MIT OpenCourseWare http://ocw.mit.edu 11.433J / 15.021J Real Estate Economics Fall 2008 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms.Recitation 2 Real Estate Economics: Rent Gradient and Land Price Sep 16, 2008 Jinhua ZhaoOutline 1 • HM1 • Richardian Theory: Rent Gradient and Land Price – Summary – House Rent – House Rent Æ Land Rent – House Rent Æ House Price – Land Rent Æ Land Price – Static vs. Growth • Full model: power and limitation – Power: numerical example by linear regression – Assumption and expansion: spatial separation • What to read a paper • TakeawayHomework 1: quick answer Q1 Q2a Q2b Q2c, part 1 Q2c, part 2 Equilibrium 1 Shock 1 Equilibrium 2 Shock 2 Equilibrium 3 P=200 P=400 P=200 P=90 P=200 R=20 R=20 R=10 R=9 R=20 S=2500 S=2500 S=5000 S=5000 S=2250 C=50 C=50 C=100 C=100 C=45Summary: Price vs. Rent; Land vs. House House 2 Land Rent 1 R(d) = ra *q + c + k(b − d) r(d) = ra + k(b − d)/ q Price 3 4 pt (d ) = ra + k(bt − d ) + kbtgPt (d) = raq + c + k(bt − d) + kbtg i iq i(i − g)qi i i i(i − g)Housing rent gradient R(d) = raq + c + k(b - d) R(d)- R(b)= k(b - d) ra: agriculture cost per acre c: annualized construction cost per house unit q: amount of acre per house unit 1/q: unit per acre (density) d = location b = boundary, farthest locatin 0 = best location, center k = annual transport cost per mile (inclusive of all: money, time)Component of Housing Rent Housing rent R(d) Location Rent: k [ b - d ] Structure Rent: c Agricultural lot Rent Ra= ra q Distance: d Border: bHouse rent Æ Land rent • House rent (per unit): R(d) = ra *q + c + k(b − d) • Land rent (per acre): a residual r(d) = ra + k(b − d)/ q Housing rent R(d) Location Rent: k [ b - d ] Land rent r(d) Location Rent: k [ b - d ]/q Structure Rent: c Agricultural lot Rent Ra= ra q Agricultural Rent: ra Distance: d Border: b Distance: d Border: b Slope: k Slope: k/qLand Supply • Land demand = function of population • In equilibrium: Land supply = Land Demand N *q =π*b2*V N *q b = πV V: water, wetland, planning controlComparative statistics: effects on house rent and land rent by changing r, k, b, q, v R(d) = ra *q + c + k(b − d) r(d) = ra + k(b − d)/ q • rN *q =π*b2*Va • k: oil price changes; transport infrastructure changes • b • V • N Æ Higher R (for everywhere except the border) • q given the same population – R(d) = raq + c + k(b – d) = raq + c + k((nq/πv)1/2 –d) – r(d) = ra + k/q(b –d) = ra + k((nq/πv)1/2 –d) /q: • With higher density (small q), • Trade off between smaller city but steeper slope • Result: higher r in the center (graphic intuition and algebra) R or r ? , which location, what timeHouse: Rent Æ Price (Spatial Capitalization of Ricardian Rent with growth) Population growth at rate 2g: Boundary [b] growth rate of g: gt t eNN 2 0 *= gt t ebb * 0 = Ricardian Rent for existing structures located at (d) in time t: House rent: Rt (d) = ra *q + c + k(bt − d) House price: PDV of future Rent Pt (d) = raq + c + k(bt − d) + kbtg i i i i(i − g) What if g=0? What if i=g or i<g? Agriculture Structure Current Future value value location location value valueExpansion of Housing Rent as the city grows in population and the border moves from b0 to bt House Rent R (d)Location rent Structure rent Agricultural rent Center b 0 bt distance (d)Spatial multipliers or capitalization rate Price / rent ratio: inverse of capitalization rate Pt (d ) = raq + c + k(bt − d ) + kbtg i i i i(i − g) Rt (d) = ra *q + c + k(bt − d) Pt (d)/ Rt (d) = 1/ i + kbtg i(i − g)Rt (d) w.r.t: g =0; g >0; g<0; g increase; g decrease w.r.t. d? d=0; d=b Edge of the city: highest return, highest riskLand Rent Æ Price • Land rent rt (d) = ra + k(bt − d)/ q • Land price p (d ) = ra + k(bt − d) + kbtg t i iq i(i − g)qVacant land price • The price of land beyond the current border? – In t years bt = b0egT d – Land at distance d> b0 will be developed in T = log(b )/ g 0 – the value of land at d p0(d ) = ria + e−iT i(ikb − tgg )q a• When g=0 Æ p0(d ) = r i • Where are the land prices most volatile as g fluctuates?Components of land prices: agriculture, Current location value, future increase in location value Land Price p0 (d)developed vacant Current Location Value Future Increase in Location Value Agricultural value Center b0 DistanceLand price: with different growth speed fast growth slow growth Land Price p0 (d) Current Location Value Future Increase in Location Value vacantdeveloped Agricultural value Center b0 DistanceSummary: Price vs. Rent; Land vs. House House Land Rent )(*( ) dk bcqrR d a −++= 1/q i qdk brr d a ) /(( ) −+= i Price )( )(( ) gi i kb g i dk b i c i r qP d tta t − +−++= g qi i kb g iq dk b i rdp tta t )( )(( ) − +−+= Discussion: 1. Impact of growth: on each of the four 2. Slope: (w.r.t: d) 3. Compensation differential 4. Special case: d=b 5. Special case: d=0 6. Special case: d>b 7. Special case: g=0Variation of real estate prices Question: Two currently identical cities with A expected to grow 10% a year; B 5% a year Rent A vs. Rent B? Price A vs. Price B? Price accounts for future; Rent does not! • Much of the variation of real estate prices over time is due to changes in the expected future growth of rental income, rather than to changes in the actual level of current rents. • Faster growing cities that are otherwise identical to slower growing cities (in population, income, and density) should have similar land and housing rents, but higher land and housing prices.A full model: simple and powerful but with limitation R(d) = ra *q + c + k(b − d) Structure N *q =π*b2*V Boundary condition Static model in equilibriumRegression: housing price variation among cities (p56) • A simple model to explain the housing price variation among cities • Three key factors: – Size of the city – Growth of the city – Construction cost • Data: – 1990, CMSAs in the US • Variables: – Price: median house price in 1990 (PRICE) – Size of the city: …