gemDualLinearProgramming: General Equilibrium Models and Linear Programming Problems (see Winston, 2003)

Description

Some examples illustrating the relationship between general equilibrium problems and (dual) linear programming problems. Some linear programming problems can be transformed into general equilibrium problems and vice versa.

Usage

gemDualLinearProgramming(...)

Value

A general equilibrium.

Arguments

...: arguments to be passed to the function CGE::sdm.

Details

These examples are similar and let us explain briefly the first example (Winston, 2003).
The Dakota Furniture Company manufactures desks, tables, and chairs. The manufacture of each type of furniture requires lumber and two types of skilled labor: finishing and carpentry. The amount of each resource needed to make each type of furniture is as follows:
desk: (8, 4, 2)
table: (6, 2, 1.5)
chair: (1, 1.5, 0.5)
Currently, 48 board feet of lumber, 20 finishing hours, and 8 carpentry hours are available. A desk sells for $60, a table for $30, and a chair for $20. Because the available resources have already been purchased, Dakota wants to maximize total revenue. This problem can be solved by the linear programming method.
Now let us regard the problem above as a general equilibrium problem. The Dakota Furniture Company can be regarded as a consumer who obtains 1 unit of utility from 1 dollar and owns lumber and two types of skilled labor. There are four commodities (i.e. dollar, lumber and two types of skilled labor) and four agents (i.e. a desk producer, a table producer, a chair producer and the consumer Dakota) in this problem. We need to compute the equilibrium activity levels and the equilibrium prices, which are also the solutions of the (dual) linear programming problems (i.e. the utility-maximizing problem of the consumer and the cost-minimizing problem of the producers).

References

Kemeny, J. G., O. Morgenstern and G. L. Thompson (1956) A Generalization of the von Neumann Model of an Expanding Economy, Econometrica, 24, pp. 115-135.

LI Wu (2019, ISBN: 9787521804225) General Equilibrium and Structural Dynamics: Perspectives of New Structural Economics. Beijing: Economic Science Press. (In Chinese)

von Neumann, J. (1945) A Model of General Economic Equilibrium. The Review of Economic Studies, 13. pp. 1-9.

Winston, Wayne L. (2003, ISBN: 9780534380588) Operations Research: Applications and Algorithms. Cengage Learning.

Examples

Run this code

# \donttest{
#### the Dakota example of Winston (2003, section 6.3, 6.6 and 6.8)
A <- matrix(c(
  0, 0, 0, 1,
  8, 6, 1, 0,
  4, 2, 1.5, 0,
  2, 1.5, 0.5, 0
), 4, 4, TRUE)
B <- matrix(c(
  60, 30, 20, 0,
  0, 0, 0, 0,
  0, 0, 0, 0,
  0, 0, 0, 0
), 4, 4, TRUE)
S0Exg <- {
  S0Exg <- matrix(NA, 4, 4)
  S0Exg[2:4, 4] <- c(48, 20, 8)
  S0Exg
}

## Compute the equilibrium by the function CGE::sdm.
ge <- CGE::sdm(A = A, B = B, S0Exg = S0Exg)

ge$p / ge$p[1]
ge$z

## Compute the equilibrium by the function sdm2.
## The function policyMeanValue is used to accelerate convergence.
ge <- sdm2(
  A = A, B = B, S0Exg = S0Exg,
  policy = policyMeanValue,
  names.commodity = c("dollar", "lumber", "lab1", "lab2"),
  names.agent = c("desk producer", "table producer", "chair producer", "consumer"),
  numeraire = "dollar"
)

ge$z
ge$p

#### an example at http://web.mit.edu/15.053/www/AMP-Chapter-04.pdf.
A <- matrix(c(
  0, 0, 0, 1,
  0.5, 2, 1, 0,
  1, 2, 4, 0
), 3, 4, TRUE)
B <- matrix(c(
  6, 14, 13, 0,
  0, 0, 0, 0,
  0, 0, 0, 0
), 3, 4, TRUE)
S0Exg <- {
  S0Exg <- matrix(NA, 3, 4)
  S0Exg[2:3, 4] <- c(24, 60)
  S0Exg
}

ge <- CGE::sdm(
  A = A, B = B, S0Exg = S0Exg
)

ge$z
ge$p / ge$p[1]

#### an example at https://web.stanford.edu/~ashishg/msande111/notes/chapter4.pdf.
A <- matrix(c(
  0, 0, 1,
  4.44, 0, 0,
  0, 6.67, 0,
  4, 2.86, 0,
  3, 6, 0
), 5, 3, TRUE)
B <- matrix(c(
  3, 2.5, 0,
  0, 0, 0,
  0, 0, 0,
  0, 0, 0,
  0, 0, 0
), 5, 3, TRUE)
S0Exg <- {
  S0Exg <- matrix(NA, 5, 3)
  S0Exg[2:5, 3] <- 100
  S0Exg
}

ge <- CGE::sdm(
  A = A, B = B, S0Exg = S0Exg
)

ge$z
ge$p / ge$p[1]

#### an example at https://utw11041.utweb.utexas.edu/ORMM/supplements/methods/lpmethod/S3_dual.pdf.
A <- matrix(c(
  0, 0, 1,
  0, 1, 0,
  1, 3, 0,
  1, 0, 0
), 4, 3, TRUE)
B <- matrix(c(
  2, 3, 0,
  1, 0, 0,
  0, 0, 0,
  0, 0, 0
), 4, 3, TRUE)
S0Exg <- {
  S0Exg <- matrix(NA, 4, 3)
  S0Exg[2:4, 3] <- c(5, 35, 20)
  S0Exg
}

ge <- CGE::sdm(
  A = A, B = B, S0Exg = S0Exg
)

ge$z
ge$p / ge$p[1]

#### the Giapetto example of Winston (2003, section 3.1)
A <- matrix(c(
  0, 0, 1,
  2, 1, 0,
  1, 1, 0,
  1, 0, 0
), 4, 3, TRUE)
B <- matrix(c(
  27 - 10 - 14, 21 - 9 - 10, 0,
  0, 0, 0,
  0, 0, 0,
  0, 0, 0
), 4, 3, TRUE)
S0Exg <- {
  S0Exg <- matrix(NA, 4, 3)
  S0Exg[2:4, 3] <- c(100, 80, 40)
  S0Exg
}

ge <- sdm2(
  A = A, B = B, S0Exg = S0Exg,
  policy = policyMeanValue,
  numeraire = 1
)

ge$z
ge$p

#### the Dorian example (a minimization problem) of Winston (2003, section 3.2)
A <- matrix(c(
  0, 0, 1,
  7, 2, 0,
  2, 12, 0
), 3, 3, TRUE)
B <- matrix(c(
  28, 24, 0,
  0, 0, 0,
  0, 0, 0
), 3, 3, TRUE)
S0Exg <- {
  S0Exg <- matrix(NA, 3, 3)
  S0Exg[2:3, 3] <- c(50, 100)
  S0Exg
}

ge <- sdm2(
  A = A, B = B, S0Exg = S0Exg,
  policy = policyMeanValue,
  numeraire = 1
)

ge$p
ge$z

#### the diet example (a minimization problem) of Winston (2003, section 3.4)
A <- matrix(c(
  0, 0, 0, 0, 1,
  400, 3, 2, 2, 0,
  200, 2, 2, 4, 0,
  150, 0, 4, 1, 0,
  500, 0, 4, 5, 0
), 5, 5, TRUE)
B <- matrix(c(
  500, 6, 10, 8, 0,
  0, 0, 0, 0, 0,
  0, 0, 0, 0, 0,
  0, 0, 0, 0, 0,
  0, 0, 0, 0, 0
), 5, 5, TRUE)
S0Exg <- {
  S0Exg <- matrix(NA, 5, 5)
  S0Exg[2:5, 5] <- c(50, 20, 30, 80)
  S0Exg
}

ge <- sdm2(
  A = A, B = B, S0Exg = S0Exg,
  policy = policyMeanValue,
  numeraire = 1
)

ge$p
ge$z

#### An example of Elizabeth Stapel (Linear Programming: Introduction. Purplemath.
## Available from https://www.purplemath.com/modules/linprog.htm):
## Find the maximal value of 3x + 4y subject to the following constraints:
## x + 2y <= 14, 3x - y >= 0, x - y <= 2, x >= 0, y >= 0

A <- matrix(c(
  0, 0, 1,
  1, 2, 0,
  0, 1, 0,
  1, 0, 0
), 4, 3, TRUE)
B <- matrix(c(
  3, 4, 0,
  0, 0, 0,
  3, 0, 0,
  0, 1, 0
), 4, 3, TRUE)
S0Exg <- {
  S0Exg <- matrix(NA, 4, 3)
  S0Exg[2:4, 3] <- c(14, 0, 2)
  S0Exg
}

ge <- sdm2(
  A = A, B = B, S0Exg = S0Exg,
  policy = policyMeanValue,
  priceAdjustmentVelocity = 0.03,
  numeraire = 1
)

ge$z
ge$p
# }

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