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Given the utility function $u(x_1,y_2)=2\sqrt{x_1}+\sqrt{x_2}$ maximize it with the given constraint $p_1 x_1+p_2 x_2=m$.

Having solved the problem, using the standard Lagrangian multiplier method, I found that $(x_1^*,x_2^*)=\left(\frac{4p_2 m}{p_1^2+4p_1 p_2} ,\frac{p_1 m}{4p_2^2+p_1 p_2}\right)$ and that $\lambda =\sqrt{\frac{4p_2+p_1}{4p_1 p_2m}}$.

Now, my question is this: How do I interpret $\lambda$ with respect to the indirect utility function $\frac{2}{p_1\lambda}+\frac{1}{p_2 \lambda}$?

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$\lambda$ is usually interpreted as the utility of money or income in these consumer optimization problems.

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The Lagrange multiplier is (up to a minus sign) the rate of change of the objective function at the new optimum point with respect to a change in the value of the constraint. In your notation here we might loosely say $\lambda = \frac{du}{dm}$. So if your constraint is how much money you have to spend, then $\lambda$ measures how much additional utility you could obtain if you had a little bit more money.

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