2. Opening the model In Session 2 you explored a simple Closed Leontief Exchange Model with five in- terdependent industries. In the closed exchange model no goods or services enter or leave a national economy. However, most nations import and export goods. For this problem you will develop an open exchange model for the newly formed nation, NationLand, then generalize the model to an economy with n sectors. (a) Suppose NationLand has 5 sectors: agriculture (A), energy (N), real estate (R), education (E), and entertainment (T). The following table represents the number of units of goods from other sectors used to produce one unit with a given sector. The table also lists the demand for each good from NationLand's trading partners.

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Chapter19: International Trade
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2. Opening the model
In Session 2 you explored a simple Closed Leontief Exchange Model with five in-
terdependent industries. In the closed exchange model no goods or services enter
or leave a national economy. However, most nations import and export goods. For
this problem you will develop an open exchange model for the newly formed nation,
NationLand, then generalize the model to an economy with n sectors.
(a) Suppose NationLand has 5 sectors: agriculture (A), energy (N), real estate
(R), education (E), and entertainment (T). The following table represents the
number of units of goods from other sectors used to produce one unit with a
given sector. The table also lists the demand for each good from NationLand's
trading partners.
Internal and External Unit Demand Levels
A
T
0.1
Internal Consumption Rates
N R E
0.05 0.025 0.15
0.02 0.0 1
0.3 0.1
0.05
0.05
0.01
Real Estate (R)
0.4
0.01 0.025
0.1 0.25
0.05 0.125 0.2
Entertainment (T) 0.01 0.05 0.15 0.25 0.05
Education (E)
0.1
0.15
Agriculture (A)
Energy (N)
Outside Demand (total)
40
10
0
15
25
Let A, EN, FR, E, and aT denote the number of units produced by the agri-
culture (A), energy (N), real estate (R), education (E), and entertainment (T)
sectors respectively. Write a system of equations that defines the number of
units each sector should produce to satisfy both internal and external demand
(total units).
(b) Rewrite your system of equations in the form M = d where M is a matrix of
coefficients. What do and d represent in this model?
2
(c) Does the system have a solution? In other words, is there a set of production
levels that would satisfy both internal and external demand? Is this solution
unique?
(d) [challenge] Adjust the parameters in the Demand Table to find levels that
lead to (i) infinitely many solutions, and (ii) no solution. Explain your choice,
verifying that your suggestions are realistic.
(e) [challenge] Now suppose NationLand has n sectors. Let x; represent the num-
ber of units produced by the i-th sector. Assume that Sector j uses aij units
from Sector i to produce one unit. Let d; be the external demand for goods
from Sector i.
i. Write a system of equations that models internal and external demand in
this n sector system. Rewrite your system of equations in the form Ma = d
where M is a matrix of coefficients.
ii. Let A = [aij] be the matrix whose entries are the internal production re-
quirements. Express M in terms of A.
iii. Find a condition you can use to quickly determine whether the model has
a unique solution.
iv. Derive a formula for the vector of x; in terms of A and the demand vector
d that holds when the system has a unique solution.
Transcribed Image Text:2. Opening the model In Session 2 you explored a simple Closed Leontief Exchange Model with five in- terdependent industries. In the closed exchange model no goods or services enter or leave a national economy. However, most nations import and export goods. For this problem you will develop an open exchange model for the newly formed nation, NationLand, then generalize the model to an economy with n sectors. (a) Suppose NationLand has 5 sectors: agriculture (A), energy (N), real estate (R), education (E), and entertainment (T). The following table represents the number of units of goods from other sectors used to produce one unit with a given sector. The table also lists the demand for each good from NationLand's trading partners. Internal and External Unit Demand Levels A T 0.1 Internal Consumption Rates N R E 0.05 0.025 0.15 0.02 0.0 1 0.3 0.1 0.05 0.05 0.01 Real Estate (R) 0.4 0.01 0.025 0.1 0.25 0.05 0.125 0.2 Entertainment (T) 0.01 0.05 0.15 0.25 0.05 Education (E) 0.1 0.15 Agriculture (A) Energy (N) Outside Demand (total) 40 10 0 15 25 Let A, EN, FR, E, and aT denote the number of units produced by the agri- culture (A), energy (N), real estate (R), education (E), and entertainment (T) sectors respectively. Write a system of equations that defines the number of units each sector should produce to satisfy both internal and external demand (total units). (b) Rewrite your system of equations in the form M = d where M is a matrix of coefficients. What do and d represent in this model? 2 (c) Does the system have a solution? In other words, is there a set of production levels that would satisfy both internal and external demand? Is this solution unique? (d) [challenge] Adjust the parameters in the Demand Table to find levels that lead to (i) infinitely many solutions, and (ii) no solution. Explain your choice, verifying that your suggestions are realistic. (e) [challenge] Now suppose NationLand has n sectors. Let x; represent the num- ber of units produced by the i-th sector. Assume that Sector j uses aij units from Sector i to produce one unit. Let d; be the external demand for goods from Sector i. i. Write a system of equations that models internal and external demand in this n sector system. Rewrite your system of equations in the form Ma = d where M is a matrix of coefficients. ii. Let A = [aij] be the matrix whose entries are the internal production re- quirements. Express M in terms of A. iii. Find a condition you can use to quickly determine whether the model has a unique solution. iv. Derive a formula for the vector of x; in terms of A and the demand vector d that holds when the system has a unique solution.
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