Deriving Production Possibilities Frontiers

Deriving Production Possibilities Frontiers

INTRODUCTION AND ASSUMPTIONS

Two-good x two-input model

Production functions

Let:

X=x(Lx, Kx) X is produced by the amount of labor and capital distributed to its

Y=y(Ly, Ky) production, ditto for Y.

Other assumptions:

Both L and K necessary for production.The two production functions are constant returns to scale. This produces several results, most important is that the curvature of isoquants are the same such that the MRTS is the same along the K/L ray from the origin.

Ktotal = Kx + Ky, full employment assumptions

Ltotal = Lx + Ly

L and K perfectly mobile between production of X or Y. In other words, L and K are homogeneous.

Production functions for X and Y differ sufficient but not necessarily if for some wage rental ratio w/r (slope of an isocost), their capital labor ratios differ.

In the diagram to the right there are two representative isoquants for the production of goods X and Y. Both are tangent to an isocost with the slope of w/r. Each tangency has a ray from the origin passing through it, the slopes of these rays are the capital labor ratios used in producing X and Y, and are labeled kX and kYrespectively. X and Y are different goods using different technologies because for the same w/r they have different capital labor ratios. In this case X is considered to be a labor intensive good and Y is a capital intensive good. The production of Y uses relatively more capital than labor then does the production of good X and vice versa.

EDGEWORTH BOX

The next step in the derivation is to shift our attention away from the production of individual goods to how a country’s endowments of labor and capital are allocated to the production of both goods. This will be done with the help of an Edgeworth Box.

Let Lt and Kt stand for the total amount of labor and capital in a particular country. In our production graphs that could be represented as the following graph. Now imagine this graph is specific for the production of good X. Let a second graph describe the production of good Y. Now rotate Y’s graph 180° so that the origin is in the upper right hand corner.

To create an Edgeworth Box we now align the two graphs together so that the horizontal and vertical distances between the two origins are the endowments of labor and capital respectively. The Edgeworth Box now allows us to study the distribution of labor and capital between the two goods maintaining full employment. Any point in the box such a point A describes the distribution of labor Lx to good X and Ly to good Y as well as the distribution of capital Kx and Ky .

We will interested in several different distributions of L and K across X and Y. In particular is the proportional distribution of inputs across the two goods. This is portrayed as the diagonal between the two origins. Halfway along the diagonal L and K are equally distributed (point w). Another example of a proportional distribution would occur one fourth along the diagonal from the X origin (black dot). Here we are allocating one fourth of L and K to X and three fourths of L and K to Y.

Recall that one assumption of the production functions for X and Y is constant returns to scale. In other words, a proportional increase in all inputs leads to the same proportion increase in output. For example a doubling of inputs doubles output. With this in mind we can begin constructing a Production Possibilities Frontier. If we allocate all resources to the production of good X, the upper right-hand corner of the Edgeworth Box, we’d be able to produce some maximal amount of good X, say Xt described by the isoquant that runs through that point. Conversely, if we allocate all resources to the production of good Y, we’d be located in the lower left-hand corner, producing some maximal amount of good Y, say Yt. Now if we allocated our resources one half to production of good X and the other half to good Y, with constant returns to scale we’ll produce one-half of Xt and one-half of Yt.

Graphing these quantities of X and Y in a Production Possibilities Frontier gives a downward sloping line between Yt and Xt. All points on this line correspond to the levels of X and Y along the diagonal in the Edgeworth Box.

The question we now turn our attention to is whether we can produce more X and/or Y located at the halfway point along the Edgeworth Box diagonal. Is it possible to increase the production of one good while holding the amount of the other constant? In this case we will increase the production of X while holding the amount of Y constant along the ½ Yt isoquant. As we move left to right along the ½ Yt holding the amount of Y constant we move to progressively higher X isoquants lying above the ½ Xt isoquant line. We continue to increase production of X until we can no longer move to a higher X isoquant without moving to a lower Y isoquant. The maximum amount of X (Xp) we can produce holding Y constant at 1/2Yt is given by the tangency of the X isoquant to the 1/2Yt isoquant. The same logic if we hold X constant and increase the amount of Y. We’d move along the 1/2Xt isoquant to the tangency of the Yp isoquant.

At the tangencies we have an important principle, Pareto efficiency. That is we cannot increase production of one good without reducing the production of the other. Note as we reallocate labor and capital increasing production we are shifting more labor and less capital to the labor intensive good, and doing the reverse for the capital intensive good.

These two Pareto efficient combinations of labor and capital correspond to points on the Production Possibilities Frontier. In the diagram to the left we have two such points. Recall that the initial distribution of L and K was half way along the diagonal in the Edgeworth Box which when plotted as a PPF resulting in output levels of 1/2Yt and 1/2Xt. Reallocating to become Pareto efficient we can increase production of one good holding the other constant. For example, holding Y at 1/2Yt we can increase production of X to Xp. The converse is also true holding X constant and increasing Y.

Returning to an Edgeworth Box, we initially started with a distribution of L and K halfway along the diagonal. That resulted through redistributing L and K to two Pareto efficient allocations. If we choose some other initial distribution we’ll have two other Pareto allocations. If we connect all the possible Pareto efficient allocations in the Edgeworth Box we get a curve between the two origins called an efficiency locus. Remapping the quantities of X and Y along the efficiency locus results in a Production Possibilities Frontier.

The student should contemplate several issues. What contributes to the concavity of a PPF? How are resources allocated? What does Pareto efficiency mean?