Consider the following macroeconomic model involving the linkage of the economies of the U.S., Canada and Mexico.
The demand for the production of the U.S. consists of consumer demand (for the output of the U.S.), investment demand, government demand and export demand. Consumer demand is a function of the level of production in the U.S. The export demand consists of exports to Canada, Mexico and the rest of the world. The exports of the U.S. to Canada are the imports of Canada from the U.S. and are a function of the level of income and production in Canada. Likewise the level of exports from the U.S. to Mexico is a function of the output of Mexico.
Let the levels of output of the U.S., Canada, and Mexico be denoted as X1, X2 and X3, respectively. Let ai,j denote the coefficient of demand for country i's product by country j. To simplify the analysis it will be assumed that all the imports of a country are for consumption.)
The condition for macroeconomic equilibrium in the U.S. is
The same applies for the demand for the outputs of Canada and Mexico; i.e.,
These equations may be put into matrix form as:
| X1 |
| X2 |
| X3 |
| = |
| I1+G1 |
| I2+G2 |
| I3+G3 |
| + |
| a1,1 a1,2 a1,3 |
| a2,1 a2,2 a2,3 |
| a3,1 a3,2 a3,3 |
| * |
| X1 |
| X2 |
| X3 |
If we let
| X1 |
| X2 |
| X3 |
| = |
| X, |
| a1,1 a1,2 a1,3 |
| a2,1 a2,2 a2,3 |
| a3,1 a3,2 a3,3 |
| = |
| A |
| and |
| I1+G1 |
| I2+G2 |
| I3+G3 |
| = |
| F |
With the above definitions the equations can be written as:
| 1 0 1 |
| 0 1 0 |
| 0 0 1 |
Thus the equations reduce to:
We can subtract AX from both sides to get
and factor out an X on the left-hand-side of the equations giving:
The solution is obtained by multiplying both sides of the equation by the inverse of (I-A); i.e.,
The "multiplier" in this case is the matrix (I-A) -1.
The preceding derivation ignored the role of imports in the computation of GDP. The National Income Accounting identity is
The matrix we need then to describe the interaction is then:
| a1,1-m1 | a1,2 | a1,3 |
| a2,1 | a2,2-m2 | a2,3 |
| a3,1 | a3,2 | a3,3-m3 |
which may be considered as A-M where the matrix M is:
| m1 | 0 | 0 |
| 0 | m2 | 0 |
| 0 | 0 | m3 |
If we call this matrix A'=A-M then the solution for the outputs of the three countries is:
The better representation of the macroeconomic model its solution is:
The Mississippi River deposits millions of tons of sediment each year onto the continental shelf of North America in the Gulf of Mexico. As the writer Mark Twain observed, if this process continued unaltered the delta of the Mississippi would extend like a fishing pole from Louisiana to the Yucatan Peninsula. But instead the delta fans out from the lower end of the Mississippi. This occurs partly because the buildup of sediment encourages multiple channels that distribute the flow. But also every thousand or so years the Mississippi finds a new major channel.
There has been tremendous economic development along the present main channel which includes the cities of New Orleans and Baton Rouge as well as industrial plants elsewhere dependent upon the Mississippi for fresh water and deep water transportation. Three million people are depending upon the present channel of the Mississippi River.
The problem is that the Mississippi is on the verge of switching to a new channel along what is now the Atchafalaya River. The pronunciation of Atchafalaya is a bit troublesome because it is the French spelling of an Indian word. In French "ch" is pronounced as "sh" is in English. Thus "tch" is used in French to denote the English "ch" sound. Therefore the pronunciation of Atchafalaya is as though it were spelled "achafalaya."
The Atchafalaya River has already captured the Red River which flows from the west and used to be a tributary of the Mississipi. Already 30 percent of the flow of the Mississippi goes into a channel called the Old River and thence into the Atchafalaya River. The configuration is roughly in the form of an H in which the the Atchafalaya-Red Rivers form the left leg and the Mississippi the other leg with the Old River being the cross branch.
The Old River Control Project of the Corp of Engineers is working to prevent the capture of 100 percent of the Mississippi by the Atchafalaya. But the Corps of Engineers doesn't want to cut off all flow through the Old River because agricultural and marine development along the Atchafalaya River would be hurt. The Corps is committed to maintaining the 30 percent diversion that now exists.
Much of the present problem exists because of the past efforts of the Corps of Engineers. Until the nineteenth century about thirty miles of the channel of the the Atchafalaya was blocked by a prehistoric log jam. The Corps and others cleared away this plug of timber. The Red River was also cleared. The Red River had been a direct tributary of the Mississippi for two millenia, but due to the clearing of the Atchafalaya it was captured by the Atchafalaya in the 1940s. Fred Bayley, the chief engineer of the Lower Mississippi Valley Division of the Corp of Engineers, put this way, "The more water the Atchafalaya takes, the bigger it gets; the bigger it gets, the more water it takes."
The Old River was once part of the Mississippi. There a meander of the river where it almost looped back upon itself. The Corps decided to eliminate the meander by cutting a channel through the narrowest part. The Corps-made channel was quickly widened by the Mississippi and the meander virtually dried up. This is why the Old River has that name.
The Corps also blocked various distributary channels until now only the Atchafalaya diverts water from the main channel.
Consider a project that requires $5 million to build a dam and canal which will provide water to irrigate desert land that has no other use. The land is used to grow cucumbers. Suppose the demand function for cucumbers were known to be
Suppose that a project produces 0.5 million pounds per year. Without the project the quantity of cucumbers available is 6 million pounds per year. With the project then the quantity is 6.5 million pounds per year.
What is the gross benefit of the project? The gross benefit is the area under the demand curve from the level of consumption without the project to the level with the project. Since the demand function is linear we can calculate the area under the curve as the area of the trapezoid. This requires the level of the market price without the project and with the project.
The price of cucumbers without the project is found by solving for p; i.e.,
The area of the trapezoid is then
Suppose the only input required to produce cucumbers, besides the the land and water which have no opportunity cost is labor and each 10,000 pounds of cucumbers requires 1 person-year of labor.
This means the project's 0.5 million pounds of cucumbers require 50 person-years of labor.
Suppose also that the supply function for labor in the area of the project is
There is an alternate way of tallying the costs and benefits of a project. It involves the consumers' and producers' surpluses, but some care must be taken to capture all of the consequences of the project. The previous analysis was in terms of the benefits of increased consumption of commodities less the costs of increased resource use. The situation may be depicted diagrammatically. The alternative is to total up the net gains of consumers, businesses, and resource owners; i.e.,
Usually what is above labeled "businesses' surplus" is referred to as producers' surplus. Shortly it will be shown why businesses' surplus is a more appropriate term.
First, let us compute consumers' surplus. It is the area of the trapezoid which is bounded by pw/o and pw above and below and the vertical axis on the left and the demand schedule on the right. Its value is equal to
The gain for the businesses selling cumcumbers comes from the higher price of cucumbers, but the loss for businesses comes from the higher price they have to pay for labor. This is not just the cost for businesses involved in the cumcumber market, instead it includeds all businesses which use labor. Producers' surplus would tend to mislead one into counting only the effects on the producers of cucumbers. Without the project the revenue received by businesses from the sale of cucumber was 6.0(1.20) = $7.2 million. After the project the revenue received by businesses is 6.5(1.15) = $7.475. Thus the gain in revenue to businesses from the sale of cucumbers is$7.475-7.20 = $275,000 per year. But the project increased the payment for labor from 2000(5500) = $11 million to 2050(5512.50) = $11,300,625 per year, an increase of $300,625. Therefore there is a net loss to businesses; i.e. the net gain is negative and equal to:
Suppose that a project produces 15 million pounds per year. Without the project the quantity of strawberries available is 160 million pounds per year.
Assume the only input required to produce strawberries, besides the the land and water which have no opportunity cost, is labor and each 10,000 pounds of strawberries requires 1 person- year of labor.
Assume that the supply function for labor in the area of the project is