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 8 units of milk or 16 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,

1)   What is the equation for the production possibility frontier for this economy?

2)   What is the consumption ratio of wheat over milk for a consumer? And what is total milk production?  

3)   Here, we use the necessary condition for the Pareto efficient coordination of consumption and production that you learned in the class. What is the condition?

4)   Show why it is the necessary condition. (you can draw a diagram).

Answer 1

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. Let us denote land as L and tota land is L°. Hence,

L° = 1000 acres.

It is told that, one acre of land produces either 8 units of milk or 16 units of wheat. Now, we have to allot the total land L° between milk and wheat production.

Let us denote the Land alloted for milk production as Lm and Land alloted for wheat production as Lw. Hence,

Lm + Lw = L° = 1000.........(1)

Hence, 8 units of milk (M) is produced with 1 acre of Land (Lm). Hence production function for milk is

M = 8.Lm

or, Lm = M/8..........(2)

Also, 16 units of wheat (W) is produced with 1 acre of Land (Lw). Hence, production function for wheat is

W = 16.Lw

or, Lw = W/16.........(3)

The utility function of the citizens of the island is

U(M,W) = M.W..........(4)

Now, let us answer the following questions.

(1) From equation (1) we got

Lm + Lw = 1000

Now putting the values of Lm and Lw from equation (2) and (3) in terms of M and W.

Hence,

Lm + Lw = 1000

or, M/8 + W/16 = 1000

or, 2M + W = 16000...........PPF

This is the production possibility frontier.

The equation of the production possibility frontier or PPF of the economy is

PPF: 2M + W = 16000.

(2) The absolute value of the slope of the PPF is

| Slope of PPF | = | -2 | = 2

Now, we will calculate the MRS or Marginal Rate of Substitution from the utility function.

From, U(M,W) = M.W, we get

Marginal Utility of milk is

MUm = dU/dM = W

And, Marginal Utility of wheat is

MUw = dU/dW = M

Hence, MRS = MUm/MUw

or, MRS = W/M

Now, at the optimal consumption in the economy, the Utility curve should be tangent to the PPF. In other words the slope of the Utility curve i.e. MRS would be equal to the slope of the PPF.

Hence, at optimum consumption

MRS = | Slope of PPF |

or, W/M = 2........(4)

This is the ratio of consumption of wheat over milk.

The consumption ratio of wheat over milk (W/M) is 2.

Now, from (4) we get

W = 2M.........(5)

Pitting W = 2M in the equation of PPF we get

2M + W = 16000

or, 2M + 2M = 16000

or, M* = 4000 units

And, W* = 2.M* = 2×4000

or, W* = 8000 units

Hence, the total milk production is M* = 4000 units.

(3) Here, we use the necessary condition for the Pareto efficient coordination of consumption and production. The condition is the following

| Slope of Indifference Curve | = | Slope of PPF |

or, MRS = | Slope of PPF |

(4) The above condition is the necessary condition for pareto optimality because,

The citizens are maximizing their utility subject to the Production Possibility Frontier. Let us draw the equilibrum in a diagram below.

From the above diagram, we can see that, at point E the Indifference curve is tangent to the Production Possibility Frontier or PPF. At this point the citizens are obtaining the maximum utility. Also at point E, the production and consumption of milk and wheat are equal. Hence the market is also cleared. This is why the above condition is the necessary condition for pareto optimality.

Hope the solution is clear to you my friend.