# $U=xy$ for all people in a small island, What are the pareto efficient allocations? [closed]

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 16 units of milk or 37 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 8,000

(c) Every consumer’s marginal rate of substitution between milk and wheat -1.

(d) None of the above is true at every pareto optimal allocation.

Hello, Is the MRS for everyone =$\frac{\mathrm{d} W}{\mathrm{d} M}=\frac{37}{16}$?

Also,What are the Pareto-efficient allocations?

1. A Pareto efficient allocation is an allocation for which it is impossible to change the allocation and make someone better off without making someone else worse off.

2. In this economy you can produce either 16,000 units of milk or 37,000 units of wheat and you exchange 16 units of milk for 37 units of wheat by changing one acre from producing milk to producing wheat. - 16 / 37 is the slope of the production possibility frontier.

3. The price of producing one unit of wheat is 1/37th of an acre. The price of producing one unit of milk is 1/16th of an acre. There are a total of 1,000 acres. The problem for the island as a whole is:

$\max_{w,m} \quad m\cdot w \quad s.t. \quad 1,000 = \frac{1}{16} m + \frac{1}{37} m$

This can be re-written as the Lagrangian $\Lambda$:

$\Lambda = m\cdot w + \lambda \left(1,000 - \frac{1}{16} m - \frac{1}{37}w\right)$

The First Order Conditions for a maximum are:

$m: w - \frac{1}{16} \cdot \lambda = 0\\$

$w: m - \frac{1}{37} \cdot \lambda = 0\\$

$\lambda: 1,000 - \frac{1}{16} m - \frac{1}{37}w = 0\\$

This can be re-written as:

$16 w = \lambda$

$37 m = \lambda$

Or,

$\frac{16}{37} = \frac{m^*}{w^*}$ (The MRS in equilibrium is therefore not equal to - 1.)

Plugging $\frac{16}{37}w^* = m^*$ back into $\lambda:$:

$w^* = 18,500$ $m^* = 8,000$