Managerial Economics and Strategy, 2/e

Appendix 4A The Marginal Rate of Substitution

We can derive Equation 4.1 using calculus. Lisa’s utility function is U(B, Z). Along an indifference curve, we hold utility fixed at $\overline{U}=U\left(B,Z\right).$ [&*obar*{U}|=|U(B, Z).&] If we increase Z, we would have to lower B to keep her on the same indifference curve. Let B(Z) be the implicit function that shows how much B it takes to keep Lisa’s utility at $\overline{U}$ [&*obar*{U}&] given that she consumes Z. Thus, we can write her indifference curve as $\overline{U}=U\left(B\left(Z\right),Z\right),$ [&*obar*{U}|=|U(B(Z), Z),&] which is solely a function of Z.

We want to know how much B must change if we increase Z, dB/dZ, given that we require her utility to remain constant by staying on the original indifference curve. ...