or greater in absolute value than, lhat of the budget line, as the condition above slates. At the corner solution, the slope of the indifference curve is either equal to. There is not an interior tangency solution, bccausc along the budget constraint the indifference curves are everywhere steeper than the budget line. (Be sure you can explain the direction of this inequality.) This solution is illustrated in figure 15.2 (b). Take the case x* >0, xi = 0įrom condition (15,3) we then have x* = mfp\, while condition (15.2) implies that ii i = X*p iįigure 15.2 Solution possibilities lor the consumer's problem giving There are, however, two further cases to consider, namely those in which one of the goods has a zero value at the optimum. This is the condition that the indifference curve be tangent to the budget constraint, as illustrated in figure 15.2 (a). Then condition (15.2) becomes u, - X*ph i = 1,2, giving the condition Since it is usually assumed in consumer theory that u, > 0-this is known as the "'non-satiation" assumption-we have the justification for always taking the budget constraint as an equality rather than a weak inequality.
It follows from the second condition that the budget constraint must be satisfied as an equality that is, it is a binding constraint. Then from the first condition we see that if'.v* > 0 and its marginal utility m, is positive, we must also have a* > 0. We need only consider cases in which at least one of the demands is positive at the optimum. We can write the concave-programming problem in a simple form as max f s.t.
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