the compensated demand functions in the objective function as follows:
\[y = {p_1}{q_1} + {p_2}{q_2}\]
\[y = {p_1} \times \sqrt {\bar u\frac{{{p_2}}}{{{p_1}}}} + {p_2} \times \sqrt {\bar u\frac{{{p_1}}}{{{p_2}}}} \]
\[y = \sqrt {\bar u{p_1}{p_2}} + \sqrt {\bar u{p_1}{p_2}} \]
\[y = 2\sqrt {\bar u{p_1}{p_2}} \]
This is the expenditure function, we want to find. It can also be written us:
\[y = 2{{\bar u}^{\frac{1}{2}}}p_1^{\frac{1}{2}}p_2^{\frac{1}{2}}\]

Solution video:

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