5.3 The Tangency Condition

In the example we looked at in the last section, the indifference curve passing through the optimal point was tangent to the PPF at that point. This is not a general rule: as we’ll see in the next chapter, there are several kinds of cases in which the optimum is not characterized by this kind of tangency condition. But for certain utility functions and PPFs, we can say that the optimal choice is characterized by two conditions:

• Tangency condition: At the optimal bundle, $MRS = MRT$
• Constraint: The optimal bundle lies along the PPF

Let’s see how we can use this to solve mathematically for the optimal bundle in the previous case. In Chapter 3, we showed that the equation for the PPF for this situation was \({x_1^2 \over 100} + {x_2^2 \over 36} = 100\) This is the constraint. The tangency condition comes from setting our expression for $MRS$ equal to the expression for the $MRT$: \(\begin{aligned} MRS(x_1,x_2) &= MRT(x_1,x_2)\\ {16x_2 \over 9x_1} &= {9x_1 \over 25x_2}\\ x_2^2 &= {9 \over 16} \times {9 \over 25} x_1^2\\ x_2 &= {9 \over 20}x_1 \end{aligned}\) We now have two equations (the constraint condition and the tangency condition) in two unknowns ($x_1$ and $x_2$). Plugging the expression for $x_2$ from the tangency condition into the constraint condition reveals the point along the PPF where $MRS = MRT$: \(\begin{aligned} {x_1^2 \over 100} + {\frac{81}{400}x_1^2 \over 36} &= 100\\ {x_1^2 \over 100} + {9x_1^2 \over 1600} &= 100\\ 25x_1^2 &= 160000\\ 5x_1 &= 400\\ x_1^\star = 80 \end{aligned}\) and therefore, from the tangency condition, \(x_2^\star = {9 \over 20}x_1^\star = 36\) To visualize what’s going on here, we can plot the tangency condition and the PPF in a single diagram; the optimal point is where they cross:

Because the tangency condition depends on the utility function, it follows (as we would hope!) that people with different utility functions would have different optimal bundles: people who like fish more would want to produce and consume more fish, and people who like coconuts more would want to produce and consume more coconuts.

We solved the above case for the Cobb-Douglas utility function \(u(x_1,x_2) = 16 \ln x_1 + 9 \ln x_2\) If we do the same exercise for the more general utility function \(u(x_1,x_2) = a \ln x_1 + b \ln x_2\) The MRS becomes \(MRS = {a \over b} \times {x_2 \over x_1}\) so the tangency condition becomes \(x_2 = \sqrt{\frac{b}{a}}\times {3 \over 5}x_1\) Note that as $a$ increases, this gets flatter, resulting in a point along the PPF with more fish: that is, the more you like good 1, the more good 1 you’ll produce when you find your optimal bundle. Likewise, as $b$ increases, this gets steeper, resulting in a point along the PPF with more coconuts: