A simple economy consists of two industries: agriculture and manifacturing. The production of 1 unit of agricultural products requires the consumption of 0.4 units of agricultural products and 0.15 units of manufactured goods. The production of 1 unit of manufactured goods requires the consumption of 0.4 units of agricultural products and 0.35 units of manufactured goods.
Find the gross output of goods needed to satisfy a consumer demand for 120 million dollars worth of agricultural products and 130 million dollars worth of manufactured products.
XXX? million dollars worth of agricultural products, and
xxx? million dollars worth of manufactured products.
xcv
To find the gross output of goods needed to satisfy the consumer demand, we need to use a system of equations.
Let's assume the gross output of agricultural products is A, and the gross output of manufactured goods is M.
According to the information given, the production of 1 unit of agricultural products requires 0.4 units of agricultural products and 0.15 units of manufactured goods. So, we can write the following equation:
A = 0.4A + 0.15M
Similarly, the production of 1 unit of manufactured goods requires 0.4 units of agricultural products and 0.35 units of manufactured goods. So, we can write the following equation:
M = 0.4A + 0.35M
We also know the consumer demand for agricultural products is 120 million dollars and for manufactured products is 130 million dollars. Since the value of a unit of agricultural products is not given, we need to introduce variables for that as well.
Let's assume the value of a unit of agricultural products is a million dollars, and the value of a unit of manufactured goods is also a million dollars.
Thus, we can write the following equations:
120A = aA + mM
130M = aA + mM
Now we have a system of equations with four unknowns (A, M, a, m) and four equations. To solve this system of equations, we can use substitution or elimination.
Using substitution, we can solve for A in terms of M from the second equation:
130M = aA + mM
130M - mM = aA
(130 - m)M = aA
A = (130 - m)M / a
Substituting this value of A into the first equation, we get:
120(130 - m)M / a = aA + mM
120(130 - m)M / a = amM / a + mM
120(130 - m)M = amM + aM
120(130 - m) = am + a
120(130 - m) = a(m + 1)
Now we can solve for the unknowns using this equation. Since we are looking for the gross output of goods in million dollars, we need to find the values of a and m.
We know the consumer demand for agricultural products is 120 million dollars, so we can set a = 1. Therefore,
120(130 - m) = m + 1
15600 - 120m = m + 1
15600 = 121m + 1
121m = 15599
m ≈ 128.925
Now we can substitute the value of m back into the equation for A to find the value of a:
A = (130 - m)M / a
A = (130 - 128.925)M / a
A ≈ 1.075M / a
So, the gross output of agricultural products is approximately 1.075 times the gross output of manufactured goods.
To find the actual values in million dollars, we need to substitute the value of M into the equations for A and M:
A = 1.075M / a
A = 1.075(130) / a
A ≈ 139.75 / a
M = m
M ≈ 128.925
Now we have the values of A and M. To find the actual values in million dollars, we need to substitute these values into the equations:
120A = aA + mM
120(139.75 / a) = a(139.75 / a) + 128.925M
(120 * 139.75) / a = 139.75 + (128.925 * 128.925)
Approximately:
- a ≈ 1.152 which corresponds to a negative value for A, which is not possible in this context.
Therefore, there is no feasible solution that satisfies the consumer demand for 120 million dollars' worth of agricultural products and 130 million dollars' worth of manufactured products in this simple economy.