Question

Taylor and stephen have pencils and erasers in exact proptions. 

Taylor 1 pencil per 2 erasers: Stephen has 2 pencils per 1 eraser.  

At the start Taylor  had 10 pencils  and 2 eraser with him; Daniel had 2 pencils and 10  erasers

Show all Pareto optimal allocations of  pencils and erasers for Taylors and Stephen Edgeworth Box.

Answer 1

To find the Pareto optimal allocations of pencils and erasers for Taylor and Stephen in the Edgeworth Box, we first need to construct the Edgeworth Box based on their initial endowments.

 

The Edgeworth Box is a graphical representation that shows all possible allocations of two individuals' goods. The x-axis represents Taylor's endowment of pencils, and the y-axis represents Stephen's endowment of pencils. The origin (0,0) represents the initial endowment point, and the total resources in the system (pencils + erasers) are fixed.

 

Given the initial endowments:

- Taylor: 10 pencils and 2 erasers

- Stephen: 2 pencils and 10 erasers

 

Now, let's construct the Edgeworth Box:

 

```

Taylor's pencils (x) | Stephen's pencils (y)

------------------------------------------

0                    | 12

1                    | 11

2                    | 10

3                    | 9

4                    | 8

5                    | 7

6                    | 6

7                    | 5

8                    | 4

9                    | 3

10                   | 2

```

 

Since both Taylor and Stephen have fixed endowments, the total quantity of pencils and erasers in the system remains constant at 12 (10 + 2) for pencils and 12 (2 + 10) for erasers.

 

Now, let's examine the Pareto optimal allocations in the Edgeworth Box. A Pareto optimal allocation is one in which no individual can be made better off without making someone else worse off. This means that any movement towards a different allocation would make at least one person worse off.

 

In the Edgeworth Box, the Pareto optimal allocations are the points on the contract curve, which represents the set of all Pareto efficient allocations. The contract curve is the locus of points where the individuals are indifferent between any two allocations on the curve.

 

```

Taylor's pencils (x) | Stephen's pencils (y)

------------------------------------------

0                    | 12

1                    | 11

2                    | 10

3                    | 9

4                    | 8

5                    | 7

6                    | 6

7                    | 5

8                    | 4

9                    | 3

10                   | 2

```

 

All the points listed above on the contract curve are Pareto optimal allocations, meaning they are efficient and no one can be made better off without making someone else worse off.

 

To summarize, the Pareto optimal allocations of pencils and erasers for Taylor and Stephen in the Edgeworth Box are any points located on the contract curve, as shown in the table above. These points represent the efficient allocations where both individuals are indifferent between any two allocations on the curve, and no one can be made better off without making someone else worse off.