2.4 The Ricardian Model Production Possibility Frontier

Using the two production functions and the labor constraint, we can describe the production possibility frontier (PPF)The set of all output combinations that could be produced in a country when all the labor inputs are fully employed. In the Ricardian model, the PPF is linear.. First, note that the production functions can be rewritten as L_C = a_LC Q_C and L_W = a_LW Q_W. Plugging these values for L_C and L_W into the labor constraint yields the equation for the PPF:

a_LC Q_C + a_LW Q_W = L.

This equation has three exogenous variables (a_LC, a_LW, and L) that we assume have known values and two endogenous variables (Q_C and Q_W) whose values must be solved for. The PPF equation is a linear equation—that is, it describes a line. With some algebraic manipulation, we can rewrite the PPF equation into the standard form for an equation of a line, generally written as y = mx + b, where y is the variable on the vertical axis, x is the variable on the horizontal axis, m is the slope of the line, and b is the y-intercept. The PPF equation can be rewritten as

$$Q_W=\frac{L}{a_{LW}}-\left(\frac{a_{LC}}{a_{LW}}\right)Q_C\text{.}$$

We plot the PPF on the diagram in Figure 2.1 "Production Possibilities" with Q_C on the horizontal axis and Q_W on the vertical axis. The equation is easily plotted by following three steps.

1. Set Q_C = 0 and solve for Q_W. In this case, the solution is $Q_W=\frac{L}{a_{LW}}$. This corresponds to the Q_W-intercept. It tells us the quantity of wine that the United States could produce if it devoted all of its labor force (L) to the production of wine.
2. Set Q_W = 0 and solve for Q_C. In this case, the solution is $Q_C=\frac{L}{a_{LC}}$. This corresponds to the Q_C-intercept. It tells us the quantity of cheese that the United States could produce if it devoted all of its labor force (L) to the production of cheese.
3. Connect the two points with a straight line.

The straight downward-sloping line is the production possibility frontier. It describes all possible quantity combinations of wine and cheese that can be achieved by the U.S. economy. A movement along the curve represents a transfer of labor resources out of one industry and into another such that all labor remains employed.

Points inside the PPF are production possibilities but correspond to underemployment of labor resources. In fact, all production possibilities regardless of whether full employment is fulfilled are referred to as the production possibility set (PPS). The PPS is represented by all the points within and on the border of the red triangle in Figure 2.1 "Production Possibilities".