Question

The accompanying table shows Tom and Ray's total willingness to pay for different amounts of public recreational areas (REC), which is measured in acres. The marginal cost of one acre of recreational area is $1,000.

Tom's willingness to pay        Ray's willingness to pay         REC (acres)

$2,000                                     $1,000                                     1

3,000                                       1,800                                       2

3,800                                       2,500                                       3

4,300                                       3,100                                       4

4,500                                       3,600                                       5

4,600                                       4,000                                       6

4,600                                       4,300                                       7

4,600                                       4,400                                       8

4,600                                       4,400                                       9

Part A: What is the socially optimal level of recreational areas? Assume you cannot purchase partial acres. Explain your answer.

Part B: What should be the tax burden to Tom and Ray so that their consumer surpluses are equal? Explain.

Answer 1

Part A : size of socially optimal recreational area is 4 acre

Part B : Tom's tax burden is $2,600 and Ray's Tax burden is $1,400

Explanation:-

Part A : Compute socially optimal recreational area

Compute Marginal willingness to pay for both Tom and Ray for each of the acre size

Marginal willingness to pay for X acres = Willingness to pay for X acres -

                                                            Willingness to pay for X-1 acres

Using this formula, the Marginal willingness to pay for both Tom and Ray are computed and populated in the table below for each of the acre size

Compute total willingness to pay for each of the acre size

Total Willingness to pay = Tom's willingness to pay + Ray's willingness to pay

Using this formula, the table below is populated for each of the acre size

Socially optimal level is one beyond which Marginal cost (MC) just goes over the Total Marginal willingness to pay

Total Marginal Willingness to Pay_{acre size} >= MC_{acre size}

Total Marginal Willingness to Pay_{4 acre size} > MC_{4 acre size}   

Total Marginal Willingness to Pay_{5 acre size} < MC_{5 acre size}

Thus socially optimal level of recreational areas is 4 acre

Part B : Compute tax burden

At 4 acre, Tom is willing to pay = $4,300

Tom's tax burden = Tax_Tom

Tom's consumer surplus = $4,300 - Tax_Tom

At 4 acre, Ray is willing to pay = $3,100

Ray's tax burden = Tax_Ray

Ray''s consumer surplus = $3,100 - Tax_Ray

Tom and Ray's consumer surplus are equal

$4,300 - Tax_Tom = $3,100 - Tax_Ray     ------------------ Eq₁

Total Cost of the 4 acre REC = 4 * MC = 4* $1,000 = $4,000

Total Tax = Total Cost

Tax_Tom + Tax_{Ray =} $4,000                  ------------------ Eq₂

Solve two equations and two variables

Adding left and right sides of Eq₁ and Eq₂

$4,300 - Tax_{Tom +} Tax_Tom + Tax_Ray = $3,100 - Tax_Ray + $4,000

Tax_Ray + Tax_Ray = $3,100+ $4,000 - $4,300

2 * Tax_Ray = $2,800

Tax_Ray = $2,800 / 2 = $1,400

Sbstituting this value of Tax_Ray in Eq₂

Tax_Tom + $1,400 = $4,000

Tax_Tom = $4,000 - $1,400 = $2,600

Tom's Tax burden is $2,600 and Ray's Tax burden is $1,400