Our thoughts return to Ambrose and his nuts and berries. Ambrose’s utility function is U(x1, x2) = 4√x1 + x2, where x1 is his consumption of nuts and x2 is his consumption of berries.

(a)  Let us find his demand function for nuts. The slope of Ambrose’s  indifference curve at (x₁, x₂) is______. Setting this slope equal to the slope of the budget line, you can solve  for x1 without even  using the budget equation. The solution is x1 =_____________.

(b)  Let us find his demand for berries. Now  we need the budget equation. In Part (a), you solved for the amount of x1 that he will demand. The budget equation  tells us that p1x1 + p2x2 = M. Plug the solution that you found for x1 into the budget equation and solve for x2 as a function of income and prices. The answer is x2 =______.

(c) When we visited Ambrose in Chapter 5, we  looked at a “boundary solution,” where Ambrose consumed only nuts and no  berries. In that example, p1 = 1, p2 = 2, and M = 9. If you plug these numbers into the  formulas we found in Parts (a) and (b), you find x1 = 16 , and x2 = ________. Since we get a negative solution for x2, it must be that the budget line x1 +2x2 = 9 is not tangent to an indifference curve when x2 ≥ 0. The best that Ambrose can do with this budget is to spend all of his  income on nuts. Looking at the formulas, we see that at the prices p1 = 1 and p2 = 2, Ambrose will demand a positive amount of both goods if and only if M > _____________.