Housing Demand © Allen C. Goodman, 2009

Equations Price and income are problematic. Simple model Q =  p Y P +  t Y T +  P +  Z What is permanent income? Why is it important? Do we move every time our income changes? Why?

Problems True equation: Q =  p Y P +  t Y T +  P +  Z What if we estimate: Q =  Y +  P +  Z If this is the case we find that our estimate of  = [  2 p /(  2 p +  2 t )]  p + [  2 t /(  2 p +  2 t )]  t where: Var (y) = Var (Y P ) + Var (Y T )  2 p +  2 t

How to get Y P Suppose that Y P is related to return on human capital H and non-human capital N. Then: Y = Y P + Y T Y = r h H (E, T, A) + r n N where E = education, T = training, A = age. Y =  h H +  e E +  a A + r n N Wage income = Y - r n N =  h H +  e E +  a A + e Wage income = Y P + Y T Predicted value of equation is Y P ; Predicted value of residual is Y T.

Problems with price Remember, the price of a dwelling unit is the multiple of pq. We suspect that unit price of housing changes over distance u, in particular that  p/  u < 0. We also guess that income rises as u . A picture.

Price biases ln q = a ln p + b ln y ln E = ln p + ln q = (1 + a) ln p + b ln y p may vary either according to: - changes in the price of land - changes relating to public services, taxes, etc., from hedonic price regressions. First problem: If p is measured with error, then the coefficient is biased toward 0. If so: ln E =  ln p + b ln y  is biased toward 0, but  = (1 + a), so a is biased toward –1.0.

Price biases Second problem: ln q = a ln p + b ln y ln E = ln p + ln q = (1 + a) ln p + b ln y We may not be able to measure ln p at all so we estimate: ln E = B ln y ln (income) ln (expenditures) True ln E = B ln y

Problem As y , p  (moving further out). If price elasticity is between 0 and –1, what happens to expenditures? A> They fall. Estimated income elasticity is biased downward. ln (income) ln (expenditures) True Estimated ln E = B ln y

Goodman-Kawai 1982

Measured Income

Permanent Income

Multi-period problems It is costly to move Suppose income . Do you automatically move? h c h1h1 c1c1 If moving costs equal m, you may not. Then what? h2h2 c2c2 y 2 - m

Multi-period problems Are we at blue point? h c h1h1 c1c1 Or somewhere else? h2h2 c2c2 y 2 - m Are we at red point?

Simple model h c h1h1 c1c1 U = u 1 (c 1, h) + D u 2 (c 2, h) Two income constraints: y 1 = c 1 + p 1 h y 2 = c 2 + p 2 h KEY is that h must be constant. L = u 1 (c 1, h) + D u 2 (c 2, h) + 1 (y 1 - c 1 - p 1 h) + 2 (y 2 - c 2 - p 2 h) h2h2 c2c2 y 2 - m

Simple model h c h1h1 c1c1 L = u 1 (c 1, h) + D u 2 (c 2, h) + 1 (y 1 - c 1 - p 1 h) + 2 (y 2 - c 2 - p 2 h) We differentiate w.r.t to c 1, c 2, and h. Eq’m condition is: MU y 1 (MRS 1 – p 1 ) = -MU y 2 (MRS 2 – p 2 ) What does this mean? h2h2 c2c2 y 2 - m

Simple model h c h1h1 c1c1 MU y 1 (MRS 1 – p 1 ) = -MU y 2 (MRS 2 – p 2 ) h2h2 c2c2 y 2 - m

Another way to think Minimize the total cost of immobility h*1h*1  = mc of immobility h*2h*2 h* Total impact is the sum of the mgl impacts **