class: center, middle, inverse, title-slide # Lecture 9 ## Hedonics: Property value models ### Ivan Rudik ### AEM 6510 --- exclude: true ```r if (!require("pacman")) install.packages("pacman") ``` ``` ## Loading required package: pacman ``` ```r pacman::p_load( tidyverse, tidylog, xaringanExtra, rlang, patchwork ) options(htmltools.dir.version = FALSE) knitr::opts_hooks$set(fig.callout = function(options) { if (options$fig.callout) { options$echo <- FALSE } knitr::opts_chunk$set(echo = TRUE, fig.align="center") options }) ``` ``` ## Warning: 'xaringanExtra::style_panelset' is deprecated. ## Use 'style_panelset_tabs' instead. ## See help("Deprecated") ``` ``` ## Warning in style_panelset_tabs(...): The arguments to `syle_panelset()` changed ## in xaringanExtra 0.1.0. Please refer to the documentation to update your slides. ``` --- # Roadmap - What can we use to infer the demand for environmental goods? - What do housing prices tell us? - When do changes in house prices give us welfare measures --- # Environmental quantity changes Last time we learned that we cannot directly compute the welfare consequences of a change in environmental quantities -- There are no markets for the environmental goods -- We don't observe the ordinary inverse demand curve -- We can't compute CS, EV, CV, etc! --- # Revealed preference approaches One way to circumvent this problem is to look at private goods that interact with the environmental good -- If there are changes in the environmental good, holding everything else fixed, that should be reflected in *some way* in changes in the price of the related private good -- This change in price can tell us something about how people value the change in the environmental good --- # Revealed preference approaches There is no market for orcas -- Suppose there's a massive decline in orcas off the Washington coast, what happens? -- We will likely see demand for sightseeing tours go down (MB of these tours went down!) -- This drops the price of tours -- A non-market good had an effect on a market price -- What does this price change mean? --- # Hedonics: Property value models Common market goods to use for revealed preference valuation are .hi-blue[properties] -- When people buy a home they are purchasing a bundle of goods: - Rooms - Bathrooms - School quality - .hi[Environmental quality] -- Homes located in pristine areas are likely to be more valuable than identical homes located near toxic facilities --- # Hedonics: Property value models Real estate is virtually ideal for measuring environmental changes -- Real estate markets are often competitive and thick -- Property purchases are large and consequential: buyers and sellers are likely to be well-informed -- It is uncontroversial that property values should reflect local attributes --- # The hedonic model Property value approaches are often called .hi-blue[hedonics] because they rely on the hedonic model -- Suppose that we have some quality-differentiated good (i.e. a home) -- This good is characterized by a set of `\(J\)` property characteristics `\(x\)` - parcel size, school quality, bedrooms, etc -- It is also characterized by an environmental good `\(q\)` --- # The hedonic model The price of a house is determined by a .hi[hedonic price function] `\(P(x,q)\)` -- `\(P\)` maps characteristics of the house and local environment to the market value of the home -- For a particular house `\(k\)` its price is `\(p_k = P(x_k,q_k)\)` -- `\(P\)` arises in equilibrium from the interaction of all buyers in sellers in the market -- Here we will assume the supply of houses is fixed in the short run so the price function arises from buyer behavior --- # The hedonic model: consumer's choice problem Households buy a single property given their budget constraint and their preferences -- Here we will assume that households are effectively just choosing `\((x,q)\)` instead of a specific house with the following objective: `$$\max_{x,q,z} U(x,q,z;s) \,\,\,\,\, s.t. \,\,\,\,\, y = z + P(x,q)$$` -- - `\(z\)` is the numeraire good (spending on other private goods) - `\(y\)` is income - `\(s\)` is the set of the household's characteristics like family size --- # Unrealistic pieces of the model One unrealistic part of this model is that we are assuming household characteristics are continuous -- Many housing characteristics are discrete (bedrooms, bathrooms, etc) -- Another is that you just can't purchase some sets of `\(x\)` (i.e. a huge lot in downtown manhattan with a farm) -- We won't touch on this in class but there is a .hi-blue[discrete choice] literature that works to alleviate these issues --- # Choosing q Another thing to note: the consumer *chooses* `\(q\)` where as before it was fixed -- The idea is that mobile households can move to get their desired level of the environmental good -- We are thus also implicitly assuming `\(q\)` varies across space so that households can sort into areas they prefer - q is really picking up .hi[local] environmental goods --- # What is `\(P(x,q)\)` In the model we are thinking of `\(P(x,q)\)` as the annual .hi-blue[rental rate], not the purchase price -- This allows us to mesh more cleanly with annual income and view the problem as static rather than dynamic -- This clearly works well for renting households -- For homeowners we are basically assuming they rent from themselves every year --- # The hedonic model: consumer's choice problem `$$\max_{x,q,z} U(x,q,z;s) \,\,\,\,\, s.t. \,\,\,\,\, y = z + P(x,q)$$` The FOCs for this problem are: `\begin{align} {\partial U \over \partial x_j} =& \lambda {\partial P \over \partial x_j} \,\,\, j=1,\dots,J \\ {\partial U \over \partial q} =& \lambda {\partial P \over \partial q} \\ {\partial U \over \partial z} =& \lambda \end{align}` Next, combine the last two FOCs --- # The hedonic model: consumer's choice problem `\begin{align} {\partial U \over \partial q} =& \lambda {\partial P \over \partial q} \\ {\partial U \over \partial z} =& \lambda \end{align}` gives us that `$${\partial P \over \partial q} = {\partial U \over \partial q} \bigg/ {\partial U \over \partial z}$$` At a utility-maximizing choice, a household equates their MRS between `\(q\)` and `\(z\)` and the marginal implicit cost of `\(q\)` --- # The hedonic model: consumer's choice problem `$${\partial P \over \partial q} = {\partial U \over \partial q} \bigg/ {\partial U \over \partial z}$$` Recall that `\(z\)` is the numeraire good so we can think of it in terms of dollars -- This means that `\({\partial U \over \partial q} \big/ {\partial U \over \partial z}\)` is the WTP for `\(q\)`, the reduction in income needed to compensate for an additional unit of `\(q\)` -- Knowledge of the hedonic price function `\(P\)` is enough to tell us about household WTP for `\(q\)`! --- # The hedonic model: bid functions Now let's dive deeper by looking at some reference utility level `\(\bar{u}\)`: `$$U(x,q,z;s) = \bar{u}$$` -- Next we will define something called a .hi-blue[bid function] `\(b(x,q,y,s,\bar{u})\)` where: `$$U(x,q,y -b(x,q,y,s,\bar{u}) ;s) = \bar{u}$$` The bid function `\(b\)` is the maximum amount the household is willing to pay for: - A house with characteristics `\(x,q\)` - Given income `\(y\)` and household characteristics `\(s\)` - Holding utility fixed --- # The hedonic model: bid functions `$$U(x,q,z;s) = \bar{u}$$` We can also invert this to solve for `\(z\)`:¹ `$$z = U^{-1}(x,q,\bar{u},s)$$` .footnote[ ¹ We can do this because `\(U\)` is monotonically increasing in `\(z\)` ] -- Income, the bid function and `\(z\)` are related by: `$$b(x,q,y,s,\bar{u}) ;s) = y - U^{-1}(x,q,\bar{u},s)$$` -- Now we have everything we need to derive a marginal WTP function for `\(q\)` --- # The hedonic model: deriving MWTP `$$U(x,q,y -b(x,q,y,s,\bar{u}) ;s) = \bar{u}$$` Differentiate with respect to `\(q\)` to get: `$${\partial U \over \partial q} + {\partial U \over \partial z} {\partial b \over \partial q} = 0$$` -- We can then rearrange to get: `$${\partial b \over \partial q} = {\partial U \over \partial q}\bigg/{\partial U \over \partial z}$$` --- # The hedonic model: deriving MWTP Recall that the bid function is separable in income: `\(b(x,q,y,s,\bar{u}) ;s) = y - U^{-1}(x,q,\bar{u},s)\)` -- This lets us re-write `\({\partial b \over \partial q}\)` as: `$$\pi^q(x,q,s,\bar{u}) = {\partial b \over \partial q} = {\partial U \over \partial q}\bigg/{\partial U \over \partial z}$$` -- Conditional on `\(x\)`, this defines our .hi-blue[compensated inverse demand function] for `\(q\)`! --- # The hedonic model: deriving MWTP `$$\pi^q(x,q,s,\bar{u}) = {\partial b \over \partial q} = {\partial U \over \partial q}\bigg/{\partial U \over \partial z}$$` By assuming `\(q\)` is a part of a larger bundle of house characteristics, we are able to characterize its demand through its relationship to the housing market -- We can then use the bid function (which maps into prices) to understand the marginal WTP for `\(q\)` -- Our ultimate empirical goal is to estimate `\(\pi^q(x,q,s,\bar{u})\)` --- # Bid functions and housing prices .pull-left[ The red line is the hedonic price function The blue lines are a single household's bid functions at different reference utility levels where `\(u_1 > u_0 > u_2\)` Higher utility `\(\rightarrow\)` lower bids because same level of `\(q\)` can be achieved with higher `\(z\)` ] .pull-right[  ] --- # Bid functions and housing prices .pull-left[ Optimal choice is where the household's bid function is tangent to the hedonic price schedule: `\(a\)` This gives us an observed consumption level `\(q_0\)`, observed price `\(P(x,q_0)\)`, and realized utility `\(u^0\)` ] .pull-right[  ] --- # Compensated MWTP .pull-left[ This plot shows the corresponding compensated MWTP curve associated with `\(b(x,q,y,s,u^0)\)` It is the slope of the bid function as `\(q\)` changes We observe `\(b\)` if we can estimate `\(P(x,q)\)` and its derivative ] .pull-right[  ] --- # Compensated MWTP .pull-left[ We can estimate `\(P(x,q)\)` using home sales prices and home attributes data The slope of `\(P(x,q)\)` is then equal to the MWTP for `\(q\)` This gives us the consumers inverse demand for `\(q\)` `$${\partial P(x,q_0) \over \partial q} = \pi^q(x,q_0,s,u^0)$$` ] .pull-right[  ]