Leontief preferences

Question

I can solve most utility maximization problems using my mathematical knowledge .... but not when it comes to Leontief preferences. I do not have a book to lean on (am self-studying), so would really like some help. How does one solve a general maximization problem like $$\max [\alpha x_1, \beta x_2, \gamma x_3] \ \text{subject to } \ \lambda_1 x_1 + \lambda_2 x_2 + \lambda_3 x_3 = M$$ where $M$ is income and $\lambda_i$ is price for good $i$?

Really, everything I know about derivatives and slopes goes all out the window with this damn thing. If somebody told me what the prices and income were, the optimal choice, when there are only a few goods, could probably be found by just applying common sense, but what about the general case? Is there no general "formula" like there is for Cobb Douglas and CES functions? Is there some go-to method that we use in these cases?

Answer 1

You are missing a $\min$ operator just before the bracket. The utility maximization problem is as follows, $$\max \ \min [\alpha x_1, ..., \gamma x_3] \\ \ \ \text{such that} \ \ \lambda_1x_1 + ... + \lambda_3x_3 = M$$ Consider the case of two goods with utility $u$ given by $u(x) = \min[\alpha x_1, \beta x_2]$. At the optimum, what do you know about the relation between $\alpha x_1$ and $\beta x_2$? They must be equal, that is $$\alpha x_1 = \beta x_2$$ For if not, then assume without loss of generality that $\alpha x_1 > \beta x_1$. What is the utility of such choices of $x_1$ and $x_2$? It must be $\beta x_2$, which means that some of your money is being spent on $x_1$ (assuming prices are strictly positive) but it is not giving you any extra utility, and so this cannot be an optimal choice of consumption.

It follows that the equality must hold at an optimum (this is obvious graphically too). Along with the budget constraint, that gives you two equations and two unknowns, which can be solved for optimal consumption. A similar approach can be applied to the case of $n$ goods.

Of course above is assuming that we are dealing with a trivial utility maximization problem, and aren't doing integer programming and the like.