Question

On a certain island there are only two goods, wheat and milk. The only scarce resource is land. There are 1,000 acres of land. An acre of land will produce either 14 units of milk or 15 units of wheat. Some citizens have lots of land; some have just a little bit. The citizens of the island all have utility functions of the form U(M, W) = MW. At every Pareto optimal allocation,

a.	the number of units of milk produced equals the number of units of wheat produced.
b.	total milk production is 7,000 units.
c.	all citizens consume the same commodity bundle.
d.	every consumer’s marginal rate of substitution between milk and wheat is -1.
e.	None of the above is true at every Pareto optimal allocation.

Answer 1

U(M, W) = MW

An acre of land will produce either 14 units of milk or 15 units of wheat. There are 1,000 acres of land. So as a whole, 14000 units of milk or 15000 units of wheat can be produced.

Slope of the PPF = -14/15

To produce 1 unit of milk, 1/14th of 1 acre land is needed

To produce 1 unit of wheat, 1/15th of 1 acre land is needed

Total land = 1000 acres

Hence, our problem is to

Max. U(M, W) = MW

s.t. 1000 = (1/14)M + (1/15)W

The Lagrangian is,

Λ = MW + λ[1000 - (1/14)M - (1/15)W]

F.O.C for utility maximization requires,

W - λ*(1/14) = 0... [setting δΛ/δM = 0]

or, W = λ/14 ... (i)

M - λ*(1/15) = 0 ... [setting δΛ/δW = 0]

or, M = λ/15 ... (ii)

1000 - (1/14)M - (1/15)W = 0 ... [ setting δΛ/δλ = 0]

MRS = W/M = 15/14

W = (15/14)M

1000 - (1/14)M - (1/15)W = 0

or, 1000 - (1/14)M - (1/15)*(15/14)M = 0

or, (2/14)M = 1000

M = 7000

W = (15/14)*7000 = 750

Citizens have same utility function , but the amout of land is different. So, commodity bundle may vary.

Hence, option b is correct