Consider an economy where two goods, x and y can be produced using machines, k,
ID: 1252361 • Letter: C
Question
Consider an economy where two goods, x and y can be produced using machines, k, or workers, l. There are a total of 100 workers and 120 machines in the economy (L ¯ = 100, K ¯ = 120). Good x can be produced by using both 2 workers and 1 machine (lx = 2, kx = 1). Good y can be produced by using both 1 worker and 4 machines (lx = 1, kx = 4).The resource costraints for this economy are thus:
lxx+lyy = L ¯ ? 2x+y=100
kxx+kyy = K ¯ ? x+4y=120
Question:
Suppose that we wanted to have equal production of both goods. Find the quantities of good x and y produced and the total number of machines and workers used. Are all of the resources used?
Explanation / Answer
Consider an economy where two goods, x and y can be produced using machines, k, or workers, l. There are a total of 100 workers and 120 machines in the economy (L ¯ = 100, K ¯ = 120). Good x can be produced by using both 2 workers and 1 machine (lx = 2, kx = 1). Good y can be produced by using both 1 worker and 4 machines (lx = 1, kx = 4).
The resource costraints for this economy are thus:
lxx+lyy = L ¯ ? 2x+y=100
kxx+kyy = K ¯ ? x+4y=120
OK. Those are just a bit difficult to read, so I'll rewrite them:
Lxx + Lyy = 2x + y = 100
Kxx + Kyy = x + 4y = 120
Question:
Suppose that we wanted to have equal production of both goods. Find the quantities of good x and y produced and the total number of machines and workers used. Are all of the resources used?
Just set up a system of equations!
2x + y = 100
x + 4y = 120 multiply by two to get: 2x + 8y = 240 (note that it's still the same equation)
2x + 8y = 240
- 2x + y = 100 (subtract the second equation)
7y = 140
y = 20
Thus x = 120 - 4y = 120 - 80 = 40
y = 20, x = 40
Total # machines used = 120
Total # workers used = 100
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