## wpiea2019063

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---

### I. Optimality conditions of producing housing services
- Setup and definitions:
  - Representative housing services producer maximizes present value with Rt interest rate, St structure quantity, Lt land quantity, PLt land price, PSt structure price.
  - Housing service production function (CES): H(St, Lt) = (φ St1−1/ρ + (1−φ) Lt1−1/ρ)ρ/(ρ−1).
  - Define user-cost terms:
    - dSt = PSt − (1−δS) PSt+1 / Rt
    - dLt = PLt − PLt+1 / Rt
  - First-order conditions:
    - dt ∂H/∂St = dSt
    - dt ∂H/∂Lt = dLt
    - ∂H/∂St = φ (Ht / St)1/ρ
    - ∂H/∂Lt = (1−φ) (Ht / Lt)1/ρ

- Key lemmas:
  - Lemma 1 (log change relation for user-cost-of-housing):
    - ln(dt1) − ln(dt0) = ln(dSt1) − ln(dSt0) + 1/ρ ( ln(St1 / Ht1) − ln(St0 / Ht0) )
    - Implication: the elasticity of substitution ρ affects inference of user-cost-of-housing without assuming expected future house prices; "the smaller the elasticity of substitution between land and structures ρ, the larger the change in user-cost-of-housing for the same change in the ratio of structure quantity to housing services quantity."
  - Lemma 2 (cointegration of land/structure value and price ratios):
    - If ln(PLt dLt) and ln(PSt dSt) are stationary, then ln( (PLt / Lt) / (PSt / St) ) and ln(PLt / PSt) are cointegrated.
    - Exact relation:
      - ln( (PLt / Lt) / (PSt / St) ) = −ρ ln( φ/(1−φ) ) + (1−ρ) ln( PLt / PSt ) + ρ( ln(PLt / dLt) − ln(PSt / dSt) )

- Empirical estimate:
  - Engle-Granger two-step cointegration result (developed land and structure indexes, 1930-2015):
    - ln( (PLt / Lt) / (PSt / St) ) = −1.653 ∗∗∗ + 0.752 ∗∗∗ ln(PLt / PSt) + εt. (standard errors: (0.00953) (0.0118))
    - Aggregate elasticity of substitution between land and structures: ρ = 0.248.
    - Alternative raw indexes yield ρ = 0.204; ρ = 0.419; ρ = 0.542, but long time series preferred.

### II. The general equilibrium model
- Objective and structure:
  - Overlapping generations growth model with endogenous land development, production of structures, CES housing services; housing demand shocks and belief shocks drive house price cycles.
  - Two final goods: general consumption goods (numeraire) and housing services.

- Production and developers:
  - General consumption production: Yt = Ktα (Zt Nt)1−α; capital depreciation δK; firm problem max {Kt, Nt} [ Ktα (Zt Nt)1−α − rt Kt − wt Nt ].
  - Land developers: continuum measure 1; land development technology AL Lt−1−γ (YLt)ω with ω ∈ (0,1) and γ > 0.
    - Developer profit: πL(PLt, Lt−1) = max {YLt} [ PLt AL Lt−1−γ (YLt)ω − YLt ].
    - Mutual fund owns developers; present value VLt = ∑∞ s=0 πL(P L t+s, L t+s−1) / ∏s−1 j=0 R t+j.
  - Housing firm: CES H(St, Lt) as above; structures depreciate at rate δS; cost PSt units of general consumption goods.

- Households:
  - Overlapping generations: nt households born in period t, live T periods, mortality risk mt−a t; no borrowing constraints; no Ponzi: bt−T+1 t ≥ 0.
  - Lifetime utility: max {c, h, b} ∑a=0 T−1 βa ∏j=0 a−1 (1 − m t t+j) u(ct+a, ht+a; φt+a).
  - Period utility (log form): u(c, h; φ) = φ ln(c) + (1−φ) ln(h).
  - Budget constraint: ct+a + dt+a ht+a + bt+a = Rt t+a bt+a−1 + wt+a la + Tt+a.

- Shocks and timing:
  - Shocks considered: housing demand shocks (εφt); belief shocks (εBt); structure price shocks (εPSt); TFP shocks (εZt); growth shocks (εGt).
  - Sample processes (selected):
    - ln(Zt) = ln(Zt−1) + gZt + vZt; gZt = gZt−1 + εGt; vZt = ρZ vZt−1 + εZt.
    - ln(PSt) = ln(PSt−1) + gPS + εPSt.
  - Assumptions: perfect foresight; shocks hitting 2000–2015 have zero ex ante probability for tractability when ρ ≠ 1.

- Modeling choices and computation:
  - No explicit borrowing constraints modeled; renting vs owning not segmented.
  - Land price endogenous; structure price exogenously given.
  - One period = one year; households live 76 periods (i.e., 20-95).
  - Computation solves 602 equations for 602 unknowns; shocks sized to match annual data targets.
  - Model extends literature by allowing ρ ≠ 1 and endogenous land supply response.

### III. Parameterization and calibration
- Key calibrated/estimated parameters:
  - Short-run land supply elasticity ω = 0.497 (NAHB; average share in 2004).
  - Long-run land supply elasticity γ = 1.74 (using ω = 0.497).
  - Depreciation rates:
    - Structures: δS = 0.03.
    - Capital: δK = 0.071.
  - Households live for 76 periods (20-95).
  - Labor endowments by 5-year age group between age 25 and 60: 0.7500, 1.0150, 1.06750, 1.12000, 1.12375, 1.1275, 1.10375, 1.0800, 0.9275; labor endowment after age 60 set to zero.
  - Share of general consumption goods in utility φt−a t = 0.856 (NIPA 1950-2017).
  - Remaining parameters α, β, φ chosen to match:
    - Ratio of nonresidential investment to general consumption output;
    - Ratio of residential investment to general consumption output;
    - Share of land in aggregate housing value.
  - Elasticity of substitution between land and structures ρ = 0.248 (estimated).

- Table 1 key entries:
  - ρ = 0.248.
  - φ (share of structures in housing services production) = 0.973.
  - ω = 0.497.
  - γ = 1.74.
  - α = 0.233.
  - Elasticity of intertemporal substitution = 1.
  - β = 0.943.
  - Share of housing services in household preference (1 − φ) = 0.144.

### IV. Model validation and steady-state properties
- Model matches long-run trends:
  - Upward trends of land’s share in house value and ratio of house value to GDP.
  - Downward trend in ratio of residential investment to GDP.
  - Long-term trends of house, land and structure prices and quantities.
- No steady-state balanced growth path exists when ρ ≠ 1 unless growth rates of land and structures align; the model analyzes transition dynamics (initial steady state 1950; long-run steady state 2250).
- Important implication:
  - If ρ = 1 (Cobb-Douglas), model would generate very stable land share between 1950 and 2015, inconsistent with data; low ρ amplifies land share volatility.

### V. Quantitative results and transmission mechanisms (2000–2015)
- Estimation approach:
  - Two-step: (1) determine parameters from data or sensitive model moments; (2) for 2000–2015 choose shocks (housing demand, belief, structure price, TFP) sized to match four aggregate variables: real output, real land price, real structure price, and ratio of residential investment to general consumption output.
  - Assumptions: economy in transition dynamics starting 1950; no shocks between 1950 and 2000 or after 2015 (except distant convergence changes); shocks 2000–2015 have zero ex ante probability.

- Shock experiments (impulse responses):
  - Housing demand shock: increase of 1 percentage point in housing preference of current and future generations (hits in 2000).
  - Belief shock: increase of 1 percentage point in housing preference of future generations (hits in 2000).
  - TFP shock: increase in total factor productivity that leads to 1 percent increase in labor productivity (hits in 2000).
  - Structure price shock: increase of 1 percent in residential structure price (hits in 2000).

- Transmission patterns and sensitivities:
  - Variables examined: user-cost-of-housing (d), ratio of structure quantity to housing services quantity, house prices, ratio of house price to user-cost-of-housing.
  - Three notable patterns:
    1. User-cost-of-housing is sensitive to housing demand shocks but not to belief shocks.
    2. Smaller elasticity of substitution between land and structures (smaller ρ) amplifies impact of housing demand shocks and belief shocks on house prices.
    3. Smaller ρ dampens response of residential investment to housing demand shocks and dampens response of house prices to structure price shocks.
  - Interpretation:
    - Strong rise in user-cost-of-housing points to housing demand shocks as important drivers.
    - Elasticity of substitution ρ is critical quantitatively for housing price and quantity responses.

- Numerical decomposition for the 2000s housing cycle:
  - "Four fifths of the house price boom in the 2000s was caused by housing demand shocks and the remaining one fifth mainly by belief shocks."
  - "Belief shocks explain the entire deviation of the house price from its trend during the downturn."
  - User-cost-of-housing was much more volatile than the CPI rent during the 2000s.

### VI. Housing-cycle analysis, robustness, and interpretation
- Empirical patterns (2000s):
  - Divergence between user-cost-of-housing and CPI rent during 2000s; CPI rent’s rental-equivalence approach may mismeasure owner-occupied user-cost-of-housing.
  - Micro evidence: divergence between implied rent and market rent in Las Vegas, Miami, and Phoenix reported by Chinco and Mayer (2015).

- Decomposition and role of shocks (Figures referenced):
  - Housing demand shocks were the main driver of the boom (2000–2006); belief shocks contributed little to user-cost-of-housing rises but were critical during the crash.
  - If only housing demand shocks and/or structure price shocks occurred, house prices would converge back to trend after 2006 without going below it.
  - Negative belief shocks matter more than negative housing demand shocks during 2007–08 downturn; additional negative sentiment between 2010–11 amplified the decline.

- Robustness tests:
  - Cobb-Douglas case (ρ = 1) vs. baseline:
    - Under Cobb-Douglas, the increase in user-cost-of-housing between 2000 and 2006 would be only 20%, down from 30% in the smaller-elasticity case.
    - Contribution of belief shocks to the 2006 deviation of national house price from trend would be about 16% under Cobb-Douglas, up from 5% in the smaller-elasticity case.
    - Even with larger elasticity, belief shocks explain only about half of the house price boom (deviation from trend).
    - House price trend grows more slowly with more elastic housing supply; deviation of house prices from trend in 2006 would be around 30% under Cobb-Douglas.
    - Conclusion: a smaller ρ and explicit housing demand shocks and structure-price shocks reduce the role of belief shocks relative to some prior studies.
  - Alternative belief-shock specification (growth shocks):
    - Replacing belief shocks about future-generation housing preference with shocks to long-term productivity growth yields similar main findings: user-cost-of-housing much more volatile than CPI rent and housing demand shocks explain most of the boom.
    - Exception: matching 2010–2011 decline would require a permanent decline in long-term productivity growth of 6 percentage points between 2010 and 2011—deemed implausible—suggesting crash pessimism relates more to future housing-demand preference shocks than growth concerns.
  - Interaction tests: two alternative definitions of shock contributions (shutting off all but one shock vs. keeping all but one alive) yield almost the same results, indicating little interaction between shocks.

- Metrics for belief-driven cycles:
  - Traditional metrics (divergence between house price and rent, price-to-rent ratio) can be driven by housing demand shocks and thus may not reliably indicate belief-driven cycles.
  - The ratio of house price to user-cost-of-housing rose only modestly during the boom; this ratio is mainly driven by belief shocks.
  - Replacing rent with user-cost-of-housing in price-to-rent metrics better captures belief shifts disconnected from contemporaneous fundamentals.

### VII. Computational and transition dynamics details
- Transition dynamics assumptions:
  - Initial steady state in 1950; long-run steady state in 2250.
  - Shocks between 2000 and 2015 with zero ex ante probability; none between 1950–2000 or after 2015 (except distant adjustment to ensure long-run convergence).
  - Algorithm solves government transfers in outer loop and (S_t, L_t, R_t) in inner loop; 602 equations solved per inner-loop iteration.

- Numerical simplifications:
  - Elasticity of intertemporal substitution set to 1 allowing linearity exploitation; use of only 2 grid points for optimal household choices.
  - Use of revised Broyden method with truncation/resume to speed computation.

- Portfolio rule (household behavior):
  - Household is homeowner if net worth > 20% of home value; renters/homeowners with negative net worth cannot invest in non-home assets; four household portfolio types defined.
  - Aggregates and portfolio value ratios defined (θ_{V H,t}, θ_{V L,t}, θ_{K,t}, θ_{B,t}).

### VIII. Conclusions and open questions
- Main conclusions:
  - Housing demand shocks account for about four fifths of the 2000s house price boom; belief shocks account for about one fifth and explain the full downturn deviation.
  - User-cost-of-housing rose strongly during the boom due to residential investment boom, limited land supply, and small elasticity of substitution between land and structures—implying a strong decline in the marginal productivity of structures.
  - Elasticity of substitution between land and structures estimated at around 0.25 from cointegration between log ratios of land value to structure value and land price to structure price under stationarity assumptions.

- Open questions highlighted:
  - What affects cyclical divergence between user-cost-of-housing and the CPI rent?
  - What is the micro-foundation of housing demand shocks?

*Source: wpiea2019063 - Section V concludes.*

### Section V concludes.

### Section V concludes.

### I. Optimality Conditions of Producing Housing Services
- Setup:
  - Representative housing services producer sells housing services at price Ht and maximizes present value:
    max {St+s, Lt+s}∞s=0 1/∏s−1 j=0 Rt+j [ dt+s H(St+s, Lt+s) − pLt+s (Lt+s − Lt+s−1) + pSt+s (St+s − (1−δS) St+s−1) ]
    - Rt: interest rate
    - St: structure quantity
    - Lt: land quantity
    - PLt: land price
    - PSt: structure price
  - Housing service production function (CES):
    H(St, Lt) = (φ St1−1/ρ + (1−φ) Lt1−1/ρ)ρ/(ρ−1)
  - Define costs:
    - dSt = PSt − (1−δS) PSt+1 / Rt
    - dLt = PLt − PLt+1 / Rt
  - First-order conditions:
    - dt ∂H/∂St = dSt
    - dt ∂H/∂Lt = dLt
    - ∂H/∂St = φ (Ht / St)1/ρ
    - ∂H/∂Lt = (1−φ) (Ht / Lt)1/ρ

- Lemma 1 (log change relation for user-cost-of-housing):
  - ln(dt1) − ln(dt0) = ln(dSt1) − ln(dSt0) + 1/ρ ( ln(St1 / Ht1) − ln(St0 / Ht0) )
  - Implication: user-cost-of-housing can be inferred without assuming expected future house prices; elasticity of substitution ρ matters:
    - "the smaller the elasticity of substitution between land and structures ρ, the larger the change in user-cost-of-housing for the same change in the ratio of structure quantity to housing services quantity."

- Lemma 2 (cointegration of land/structure value and price ratios):
  - If ln(PLt dLt) and ln(PSt dSt) are stationary, then ln( (PLt / Lt) / (PSt / St) ) and ln(PLt / PSt) are cointegrated.
  - Exact relation:
    - ln( (PLt / Lt) / (PSt / St) ) = −ρ ln( φ/(1−φ) ) + (1−ρ) ln( PLt / PSt ) + ρ( ln(PLt / dLt) − ln(PSt / dSt) )

- Empirical cointegration and estimated elasticity:
  - Using developed land and structure value and price indexes (1930-2015) and Engle-Granger two-step test:
    - ln( (PLt / Lt) / (PSt / St) ) = −1.653 ∗∗∗ + 0.752 ∗∗∗ ln(PLt / PSt) + εt. (standard errors: (0.00953) (0.0118))
    - Therefore aggregate elasticity of substitution between land and structures ρ = 0.248.
  - Note: alternative raw indexes yield different ρ estimates (0.204; 0.419; 0.542) but long time series preferred.

### II. The General Equilibrium Model
- Objective:
  - Develop an overlapping generations growth model with endogenous land development, production of structures, and CES housing services; housing demand shocks and belief shocks are key drivers of house price cycles.

- Model elements:
  - Two final goods: general consumption goods (numeraire) and housing services.
  - General consumption production:
    - Yt = Ktα (Zt Nt)1−α
    - Capital depreciates at rate δK; labor supplied inelastically.
    - Representative firm maximizes profit: max {Kt, Nt} [ Ktα (Zt Nt)1−α − rt Kt − wt Nt ]
  - Land developers:
    - Continuum measure 1, land development technology: AL Lt−1−γ (YLt)ω
      - AL: constant scaler
      - Lt−1: aggregate land at end of previous period
      - YLt: general consumption goods used for land development
    - Short- and long-run land supply elasticities governed by ω and γ:
      - ω ∈ (0,1) implies decreasing returns in short run
      - γ > 0 captures scarcity of high-quality land
    - Individual land developer profit maximization:
      - πL(PLt, Lt−1) = max {YLt} [ PLt AL Lt−1−γ (YLt)ω − YLt ]
    - Representative mutual fund owns developers; present value:
      - VLt = ∑∞ s=0 πL(P L t+s, L t+s−1) / ∏s−1 j=0 R t+j
  - Housing firm:
    - Produces housing services Ht from St and Lt with CES: H(St, Lt) = (φ St1−1/ρ + (1−φ) Lt1−1/ρ)ρ/(ρ−1)
    - Structures depreciate at rate δS; one unit of structures costs PSt units of general consumption goods.
    - Firm maximizes net present value choosing {St+s, Lt+s}.

- Households:
  - Overlapping generations: nt households born in period t, live T periods, mortality risk mt−a t.
  - No borrowing constraints; no Ponzi: bt−T+1 t ≥ 0.
  - Lifetime utility maximization:
    - max {c, h, b} ∑a=0 T−1 βa ∏j=0 a−1 (1 − m t t+j) u(ct+a, ht+a; φt+a)
    - Budget constraints: ct+a + dt+a ht+a + bt+a = Rt t+a bt+a−1 + wt+a la + Tt+a; initial and terminal net worth conditions.
    - Period utility (log form): u(c, h; φ) = φ ln(c) + (1−φ) ln(h)
  - Government transfer Tt+a redistributes net worth of deceased.

- Shocks considered (baseline and robustness):
  - Housing demand shocks (εφt)
  - Belief shocks (εBt)
  - Structure price shocks (εPSt)
  - TFP shocks (εZt)
  - Growth shocks (εGt)
  - Processes (selected):
    - ln(Zt) = ln(Zt−1) + gZt + vZt; gZt = gZt−1 + εGt; vZt = ρZ vZt−1 + εZt
    - ln(PSt) = ln(PSt−1) + gPS + εPSt
    - φ processes detailed for cohorts (see model for exact timing equations)
  - Assumptions:
    - Agents have perfect foresight; shocks have zero probability ex ante to ensure tractability when ρ ≠ 1.

- Modeling choices and remarks:
  - No explicit borrowing constraints modeled; renting vs owning not segmented.
  - Land price endogenous; structure price exogenously given.
  - One period = one year; households live 76 periods (i.e., 20-95).
  - Computation involves solving 602 equations for 602 unknowns; shocks sized to match annual data targets.
  - Model extends existing literature by allowing ρ ≠ 1 and endogenous land supply response.

- Parameterization and calibration:
  - Land supply short-run elasticity ω equals share of general consumption goods in developed land value: ω = 0.497 (from National Association of Home Builders; corresponds to average share in 2004).
  - Long-run land supply elasticity γ estimated from optimality conditions: γ = 1.74 (using ω = 0.497).
  - Depreciation rates:
    - Structures: 3 percent (determined from structure indices excluding land development)
    - Capital: 7.1 percent
  - Households live for 76 periods (20-95).
  - Labor endowments (by 5-year age group between age 25 and 60): 0.7500, 1.0150, 1.06750, 1.12000, 1.12375, 1.1275, 1.10375, 1.0800, 0.9275. Labor endowment after age 60 set to zero.
  - Share of general consumption goods in utility φt−a t = 0.856 (average share of non-housing consumption in total private consumption, NIPA 1950-2017).
  - Three remaining parameters (share of capital α, household discount rate β, share of structures φ in housing production) chosen to match:
    - Ratio of nonresidential investment to general consumption output
    - Ratio of residential investment to general consumption output
    - Share of land in aggregate housing value
    - Averages computed over transition dynamics 1950–2015 (no steady-state balanced growth path when ρ ≠ 1).

- Model validation:
  - Model matches long-run trends: upward trends of land’s share in house value and ratio of house value to GDP; downward trend in ratio of residential investment to GDP; long-term trends of house, land and structure prices and quantities.
  - If ρ = 1, model would generate very stable land share between 1950 and 2015, inconsistent with data.

### III. Quantitative Results
- Approach:
  - Two-step: (1) determine parameters from data or sensitive model moments; (2) for 2000–2015 choose shocks (housing demand, belief, structure price, TFP) sized to match four aggregate variables: real output, real land price, real structure price, and ratio of residential investment to general consumption output.
  - Assumptions for quantitative experiment between 2000 and 2015:
    - Economy in transition dynamics starting 1950 (AppendixA5)
    - No shocks between 1950 and 2000 or after 2015 (except distant changes ensuring long-run convergence)
    - Shocks hitting 2000–2015 have zero ex ante probability

- Transmission mechanisms and key patterns:
  - Variables examined: user-cost-of-housing, ratio of structure quantity to housing services quantity, house prices, ratio of house price to user-cost-of-housing.
  - Three notable patterns:
    1. User-cost-of-housing is sensitive to housing demand shocks but not to belief shocks.
    2. Smaller elasticity of substitution between land and structures (smaller ρ) amplifies impact of housing demand shocks and belief shocks on house prices.
    3. Smaller ρ dampens the response of residential investment to housing demand shocks and dampens response of house prices to structure price shocks.
  - Interpretation:
    - Strong rise in user-cost-of-housing points to housing demand shocks as important drivers.
    - Elasticity of substitution ρ is critical quantitatively for housing price and quantity responses.

- Shock experiment details:
  - Housing demand shock: increase of 1 percentage point in housing preference of current and future generations (hits in 2000).
  - Belief shock: increase of 1 percentage point in housing preference of future generations (hits in 2000).
  - TFP shock: increase in total factor productivity producing general consumption goods that leads to 1 percent increase in labor productivity (hits in 2000).
  - Structure price shock: increase of 1 percent in residential structure price (hits in 2000).

- Findings from impulse experiments:
  - Figure 2 (referenced) shows:
    - Smaller ρ strongly amplifies responses of user-cost-of-housing and house price to housing demand shocks.
    - Smaller ρ significantly dampens response of ratio of structure quantity to housing services quantity to housing demand shocks.
  - Belief shocks:
    - Negligible immediate impact on user-cost-of-housing or ratio of structure quantity to housing services quantity in current period because share of future generations in aggregate housing services consumption rises slowly.
  - Structure price impacts:
    - Impact on house prices is significant, consistent with the fact that the share of structures in house value exceeded 60 percent around (text truncated in source).

*Source: wpiea2019063 - Section V concludes.*

### 2000.  However, a smaller elasticity of substitution between land and structures

### wpiea2019063 - 2000.  However, a smaller elasticity of substitution between land and structures

### Key findings
- Four fifths of the house price boom in the 2000s was caused by housing demand shocks and the remaining one fifth mainly by belief shocks.
- Belief shocks explain the entire deviation of the house price from its trend during the downturn.
- User-cost-of-housing was much more volatile than the CPI rent during the 2000s.
- The elasticity of substitution between land and structures is around 0.25 in the US economy at the national level.
- A small elasticity of substitution between land and structures amplifies the impact of housing demand shocks on house prices; previous studies often assume the elasticity equals 1.

### Housing-cycle analysis (2000s)
- Divergence: Figure 3 shows a strong divergence between user-cost-of-housing and the CPI rent during the 2000s.
- Measurement issues with CPI rent:
  - CPI rent uses a rental-equivalence approach that may mismeasure owner-occupied user-cost-of-housing.
  - Average renter differs from average homeowner (younger, less likely married, more financially constrained, more exposed to moving shocks).
- Micro evidence: Chinco and Mayer (2015) report divergence between implied rent (user-cost-of-housing) and market rent in Las Vegas, Miami, and Phoenix.
- Drivers of user-cost-of-housing rise (2000–2006):
  - Mainly caused by housing demand shocks and shocks to structure prices; belief shocks contributed very little to changes in user-cost-of-housing.
- Decomposition (Figure 4):
  - Housing demand shocks were the main driver of the house price boom in the 2000s.
  - Belief shocks played a critical role during the crash, explaining the entire downturn deviation.
  - The deviation caused by belief shocks was remarkably stable between 2002-07 (inconsistent with a rational bubble growing at the real interest rate).
  - If only housing demand shocks and/or shocks to structure prices occurred, house prices would converge back to trend after 2006 without going below it.
  - Negative belief shocks played a larger role than negative housing demand shocks in driving down house prices between 2007-08; another strong deterioration in sentiment occurred between 2010-11.

### Model mechanism and calibration
- User-cost-of-housing in the model equals the cost of structures divided by the marginal productivity of structures in producing housing services (optimality condition).
- The model matches:
  - The ratio of structure quantity to house quantity.
  - User-cost-of-structures.
  - The ratio of house value to business value-added.
- Structure price process: log real structure prices can be characterized by a linear trend plus a term equal to the logarithm of oil prices multiplied by a constant; oil futures often give fairly flat time profiles for future oil prices.

### Robustness tests
- Larger elasticity of substitution (Cobb-Douglass case):
  - Under Cobb-Douglass, the increase in user-cost-of-housing between 2000 and 2006 would be only 20%, down from 30% in the smaller-elasticity case.
  - The contribution of belief shocks to the 2006 deviation of national house price from trend would be about 16% under Cobb-Douglass, up from 5% in the smaller-elasticity case.
  - Even with larger elasticity, belief shocks explain only about half of the house price boom (measured by deviation from trend).
  - The house price trend grows more slowly with more elastic housing supply; the deviation of house prices from trend in 2006 would be around 30% under Cobb-Douglass.
  - Conclusion: both a smaller elasticity of substitution and explicit consideration of housing demand shocks and shocks to structure prices reduce the role of belief shocks relative to some prior studies.
- Alternative belief-shock specification (growth shocks):
  - Replacing shocks to future-generation housing preference with shocks to long-term productivity growth rate yields similar main findings: user-cost-of-housing much more volatile than CPI rent, and housing demand shocks explain most of the boom.
  - Exception: discrepancy for user-cost-of-housing between 2010 and 2011 due to a modeled long-term growth-rate decline that changes interest rates substantially.
  - To match the 2010–2011 house price decline in a closed-economy model, a large permanent decline in long-term productivity growth of 6 percentage points between 2010 and 2011 was needed—an implausibly large decline—suggesting the crash pessimism relates more to future housing-demand preference shocks than to growth concerns.
- Interaction tests: two alternative definitions of shock contributions (shutting off all but one shock vs. keeping all but one alive) yield almost the same results, suggesting little interaction between shocks.

### Metrics for belief-driven house price cycles
- Traditional metrics (divergence between house price and rent, and price-to-rent ratio) can be driven by housing demand shocks and thus may not reliably indicate belief-driven cycles.
- The ratio of house price to user-cost-of-housing rose only modestly during the boom; its driver is mainly belief shocks.
- Replacing rent with user-cost-of-housing in the price-to-rent ratio yields a metric that better captures belief shifts that are disconnected from contemporaneous fundamentals.

### Conclusions and open questions
- Main conclusions:
  - Housing demand shocks account for about four fifths of the 2000s house price boom; belief shocks account for about one fifth and explain the full downturn deviation.
  - User-cost-of-housing rose strongly during the boom due to a residential investment boom, limited land supply, and a small elasticity of substitution between land and structures—implying a strong decline in the marginal productivity of structures.
  - The elasticity of substitution is estimated at around 0.25, obtained from a co-integration relationship between log ratios of land value to structure value and land price to structure price under the assumption of a stationary price-to-user-cost ratio.
- Open issues not resolved in the paper:
  - What affects a cyclical divergence between user-cost-of-housing and the CPI rent?
  - What is the micro-foundation of housing demand shocks?

*Source: IMF working paper content unit wpiea2019063 (excerpt).*

### REFERENCES

### REFERENCES

### Major thematic areas covered by the references
- Housing markets, housing prices, and land vs structure valuation
- Asset-price bubbles, speculation, and behavioral drivers of housing booms and busts
- Measurement issues in price indexes and user cost vs rent equivalence
- Macro-financial linkages: housing finance, credit cycles, and DSGE approaches
- Urban economics, agglomeration, spatial misallocation, and supply constraints
- Empirical and theoretical methods for analyzing housing demand, supply, and macroeconomic effects

### Representative works and their focus (as cited)
- Behavioral and bubble literature
  - Akerlof and Shiller (2010). Animal spirits: How human psychology drives the economy, and why it matters for global capitalism.
  - Barberis et al. (2018). “Extrapolation and bubbles.” Journal of Financial Economics.
  - Scheinkman and Xiong (2003). “Overconfidence and speculative bubbles.” Journal of Political Economy.
  - Shiller (2000). “Irrational exuberance.” Philosophy & Public Policy Quarterly.
  - Glaeser (2013). “A nation of gamblers: Real estate speculation and American history.” American Economic Review, 103(3): 1–42.
  - Glaeser and Nathanson (2015). “Housing bubbles.” In Handbook of regional and urban economics. Vol. 5, 701–751. Elsevier.

- Housing valuation, land vs structure, and aggregate value dynamics
  - Davis and Heathcote (2005; 2007). “HOUSING AND THE BUSINESS CYCLE*.” International Economic Review, 46(3): 751–784; “The price and quantity of residential land in the United States.” Journal of Monetary Economics, 54(8): 2595–2620.
  - Capozza and Helsley (1989). “The fundamentals of land prices and urban growth.” Journal of urban economics, 26(3): 295–306.
  - Liu, Wang, and Zha (2013). “Land-price dynamics and macroeconomic fluctuations.” Econometrica, 81(3): 1147–1184.
  - Landvoigt, Piazzesi, and Schneider (2015). “The housing market(s) of San Diego.” American Economic Review, 105(4): 1371–1407.

- Housing finance, macroeconomic implications, and credit channels
  - Bernanke and Gertler (2001). “Should central banks respond to movements in asset prices?” American Economic Review, 91(2): 253–257.
  - Kiyotaki and Moore (1997). “Credit cycles.” Journal of political economy, 105(2): 211–248.
  - Favilukis, Ludvigson, and Van Nieuwerburgh (2017). “The macroeconomic effects of housing wealth, housing finance, and limited risk sharing in general equilibrium.” Journal of Political Economy, 125(1): 140–223.
  - Justiniano, Primiceri, and Tambalotti (2015). “Credit supply and the housing boom.” National Bureau of Economic Research.

- Modeling approaches and DSGE evidence
  - Iacoviello (2010). “Housing in DSGE models: Findings and new directions.” In Housing Markets in Europe. Springer.
  - Iacoviello and Neri (2010). “Housing market spillovers: evidence from an estimated DSGE model.” American Economic Journal: Macroeconomics, 2(2): 125–64.
  - Kaplan, Mitman, and Violante (2017). “The housing boom and bust: Model meets evidence.” National Bureau of Economic Research.

- Measurement, CPI bias, rents, and user cost debates
  - Boskin et al. (1998). “Consumer prices, the consumer price index, and the cost of living.” Journal of economic perspectives, 12(1): 3–26.
  - Lebow and Rudd (2003). “Measurement error in the consumer price index: where do we stand?” Journal of economic literature, 41(1): 159–201.
  - Gordon and Van Goethem (2004). “A Century of Downward Bias in the Most Important Component of the CPI: The Case of Rental Shelter, 1914-2003.” Hard-to-Measure Goods and Services: Essays in Memory of Zvi Griliches, Studies in Income and Wealth, (67).
  - Garner and Verbrugge (2009). “Reconciling user costs and rental equivalence: Evidence from the US consumer expenditure survey.” Journal of Housing Economics, 18(3): 172–192.
  - Verbrugge (2008). “The puzzling divergence of rents and user costs, 1980–2004.” Review of Income and Wealth, 54(4): 671–699.

- Urban economics, supply constraints, and spatial misallocation
  - Glaeser and Gyourko (2018). “The economic implications of housing supply.” Journal of Economic Perspectives, 32(1): 3–30.
  - Hsieh and Moretti (2015). “Housing constraints and spatial misallocation.” National Bureau of Economic Research.
  - Nathanson and Zwick (2017). “Arrested development: Theory and evidence of supply-side speculation in the housing market.” National Bureau of Economic Research.

### Figures and empirical patterns reported in the source
- Figure 1: Ratios of Value of Residential Land to Value of Residential Structures, 1950 – 2016, and Long-run Relationship Between this Ratio and the Ratio of Price of Land to Price of Structures
  - Data sources: Flow of Funds, Davis and Heathcote (2007), and the author’s calculations.
  - Definitions preserved exactly:
    - VL: the aggregate value of residential land
    - VS: the aggregate value of residential structures
    - PL: the price of land
    - PS: the price of structures
  - Key descriptive statements preserved exactly:
    - “Figure suggests that there is an upward trend for the ratio of the aggregate value of residential land to the aggregate value of residential structures, and a strong positive correlation between the ratio of the aggregate value of residential land to the aggregate value of residential structures and the ratio of the price of land to the price of structures.”
    - “Panel a is based on data from the Flow of Funds for 1950–2016. The pattern would be similar if the indexes constructed by Davis and Heathcote (2007) were used instead.”
    - “Panel b is based on the indexes of price and value of residential land and residential structures from Davis and Heathcote (2007) (which is further extended by Morris Davis to 2016Q1) and covers the 1930-2016 period.”
  - Numeric detail preserved:
    - slope = 0.799***

- Time-series plot axis labels and values (as presented)
  - x-axis range labels: 1940 1960 1980 2000 2020 (year)
  - y-axis numeric tick labels for panel a: 0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9 (ratio of VL to VS)
  - y-axis numeric tick labels for panel b: -2.5 -2 -1.5 -1 -0.5 0 0.5 (log(ratio of PL to PS)); and -2.5 -2 -1.5 -1 -0.5 0 0.5 (log(ratio of VL to VS))

- Impulse-response style panels and labels (as presented)
  - Section headings: 1. Housing demand shocks; 2. Belief shocks; 3. Shocks to structure prices
  - Panel labels and axis tick marks preserved exactly for the shock panels:
    - Panel 1a label: user-cost-of-housing (d); EOS = 0.2; EOS = 1
    - Panel 1b label: ratio of structure Q to house Q
    - Panel 1c label: house price
    - Panel 1d label: ratio of house price to d
    - Panel 2a label: user-cost-of-housing (d); EOS = 0.2; EOS = 1
    - Panel 2b label: ratio of structure Q to house Q
    - Panel 2c label: house price
    - Panel 2d label: ratio of house price to d
    - Panel 3a label: user-cost-of-housing (d); EOS = 0.2; EOS = 1
    - Panel 3b label: ratio of structure Q to house Q
    - Panel 3c label: house price
    - Panel 3d label: ratio of house price to d
  - Example axis tick labels preserved exactly:
    - For period axis: 0 5 10 15 (period)
    - For log change (percent) axes: examples include ranges and ticks such as 1, 2, 3, 4; 1, 1.5, 2, 2.5; -1.5 -1 -0.5 0 0.5; -0.5 0 0.5 1 1.5; 0 0.2 0.4 0.6 0.8; 0.5 1 1.5 2 2.5; 0.4 0.6 0.8 1 1.2; 0.5 0.6 0.7 0.8 0.9; -0.4 -0.3 -0.2 -0.1 0; 0.2 0.4 0.6 0.8; -0.2 -0.1 0 0.1 0.2

*Source: REFERENCES (as provided in the source PDF).*

### 4.  TFP shocks

### wpiea2019063 - 4.  TFP shocks

### Impulse responses of key housing variables
- Figure 2: Impulse responses to shocks (panel descriptions preserved)
  - Panel 1: responses to a 1 percentage point increase in the weight of housing services in the household utility function for both current and future generations.
  - Panel 2: responses to a 1 percentage point increase in the weight of housing services in utility function for future generations only.
  - Panel 3: responses to a 1 percent increase in the price of residential structures.
  - Panel 4: responses to a 1 percent increase in the level of productivity of nonhousing goods.
- Elasticity of substitution between land and structures cases plotted:
  - Blue curve: elasticity = 1.
  - Red curve: elasticity = 0.248.

### Model fit and time series (2000–15)
- Figure 3: Model vs. data (2000 = 100)
  - Panel a: user-cost-of-housing index shown (index levels from 100 to 130).
  - Panel b: ratio of structure quantity to house quantity (index levels from 100 to 106).
  - Panel c: ratio of house value to the output of non-housing goods (ratio values from 140 to 220).
- Model interpretation notes:
  - User-cost-of-housing is matched based on the first-order condition that user-cost-of-housing equals user-cost-of-structures divided by the marginal productivity of structures in producing housing services.
  - Because the quantity of house stock rather than quantity of housing service is observed, panel b plots the ratio of quantity of residential structures to quantity of house stock to suggest a good match with the ratio of structures to housing services.
  - Return to housing is not directly observed; panel c shows the model matches the ratio of house value to business value-added, which depends on the return to housing.

### Drivers of user-cost-of-housing and house price (2000–15)
- Figure 4 findings:
  - Both strong increases in user-cost-of-housing and house price are mainly driven by housing demand shocks.
  - Belief shocks explain the entire deviation of the house price from its trend during the downturn.
- Figure 5 (Cobb-Douglas aggregate housing service production function):
  - Plots user-cost-of-housing and house price under different shock configurations when aggregate housing services production function is Cobb-Douglas.
  - The share of land in the production technology is chosen so that the share of land in aggregate house value in 2000 equals its data counterpart.
- Figure 6 (different belief shock forms):
  - “Belief: preference” = belief shock as shocks to the housing preference of future generations.
  - “Belief: growth” = belief shock as shocks to long-term productivity growth rate.
  - Panel a plots user-cost-of-housing under different configurations and the CPI housing component.
  - Panel b plots house price under different shocks and the data counterpart.

### Ratio of house price to user-cost-of-housing (d)
- Figure 7:
  - Panel a: ratio of house price to user-cost-of-housing (d) measured by CPI housing (index range 90 to 130).
  - Panel b: trend vs. all shocks and contributions from housing demand, belief, structure price, and TFP.
  - Reiterates: strong increases in both user-cost-of-housing and house price mainly driven by housing demand shocks; belief shocks explain entire deviation of house price from trend during downturn.

### Key model parameters (Table 1)
- Housing and Nonhousing Goods Production Technology
  - Elasticity of substitution between land and structures in producing housing services (ρ): 0.248 — Estimated from the prices and quantities of developed land and residential structures.
  - Share of structures in the housing services production function (φ): 0.973 — Determined inside the model.
  - Short-term elasticity of land supply (ω): 0.497 — Estimated from the land development cost breakdown provided by the National Association of Home Builders.
  - Long-term elasticity of land supply (γ): 1.74 — Estimated from the prices and quantities of developed land.
  - Share of capital in the production function of nonhousing goods (α): 0.233 — Determined inside the model.
- Household Preferences
  - Elasticity of intertemporal substitution: 1 — Standard in the literature.
  - Household discount rate (β): 0.943 — Determined inside the model.
  - Share of housing services in household preference function (1 - φ): 0.144 — Estimated from the National Income and Product Account.
- Other Parameters
  - Depreciation rate of residential structures (δS): 0.03 — Estimated from the residential fixed asset table.
  - Depreciation rate of capital (δK): 0.071 — Estimated from the non-residential fixed asset table.
- Calibration note:
  - φ, α, and β are determined by ensuring consistency between the model and data counterparts for: (1) ratio of nonresidential investment to output of nonhousing goods, (2) ratio of residential investment to output of nonhousing goods, and (3) share of land in aggregate housing value between 1950 and 2015.

### Data sources and construction notes (Online Appendix excerpts)
- Main data sources listed (as used in the paper):
  - Quality-adjusted land and structure price and quantity indexes by Davis and Heathcote (2007), extended to 2016Q1 by Morris Davis.
  - TFP series constructed by John Fernald and co-authors.
  - Labor efficient units over the life cycle provided by Hanse (1993).
  - Detailed residential fixed asset table compiled by BEA.
  - Cost breakdown of land development and construction by NAHB.
  - CPI housing component by Bureau of Labor Statistics.
  - National Income and Product Accounts (NIPA) by BEA.
  - Population data from the United States census bureau.
  - Life tables from the centers of disease control and prevention (CDC).
  - World Population Prospects (2015 revision).
- Construction of developed land price index
  - Key equation used to back out land price inflation (Equation (A1)):
    - pLt / pL,t−1 = pHt / pH,t−1 − (1 − VL,t−1 / VH,t−1) pSt / pS,t−1 VL,t−1 / VH,t−1
    - where pHt = quality-adjusted housing price index by Davis and Heathcote, pSt = deflator of structures excluding land development (proxied by deflator of “owner occupied, 1-4 units, additions and alterations”).
  - Weight VL,t−1 / VH,t−1 (share of developed land in house value in period t−1) constructed in three steps involving an assumed constant χ linking raw land shares and a perpetual inventory method (Equation (A2)):
    - VLt = VL,t−1 pLt / pL,t−1 + VNew,raw land t + VNew, land development t
  - Assumption: VNew, land development t = ωt (VNew,raw land t + VNew, land development t) with ωt = 0.497.
  - Assumed VNew,raw land t / VResidential structures of new homes t = 0.106.
  - χ solved from fixed point (Equation (A3)) yields χ = 1.137, implying aggregate developed land value is 13.7% higher than aggregate raw land value.

### Land and construction cost breakdowns (NAHB and historical construction shares)
- Table A2 — Finished Lot Cost Breakdown for Single-family Units in the United States in 2004 (% of Lot Cost)
  - Raw Lot Cost: 0.503
  - Development Costs:
    - a. cost of processing approvals: 0.059
    - b. site preparation: 0.071
    - c. site improvement: 0.166
      - paving: 0.039
      - water and sewer: 0.052
      - erosion and sediment: 0.008
    - d. impact analysis: 0.005
    - e. water/electric hook-up: 0.018
    - f. land dedication or fee in lieu: 0.001
    - g. bonding/escrow fee: 0.005
    - h. financing cost: 0.032
    - i. tree preservation and planting: 0.017
    - j. wetland preservation and planting: 0.023
      - value of unbuilt land: 0.019
      - cost of mitigation: 0.002
    - k. value of land left unbuilt as green space or park: 0.004
    - l. other costs: 0.095
- Table A3 — Construction Cost Breakdown of Single Family Homes in the United States (1969-2011)
  - Finished Lot Cost (by year): 0.21, 0.24, 0.24, 0.24, 0.24, 0.26, 0.25, 0.20, 0.22
  - Total Construction Cost (by year): 0.55, 0.45, 0.53, 0.55, 0.51, 0.52, 0.48, 0.59, 0.59

*Source: the author’s calculations (wpiea2019063 - 4.  TFP shocks).*

### 3.  Financing Cost0.070.150.020.020.020.020.020.020.02

### wpiea2019063 - 3.  Financing Cost0.070.150.020.020.020.020.020.020.02

### Financing costs and construction cost shares
- Financing Cost: 0.07 0.15 0.02 0.02 0.02 0.02 0.02 0.02 0.02
- Other costs: 0.17 0.16
- 4. Overhead and General Expenses: 0.06 0.06 0.06 0.06 0.07 0.05 0.05
- 5. Marketing Cost: 0.02 0.01 0.02 0.02 0.03 0.01 0.02
- 6. Sales Commission: 0.03 0.03 0.04 0.03 0.04 0.03 0.03
- 7. Profit: 0.09 0.09 0.12 0.10 0.11 0.09 0.07
- Total: 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00
- Source: NAHB Constructioin Cost Surveys, 1969 - 2011.

### A3. Definition of Perfect Foresight Equilibrium
- A perfect foresight equilibrium in period t given K_t, L_{t−1}, S_{t−1}, {Z_{t+s}, P^S_{t+s}, φ_{t+s−a}, a_{t+s}, n_{t+s−a}, t+s+a, ... }_{s=0,...,∞; a=0,...,T−1} is defined by:
  - Prices {R_{t+s}, P^L_{t+s}, P^S_{t+s}, d_{t+s}, w_{t+s}, r_{t+s}}_{s=0,...,∞}
  - Allocations {K_{t+s}, N_{t+s}, L_{t+s}, S_{t+s}, H_{t+s}, Y^L_{t+s}, c_{t+s+t+s+a}, h_{t+s+t+s+a}, b_{t+s+t+s+a}}_{s=0,...,∞; a=0,...,T−1}
- Conditions for equilibrium:
  - (a) K_{t+s} and N_{t+s} solve problem (1).
  - (b) Y^L_{t+s} solves problem (2), given P^L_{t+s} and L_{t+s−1}.
  - (c) {L_{t+s}} and {S_{t+s}} solve problem (5), given {R_{t+s}, P^L_{t+s}, P^S_{t+s}}.
  - (d) {c_{t+s+t+s+a}, h_{t+s+t+s+a}, b_{t+s+t+s+a}}_{a=0,...,T−1} solve problem (6), given {R_{t+s+a−1}, w_{t+s+a}, d_{t+s+a}, T_{t+s+a}}.
- Market clearing conditions:
  - (i) Nonhousing goods:
    ∑_{a=0}^{T−1} n_{t+s−a}^{t+s} c_{t+s−a}^{t+s} + Y^L_{t+s} + P^S_{t+s}[S_{t+s} − (1−δ_S)S_{t+s−1}] + [K_{t+s+1} − (1−δ_K)K_{t+s}] = K_{t+s}^α (Z_{t+s} N_{t+s})^{1−α}.
  - (ii) Housing services:
    ∑_{a=0}^{T−1} n_{t+s−a}^{t+s} h_{t+s−a}^{t+s} = H(S_{t+s}, L_{t+s}).
  - (iii) Developed land:
    L_{t+s} − L_{t+s−1} = A^L L_{t+s−1}^{−γ} (Y^L_{t+s})^ω.
  - (iv) Labor:
    N_{t+s} = ∑_{a=0}^{T−1} n_{t+s−a}^{t+s} l_a.
  - (v) Funds:
    K_{t+s+1} + V^L_{t+s+1} + V^H_{t+s+1} R_{t+s} = ∑_{a=0}^{T−1} n_{t+s−a}^{t+s} b_{t+s−a}^{t+s}.
- No-arbitrage (ex ante):
  R_{t+s} = α K_{t+s}^{α−1} (Z_{t+s+1} N_{t+s+1})^{1−α} + 1 − δ_K.
- Government budget constraint:
  T_{t+s} = R_{t+s} [∑_{a=0}^{T−1} n_{t+s−a−1}^{t+s−1} m_{t+s−a−1}^{t+s−1} b_{t+s−a−1}^{t+s−1}] / [∑_{a=0}^{T−1} n_{t+s−a}^{t+s}].
- Population dynamics:
  - n_{t+s}^{t+s} = n_{t+s}
  - n_{t+s}^{t+s+a} = (1 − m_{t+s}^{t+s+a−1}) n_{t+s}^{t+s+a−1}
- Note: no-arbitrage holds ex ante; unexpected shocks can make returns differ ex post.

### A4. No Steady State Balanced Growth Path — main result and Proposition A1
- Main feature: No steady state balanced growth path exists if the elasticity of substitution between land and structures is different from 1, unless the growth rate of land quantity equals that of structure quantity.
- Proposition A1: A steady state exists if and only if the following conditions are satisfied:
  - ρ = 0 or μ_t = (1 − ω/(1+γ)) (g^Z_t + g^N_t).
- Proof sketch and implications:
  - Steady-state algebra shows that if g^H ≠ g^L, then (S_{t−1}/L_{t−1})^{−ρ} must be constant; this requires ρ = 0 or g^S = g^L.
  - If g^S = g^L and g^H = g^L, then μ_t must satisfy μ_t = (1 − ω/(1+γ)) (g^Z_t + g^N_t).
  - Further derivations establish:
    - (i) interest rate R_t must be constant (g^R = 0) in a steady state;
    - (ii) wage w_t grows at same rate as Z_t;
    - (iii) government transfer to individual households must grow at same rate as productivity Z_t, otherwise contradictions arise in aggregate consumption growth conditions.
- Household problem transformed with normalized variables shows dependence of consumption on ̄w, R^g_Z, T_{t+a}/Z_{t+a}, g_T − g_Z, a; steady-state consistency requires g_T = g_Z.

### A5. Definition of transition dynamics and modeling assumptions
- Calibration and observation:
  - Calibrated parameters imply the U.S. economy was not on a steady state balanced growth path between 2000 and 2015 once housing dynamics are considered.
  - Study assumes housing-related shocks during 2000–2015 caused the economy to move between transition dynamics rather than fluctuate around one steady state.
- Assumptions for transition analysis:
  - (i) Initial steady state in 1950.
  - (ii) Long-run steady state in 2250.
  - (iii) A shock in 1950 initiates transition dynamics; multiple shocks with zero ex ante probability occur between 2000 and 2015.
  - (iv) No other shocks between 1950–2000 or after 2015 (except a distant change causing convergence to long-run steady state).
- Baseline shock types between 2000 and 2015:
  - Productivity shocks.
  - Housing demand shocks.
  - Belief shocks.
  - Shocks to structure prices.
- Robustness test: replace belief shocks about housing preference with belief shocks about long-term productivity growth.
- Perfect foresight implication: all assets pay same return ex ante; portfolio choice undetermined ex ante -> author specifies a simple portfolio rule to handle ex post divergence in returns.

Part structure for transition dynamics analysis:
- (1) Characterization of the steady state balanced growth path (Propositions A2 and A3).
- (2) Simple rule about household portfolio choices.
- (3) Characterization of the initial steady state.
- (4) Characterization of the long-run steady state.
- (5) Characterization of the transition dynamics between 1950 and 2250.

### Part 1. Steady state balanced growth path — Propositions A2 and A3 (relationships)
- Proposition A2: In any steady state balanced growth path, growth rates satisfy:
  - g^H_t = φ g^S_t + (1 − φ) g^L_t
  - g^d_t = g^Z_t + g^N_t − g^H_t
  - g^L_t = ω/(1 + γ) g^A_t + g^N_t
  - g^{P^L}_t = (1 − ω/(1 + γ)) (g^A_t + g^N_t)
  - g^S_t = g^A_t + g^N_t − g^μ_t
  - g^K_t = g^A_t + g^N_t
  - g^w_t = g^A_t
- Definitions for transformed variables in steady state balanced growth path (notation for R_s, w_s, T_s, h_{a,s}, b_{a,s}, c_{a,s}, n_{a,s}, e_s, H_s, X_s).
- Proposition A3: Transformed variables satisfy system of steady-state equations, including:
  - 1 − (1−δ_K)/R_s = α K_s^{α−1}
  - w_s = (1 − α) K_s^α
  - n_{a,s} expressions via product of M_i and e^{−i g_n}
  - e_s, H_s, T_s definitions and relations involving P^L_t L_t/(Z_t N_t), P^S_t S_t/(Z_t N_t), V_{H,s}, V_{L,s}
  - Aggregate household problem in transformed variables with budget constraints and non-negativity b_{T−1,s} ≥ 0.

### Part 2. Household portfolio choices — rule and implications
- Definition: household owns home if owns at least h_t/H_t fraction of representative housing service producer; home value v^H_t(h_t) = h_t/H_t V^H_t.
- Three aspects of the portfolio rule:
  - (1) Household of age a in period t is homeowner if net worth b_{t−a}^t > 20% of v^H_t(h_{t−a}^t).
  - (2) If renter with negative net worth or homeowner with net worth minus v^H_t(h_t) negative, the household does not invest in:
    - (i) loans lent to households,
    - (ii) properties leased to renters,
    - (iii) physical capital,
    - (iv) shares of mutual fund owning land developers.
    - Such households only borrow from other households.
  - (3) Renters with positive net worth and homeowners who do not only borrow (i.e., net worth exceeds home value) have the same portfolio.
- Four household portfolio types implied:
  - (i) renters who only borrow;
  - (ii) homeowners who only borrow;
  - (iii) renters who invest in assets;
  - (iv) homeowners who invest in non-home assets.
- Dummies:
  - I_{own,a}^t = 1 iff b_{t−a}^t > 0.2 h_{t−a}^t/H_t V^H_t.
  - I_{inv,a}^t = 1 iff (i) I_{own,a}^t = 1 and b_{t−a}^t > h_{t−a}^t/H_t V^H_t, or (ii) I_{own,a}^t = 0 and b_{t−a}^t > 0.
- Aggregates:
  - B_t = ∑_{a=0}^{T−1} {(b_{t−a}^t − h_{t−a}^t/H_t V^H_t) I_{own,a}^t (1 − I_{inv,a}^t) + b_{t−a}^t (1 − I_{own,a}^t)(1 − I_{inv,a}^t)}
  - NW_{Inv,t} = ∑_{a=0}^{T−1} {b_{t−a}^t I_{own,a}^t I_{inv,a}^t + b_{t−a}^t (1 − I_{own,a,t}) I_{inv,a}^t}
  - V_{H,Inv,t} = ∑_{a=0}^{T−1} h_{t−a}^t/H_t V^H_t I_{own,a}^t I_{inv,a}^t
  - V_{H,own,t} = ∑_{a=0}^{T−1} h_{t−a}^t/H_t V^H_t I_{own,a}^t
  - V_{H,rent,t} = ∑_{a=0}^{T−1} h_{t−a}^t/H_t V^H_t (1 − I_{own,a}^t)
- Portfolio value ratios:
  - θ_{V H,t} = V_{H,rent,t} / (V_{H,rent,t} + V_{L,t} + K_{t+1} − B_t)
  - θ_{V L,t} = V_{L,t} / (V_{H,rent,t} + V_{L,t} + K_{t+1} − B_t)
  - θ_{K,t} = K_{t+1} / (V_{H,rent,t} + V_{L,t} + K_{t+1} − B_t)
  - θ_{B,t} = −B_t / (V_{H,rent,t} + V_{L,t} + K_{t+1} − B_t)
- Individual household asset allocations expressed as functions of I_{own,a,t}, I_{inv,a,t}, θ ratios, and b_{t−a}^t.

### Part 3. The initial steady state (calibration targets and numbers)
- Mortality rate by 5-year age group (initial steady state) from 1900–1949 average; mortgage rate by 5-year age group used as mortality proxy: {0.72%, 0.76%, 0.93%, 1.32%, 2.09%, 3.20%, 5.03%, 7.26%, 10.61%, 15.10%, 22.08%, 32.13%, 44.49%, 59.37%, 72.73%, 100%} then linearly interpolated to 1-year age groups.
- Population growth rate (geometric average 1900–1949): 1.018%.
- Growth rate of real price of structure (average 1930–1950): 1.634%.
- Growth rate of labor productivity (average 1950–2015): 1.99%.

### Part 4. The long-run steady state (calibration targets and numbers)
- Demographics based on medium fertility variant of 2015 World Population Prospects.
- Mortality rates by 5-year age group (projected 2100): {0.08%, 0.09%, 0.11%, 0.14%, 0.22%, 0.36%, 0.57%, 0.88%, 1.32%, 2.17%, 3.77%, 6.76%, 12.63%, 23.34%, 39.81%, 100%}; interpolated to 1-year age groups.
- Population growth rate (geometric average projected 2050–2095): 0.122%.
- Growth rate of real structure prices (average 1950–2015): 0.674%.
- Growth rate of productivity chosen such that g^μ_t = (1 − ω/(1+γ))(g^Z_t + g^N_t) holds and is equal to 0.7%.

### Part 5. Transition dynamics between 1950 and 2250 (setup and parameter paths)
- Demographic structure for transition dynamics: estimated and forecasted values from World Population Prospects (medium fertility variant).
  - Mortality rates between 1950 and 2015: estimated values.
  - Mortality rates between 2015 and 2100: forecast values.
  - Mortality rates between 2100 and 2250: same as forecast values of 2100.
- Growth rates:
  - Productivity growth rate during transition set equal to long-run steady state: 1.99% until after 2200.
  - Real structure price growth: 0.647% (same as long-run steady state).
  - After 2200, productivity growth rate declines to 0.7% so that g^μ_t = (1 − ω/(1+γ))(g^Z_t + g^N_t) holds.
- After the decline in productivity growth rate, economy converges rapidly to long-run steady state.

### A6. Algorithm to compute transition dynamics
- Solution approach:
  - Outer loop: solves (T_t)_{t=1950,...,T_Long run} such that government budget constraint is satisfied in every period from 1950 to T_Long run.
  - Inner loop: solves (S_t, L_t, R_t)_{t=1950,...,T_Long run} using market clearing conditions for developed land and funds, taking (T_t, Z_t, P^S_t) as given.
- Algorithm applicability: can be applied to compute new transition dynamics in response to shocks.
- Subsections:
  - Part 1: solving (S_t, L_t, R_t) given (T_t, Z_t, P^S_t).
  - Part 2: solving (T_t)_{t=1950,...,T_Long run}.

*Source: wpiea2019063 - chapter content as provided.*

### Part 1. Solve

### Part 1. Solve

### Model setup and equilibrium conditions
- Wage, depreciation, and price relationships (first-order conditions of general goods and housing service producers):
  - w_t = (1−α) Z_t (R_t − (1−δ_K) α)^{α/(α−1)}
  - d_H_t = ( (P_S_t − (1−δ_S) P_S_{t+1} / φ) ( φ + (1−φ) (S_t L_t)^{ρ} )^{1+1/ρ} )
  - P_L_t = ∑_{j=0,...,∞} (1 / ∏_{s=t}^{t+j−1} R_s^{1−φ}/φ (S_{t+j} L_{t+j})^{1+ρ} (P_S_{t+j} − (1−δ_S) P_S_{t+j+1} / R_{t+j}) )
- Long-run steady state assumptions:
  - R_t = R_{t−1}, S_t L_t = S_{t−1} L_{t−1} e^{g_{S,LR} − g_{L,LR}} for t > T_{Long run}
  - g_{S,LR} and g_{L,LR} solved based on proposition A.2
- Aggregate housing services and demand for developed land:
  - L_D_t = ∑_{a=0}^{T−1} h_{t−a}^t({w_t,d_t,R_t,P_L_t,P_S_t}_{t=1950,...,∞}, Ω_{1950}) { φ (S_t L_t)^{1−1/ρ} + (1−φ) }^{ρ/(ρ−1)}
  - Ω_{1950} is the relevant information in the initial steady state in 1950
- Supply of developed land:
  - Static, depends only on aggregate land in previous period and current land price
  - Solve {L_S_t}_{t=1950,...,2250} iteratively as a function of {P_L_t}_{t=1950,...,∞} and L_{1949}
- Market clearing and unknowns:
  - Equations: L_D_t = L_S_t for t = 1950,...,2250 (301 equations)
  - Funding market clearing between 1950 and 2250 gives another 301 equations
  - Total 602 equations solve unknowns {S_t L_t, R_t}_{t=1950,...,2250}

### Computational strategies and simplifications
- Major computational burdens:
  - Solving 602 equations per inner-loop iteration
  - Solving 301 generations’ optimal choice problems, each living 80 periods (ages 20 - 99)
  - Calibrating yearly shock sizes to match data targets for 2000–2015 housing cycle
- Tractable simplifications implemented:
  - Set the elasticity of inter-temporal substitution (EIS) = 1 to exploit linearity of policy functions in wealth; allows use of only 2 grid points for optimal household choices
  - Use a revised Broyden method that stops after a certain number of iterations and resumes from the place with the best solution to substantially speed computation

### Solving government transfers
- After optimal household choices are computed, derive total net worth of households who die at end of previous period
- Use that to set up government budget constraints and solve {T_t}_{t=1950,...,T_{LR}}

### Model performance: explaining housing trends
- Low land-structures substitutability inference:
  - Inferred from strong positive co-movement between ratio of land value to structure value and ratio of land price to structure price in the long run
  - Plays a critical role in explaining housing trends and transmission of shocks to land share
- Comparison of elasticities of substitution:
  - Elasticity of substitution (EOS) between land and structures used in baseline: EOS = 0.248
  - Counterfactual with EOS = 1 leads to collapse in model’s ability to explain land share dynamics
  - If EOS = 1 (Cobb-Douglas), land share becomes much less volatile and stable in the long run, inconsistent with data
- Empirical consistency (Figure A2 summary):
  - Model reproduces:
    - Downward trend of ratio of residential investment to GDP
    - Upward trends of land share in house value and ratio of house value to business value-added
    - Long-term trends of price and quantity of house, residential land, and residential structure
  - Panel 4 shows low EOS amplifies impact of shocks on land share and is necessary to match land share volatility
  - In panels 4b and 4c: shocks under "EOS = 1" are the same as shocks under "EOS = 0.248"; these shocks are chosen to match the housing cycle in the 2000s

### Data and calibration notes appearing in figures
- Demographics:
  - Population share by age group plotted for years: 1950, 1975, 2000, 2025, 2050, 2075
  - Age group ranges shown 20–80 in plots
  - Age distributions for 1d, 1e, 1f based on medium fertility rate scenario from World Population Prospects in 2011
- Macroeconomic and housing series (indices and shares):
  - Output index (2000 = 100) shown for years 1960–2010
  - Land share, ratio of residential investment to output, ratio of non-residential investment to output, ratio of non-residential fixed asset to output, ratio of house value to output shown for 1960–2010
  - Price and quantity indexes (2000 = 100) for house price, land price, structure price, ratio of house quantity to GDP, ratio of land quantity to GDP, ratio of structure quantity to GDP shown for 1950–2050
- Notes on series:
  - In panels 2c, 2d, 2e, 2f, "output" refers to non-housing goods; data counterpart is value-added of business sector
  - Except for panel 2a, no normalization applied to other series in panel 2
  - The model and data series are compared directly in the figures

*Source: wpiea2019063 - Part 1. Solve (IMF working paper PDF).*

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_Source: https://www.imf.org/-/media/files/publications/wp/2019/wpiea2019063.pdf_