Unregulated International Trade Unjustified By Comparative Advantage (Part 3)

3.0 Production Possibilities Frontier

3.1 Alpha Techniqe

Suppose 64 barrels ale are produced with the ale-producing process. And suppose 64 bushels corn are produced with the first corn-producing process. This is a matter of scaling the first two processes defined in Table 2-1. The quantity flows shown in Table 3-1 result. In a stationary state, the 64 barrels ale produced just replace the ale used up in in the corn-producing process. Likewise, eight of the 64 bushels corn produced replace the corn used up as a capital good in the ale-producing process. So the net output of this economy with these quantity flows is 56 bushels corn. These processes at this scale use all of the 320 person-years available from the labor force. They do not use all of the available land, but production cannot be increased in these proportions. The labor force provides a binding constraint.

Table 3-1: Quantity Flows for The Alpha Technique Producing Only Corn (14 2/3 Acres Land Unused)

INPUTS HIRED AT START OF YEAR	ALE INDUSTRY	CORN INDUSTRY
Labor	64 Person-Years	256 Person-Years
Land	72 Acres	53 1/3 Acres
Ale	0 Barrels	64 Barrels
Corn	8 Bushels	0 Bushels
OUTPUTS	64 Barrels Ale	64 Bushels Corn
NET OUTPUTS	0 Barrels Ale	56 Bushels Corn

Tables 3-2 and 3-3 show the results of other levels of production with the two techniques comprising the Alpha technique. In Table 3-2, the available labor force and the available land are fully used. In Table 3-3 some labor remains unused, but the constraint imposed by the fixed amount of land is binding.

Table 3-2: Quantity Flows for The Alpha Technique Producing Both Ale and Corn

INPUTS HIRED AT START OF YEAR	ALE INDUSTRY	CORN INDUSTRY
Labor	80 Person-Years	240 Person-Years
Land	90 Acres	50 Acres
Ale	0 Barrels	60 Barrels
Corn	10 Bushels	0 Bushels
OUTPUTS	80 Barrels Ale	60 Bushels Corn
NET OUTPUTS	20 Barrels Ale	50 Bushels Corn

Table 3-3: Quantity Flows for The Alpha Technique Producing Only Ale (149 9/59 Person-Years Labor Unused)

INPUTS HIRED AT START OF YEAR	ALE INDUSTRY	CORN INDUSTRY
Labor	113 53/59 Person-Years	56 56/59 Person-Years
Land	128 8/59 Acres	11 51/59 Acres
Ale	0 Barrels	14 14/59 Barrels
Corn	14 14/59 Bushels	0 Bushels
OUTPUTS	113 53/59 Barrels Ale	14 14/59 Bushels Corn
NET OUTPUTS	99 39/59 Barrels Ale	0 Bushels Corn

I have shown some quantity flows with the Alpha technique being exclusively used. Figure 3-1 shows all such possible quantity flows in which at least one of the land and labor constraints is binding. The intersection of the graphed locus with the ordinate is defined by Table 3-1. The point with a net output of 20 barrels ale and 50 bushels corn is defined by Table 3-2. And the intersection with the abscissa is defined by Table 3-3. The straight line connecting the intersection with the ordinate and the point (20 barrels, 50 bushels) represents a linear combination of the quantity flows shown in Tables 3-1 and 3-2. The other line segment represents a linear combination of the quantity flows shown in Tables 3-2 and 3-3. Any point on this locus or in its interior can be achieved by this economy operating the Alpha technique, given the endowments of labor and land.

Figure 3-1: Production Possibilities Curve For Alpha Technique

3.2 The Frontier

I relegate to the appendix the specification of stationary state quantity flows for the Beta technique. The analysis of those flows produce a locus like that shown in Figure 3-1, but with different intercepts and a different point (a net ouput of 80 barrels and 20 bushels) corresponding to the full employment of both labor and land.

Figure 3-2 shows the Production Possibilities Frontier. The Alpha technique is exclusively used in the portion of the frontier to the left of point a. Less land is employed here than in the endowment of the economy; hence land services are free for this portion of the PPF. The Beta technique is exclusively used in the portion of the frontier below point b. Labor is not a binding constraint for this portion of the PPF; labor services are free. The line segment connecting points a and b represents a linear combination of the Alpha and Beta techniques. Both land and labor are binding constraints along this segment, including at points a and b.

Figure 3-2: Production Possibilities Frontier

The portion of the frontier between points a and b, inclusive, is the focus of the remainder of this series of posts. The relative combinations of ale and corn, produced net, are different along this segment because of differences in the amount of ale and corn produced by the two techniques. Notice the slope of this segment is one-half bushels per barrels. The slope reflects the rate of transformation possible in comparing two autarkic equilibria. Consider two autarkic economies in stationary states facing this technology and these endowments, and suppose both labor and land are fully employed in these economies. One more bushel of corn is consumed in one economy for every two more barrels of ale consumed in the other.

In the next post, I consider prices for this economy, without and with trade. I don't have ask good leading questions here. Consider equilibria (in which both ale and corn are produced by cost-minimizing firms) without trade:

• If the interest rate is 2% and rent is 1/10 bushels per acre, how would you figure out what the wage and price of ale must be?
• For every feasible interest rate, what equations must the wage, the rent, and the price of ale satisfy such that cost-minimizing firms will be willing to adopt a linear combination of both techniques? How does the price of ale for this switch point compare to the slope of the Production Possibilities Frontier between points a and b?

Appendix 3.A Quantity Flows For The Beta Technique

Table 3-4: Quantity Flows for The Beta Technique Producing Only Corn (73 1/3 Acres Land Unused)

INPUTS HIRED AT START OF YEAR	ALE INDUSTRY	CORN INDUSTRY
Labor	21 1/3 Person-Years	298 2/3 Person-Years
Land	24 Acres	42 2/3 Acres
Ale	0 Barrels	21 1/3 Barrels
Corn	2 2/3 Bushels	0 Bushels
OUTPUTS	21 1/3 Barrels Ale	42 2/3 Bushels Corn
NET OUTPUTS	0 Barrels Ale	40 Bushels Corn

Table 3-5: Quantity Flows for The Beta Technique Producing Both Ale and Corn

INPUTS HIRED AT START OF YEAR	ALE INDUSTRY	CORN INDUSTRY
Labor	96 Person-Years	224 Person-Years
Land	108 Acres	32 Acres
Ale	0 Barrels	16 Barrels
Corn	12 Bushels	0 Bushels
OUTPUTS	96 Barrels Ale	32 Bushels Corn
NET OUTPUTS	80 Barrels Ale	20 Bushels Corn

Table 3-6: Quantity Flows for The Beta Technique Producing Only Ale (110 Person-Years Labor Unused)

INPUTS HIRED AT START OF YEAR	ALE INDUSTRY	CORN INDUSTRY
Labor	112 Person-Years	98 Person-Years
Land	126 Acres	14 Acres
Ale	0 Barrels	7 Barrels
Corn	14 Bushels	0 Bushels
OUTPUTS	112 Barrels Ale	14 Bushels Corn
NET OUTPUTS	105 Barrels Ale	0 Bushels Corn