We need to construct measures of output, capital, the stock of houses (Y , K, H), and their investment counterparts according to an appropriate criterion. We use data from the National Income and Product Accounts (henceforth NIPA) and the Fixed Assets Tables (henceforth FAT), both from the Bureau of Economic Analysis. We deﬁne capital as the sum of non-residential private ﬁxed assets plus the stock of inventories plus consumer durables. Investment in capital, Ik, is deﬁned accordingly. H is private residential stock and Ih is private residential investment. Finally, we need a measure of output, Y . In our benchmark economy, output consists of labor income plus income from non-residential capital: Y = F(K,L) = wL+rK = C +Ik+Ih. Thus, output is measured as GDP minus housing services.6 We proceed as Cooley and Prescott (1995) to calculate the capital share of our economy. We do not make any imputation to output for government owned capital since are focus is on privately held wealth. The implied share of capital in output is 0.26. The capital-output ratio is 1.64 and the housing-output ratio is 1.07.7 We set the depreciation rate of capital so that it matches the investment-capital ratio, 0.12. The implied steady state interest rate
is 3.91 percent.8 Finally, we need a measure of GDP in our model economy. GDP is simply output, Y , plus housing services. We follow Cooley and Prescott (1995) and set GDP=Y + iH, where i = r + δh is the implicit rental price for housing services. The resulting capital-GDP ratio is 1.51 and the corresponding housing-GDP ratio is 0.98. The aggregate ratio (K + H)/GDP is 2.49. The share of capital income to GDP in our model, once we impute housing services, is 31 percent, slightly lower than that estimated by Prescott (1986). For preferences over consumption of the nondurable good and housing services, we follow Luengo-
Prado (2006) and use the separable utility function u(c, s(·)) = c1−σ 1−σ + γ s(·)1−σ 1−σ . We assume that housing services are proportional to the housing stock and set the constant of proportionality to one. σ, the risk aversion parameter is 2. The calibration of γ and δh is not straightforward due to the presence of adjustment costs. In the steady state, Ih is δh H plus the aggregate adjustment cost. We choose values for γ and δh to jointly match the ratio of housing to nondurable consumption and the housing-output ratio in NIPA (H/C = 1.40 and H/Y = 1.07, respectively). This implies γ = 0.166 and δh = 0.0367. The discount factor, β = 0.9006, is such that the net interest rate in the steady state is 3.91 percent.
We use a down payment of 20 percent, slightly below the 25 percent average down payment for the period 1963-2001 reported by the Federal Housing Finance Board. Thus, individuals can borrow up to 80 percent of the value of the durable.9 While in reality households may be able to acquire houses with lower down payments, it is also the case that these households face higher marginal borrowing costs (including a higher interest rate and the purchase of mortgage insurance). To keep the model tractable, the down payment parameter is the same for all consumers and the borrowing rate is not a function of θ. We report results for higher and lower down payments in section 5 to assess the robustness of our results.
We consider non-convex costs of adjustment in the market for houses, which result in infrequent changes of the residential stock. We assume households pay the adjustment cost every time the value of the stock changes. The idea underlying this assumption is that a household can buy a house of any desired size, but once it has been bought the stock is illiquid. In order to change the house, the household needs to sell the stock and selling it entails transaction costs. We assume that right after consuming housing services the stock depreciates at the rate δh, and that if a household lets the house depreciate, the household must pay the adjustment cost (i.e., we force households to do maintenance of the stock).10 In particular, the speciﬁcation of the adjustment cost is:τ (h , h) = Iρ(1 − δh) h, (9)where I = 0 if h = h, and 1 otherwise. This cost can be seen as a loss in the selling price when changing the housing stock. Note that once the household decides to change the stock, the adjustment cost is proportional to the inherited level of residential assets, ρ (1 − δh) h. With this speciﬁcation, the transaction cost does not quickly diminish in importance as households become wealthier, as with a purely ﬁxed cost. In our benchmark case, we set ρ equal to 5 percent (the typical fee charged by real estate brokers in the U.S. economy is around 6 percent). Computational details on how to compute the model are given below.
In an economy with no transaction costs, a zero down payment, and a perfect rental market, the return to ﬁnancial assets is the same as the market return for housing. Therefore, the household portfolio composition cannot be determined. Additionally, the consumption of housing services is not tied to the household’s holdings of residential assets. Households can acquire additional housing
services or sell housing services to others in the rental market. In this case, the price of housing services aﬀects the composition of the consumption basket (nondurable consumption vs. housing services) but not the savings decision. Thus, the household problem can be written in terms of two state variables, earnings and total assets (a + h). More details are given in Appendix A. Our calibration strategy is such that both the benchmark economy and the one-asset economy produce the same capital-output, housing-output and housing-nondurable ratios. The parameter in the utility function, γ = 0.161, is chosen to match the ratio of residential stock to nondurable consumption in the data, H/C = 1.40.11 The depreciation rate of houses, δh, is set so that it matches the housing investment-stock ratio in the data, 0.043. The discount factor, β = 0.904 is chosen so that the ratio of total wealth to GDP, (K + H)/GDP is equal to 2.49. The share of capital and the depreciation rate of capital do not change. Table 3 summarizes the calibration parameters for both the benchmark economy and the one-asset economy (the other rows in the The wealth distribution Table 4 shows wealth distribution and wealth composition statistics for the benchmark economy that we can compare to the ones from the data summarized in Table 1. Wealth is unequally distributed with a Gini index of 0.801 (this is because of our calibration strategy). In our model as in the data, houses are more equally distributed than ﬁnancial assets. The Gini coeﬃcient for houses is 0.483 (0.649 in the data), while the Gini coeﬃcient for ﬁnancial assets is 0.93 (0.945 in the data).12 Also, houses represent a smaller proportion of wealth for the rich (as in the data). Since the return to housing is the marginal utility of the services it renders and marginal utility is decreasing, this is not surprising. The model fares remarkably well in the wealth-composition dimension given that we abstract from several factors that may aﬀect the composition of a household’s portfolio,such as taxes, house price changes and life-cycle eﬀects. For instance, our model predicts that households in the bottom 40 percent of the wealth distribution hold, on average, 339 percent of their wealth as houses, whereas this number is 280 percent in the data. For the top quintile, the predicted ratio in the model is 24, while it is 27 in the data. Next, Table 4 presents wealth distribution statistics for the one-asset economy. In this case, we cannot distinguish between ﬁnancial assets and houses and concentrate on total wealth. The Gini index for wealth is slightly lower in the benchmark economy than in the one-asset economy, 0.801 and 0.816, respectively. Inequality is lower in the benchmark economy (which does not allow for a rental market) because all households have some wealth in the form of the required down payment. The diﬀerence between both economies is small because the frictions of our model (down payments and adjustment costs) mainly aﬀect the poor who only account for a small fraction of aggregate wealth. In the one-asset economy, both the down payment and the adjustment cost are zero (as opposed to 20 and 5 percent,respectively, in the benchmark case). That is, there is more credit and more liquidity in the one-asset economy than in the benchmark economy. Lowering the down payment only aﬀects households who are constrained at the margin, typically poor households. Moreover, the larger the house a individual owns, the higher the possible loan. That is, collateralized loans of this type provide more credit to households who may need it less. In addition, wealthy households do not need to change their houses as often as less wealthy ones, the reason being thatthe variance of their income is lower than that of poor households because earnings represent a lower fraction of their income. Aside from houses being illiquid, we have made two assumptions that may be important for understanding the diﬀerences between the benchmark economy and the one-asset economy. The ﬁrst is the absence of a housing rental market. The second is the minimum house size that a household can purchase. We analyze each in turn.