An implicit assumption in the assignment is that the many towns referred to in part (c) -- call them "reference towns" -- all have marginal social cost = $3000/acre and all have 150 residents.
I''m making that assumption explicit here. Also you''ve probably already figured out it makes no difference whether these reference towns have a head tax or a property tax so I''m not giving up much by confirming that. It''s only in the model town that a head tax and a property tax yield different results.
It''s convenient to label houses as either big (with value Vb = $1 million in parts a and b) or small (with value Vs = $500,000 in parts a and b).
To solve the original version of part (c) you will need to develop four equations in four unknowns: big house value Vb; small house value Vs; equilibrium park consumption
Problem:
Note: this assignment is based on the local public goods model in Chapters 15, 16 of the text and the Notes on Urban Public Finance. Letter symbols used below are defined as in the text and notes.
A town has 150 houses. 100 of them are valued at $500,000 each and the other 50 are valued at $1 million each. Every house buyer finances the purchase with an interest-only mortgage at 5 percent interest, repaying the mortgage principal only when the house is sold; for example the owner of a million dollar house pays $50,000 / year interest. When properties sell, buyers and sellers expect no change to future house values.
Your answer to each part should include a diagram (or diagrams where relevant) to illustrate your results. If you are unable to determine a numerical result, a clear and correct diagram will earn partial credit. If you conclude that a part of the question can't be solved numerically explain why you reached this conclusion.
The town's public good is a park, with MSC = $3000 / acre. All 150 residents (one resident per house) have the same marginal benefit function: MB = 45 - 0.5A.
(a) The town initially uses a head tax. Determine the equilibrium park consumption and determine taxpayer surplus for each resident.
(b) The town then changes to a property tax, and assessed values remain the same as with the head tax. (Note that assessed values, which are estimated house values the town uses for property taxation, are not necessarily values that house buyers would actually pay.)
Determine the equilibrium park consumption and determine taxpayer surplus for each resident.
(c) There are many nearby towns with all residents having MB = 45 - 0.5A, and within this group of towns there are many that have every house valued at $1 million; also within this group of towns there are many that have every house valued at $500,000. The towns with million dollar houses use property taxes and the towns with $500,000 houses use head taxes.
The town considered in parts (a) and (b) decides to revalue its property assessments in line with what house buyers will pay now that the property tax is in effect. Determine the equilibrium park consumption and determine taxpayer surplus for each resident.
(d) Compare your part (c) results with your results in parts (a) and (b) from the standpoint of efficiency.
(e) Considering separately the part (b) equilibrium and the part (c) equilibrium, compare the taxpayer surplus and housing costs of two categories of residents: (i) residents who bought the more expensive homes when the head tax was still in effect; and (ii) residents who bought the more expensive homes after the head tax was replaced with the property tax.