## 1. Calibrated Parameter

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### Background and research question
- Housing market imbalances in Canada are a key source of systemic risk, driven by rapidly-rising house prices coupled with rising household indebtedness.
- Central question: Which policy—macroprudential policy (LTV limit) or property-transfer tax—best satisfies policymakers’ overall objectives?
- Approach: Develop a simple DSGE framework with two household types, two production sectors, three policy instruments (interest rate, LTV ratio, property-transfer tax), estimate by Bayesian methods on Canadian quarterly data (1993–2017), and conduct optimal-policy experiments under specified operational objectives for the central bank, macroprudential authority, and property-tax authority.

### Model structure and calibration (selected)
- Households:
  - Two types: ω borrowers and (1−ω) savers; borrowers are impatient (β_b), savers are patient (β_s).
  - Borrowers face an LTV-based borrowing constraint: R_t B_t^b ≤ (1−χ)(1−δ) E_t[ Q_{t+1} H_t^b / Π_{C,t+1} ] ε_t^{LTV}.
  - Property-transfer tax shock ε_t^τ ≡ exp(TAX_t); LTV shock ε_t^{LTV} ≡ exp(LTV_t).
- Sectors and prices:
  - Two production sectors: housing (H) and non-housing consumption (C).
  - Calvo pricing with backward- and forward-looking components; New Keynesian Phillips curves for consumption and housing (equations (14) and (15)).
- Monetary policy:
  - Taylor-type rule: r̂_t = ρ_r r̂_{t−1} + (1−ρ_r)(φ_π π̂_{C,t} + φ_y Δŷ_t) + ν_t^r.
- Exogenous shocks:
  - AR(1) processes for technology, cost-push, preference, foreign, and government shocks; LTV and tax policy shocks set to zero for estimation sample.

- Calibrated steady-state parameters (Table 1):
  - β_s = 0.99
  - β_b = 0.98
  - δ = 0.01
  - (1−χ) LTV steady state = 0.60
  - τ property tax steady state = 0.05
  - α degree of openness = 0.30
  - γ share of housing in utility = 0.30

- Selected estimated parameters (posterior means and [10%, 90%] intervals):
  - σ = 1.05 [0.97, 1.12]
  - ψ = 1.97 [1.81, 2.13]
  - h = 0.69 [0.67, 0.71]
  - ω = 0.61 [0.59, 0.63]
  - ζ = 1.13 [1.06, 1.21]
  - η = 1.09 [1.01, 1.17]
  - ε = 4.05 [3.89, 4.21]
  - ρ_r = 0.76 [0.72, 0.79]
  - φ_π = 2.10 [1.83, 2.36]
  - φ_y = 0.21 [0.13, 0.29]
  - θ_C = 0.71 [0.67, 0.74]; θ_H = 0.50 [0.47, 0.54]
  - ι_C = 0.80 [0.72, 0.89]; ι_H = 0.79 [0.71, 0.88]

### Data and estimation
- Quarterly data: 1993–2017.
- Measurement variables: real GDP per capita, real consumption per capita, real residential investment per capita, employment per capita, headline CPI inflation, house price inflation, overnight policy interest rate.
- Variables expressed as log deviations from steady states (HP filter for reals; sample averages for nominals).
- Bayesian estimation using Metropolis-Hastings: 500,000 draws, first half discarded for convergence.

### Policy objectives and loss functions
- Central bank loss L_cb (minimize inflation, output variability, interest-rate volatility):
  - L_cb = 1/(Ω_π^{cb} σ_π^2 + Ω_y^{cb} σ_y^2 + Ω_{Δr}^{cb} σ_{Δr}^2) (Ω_π^{cb} + Ω_y^{cb} + Ω_{Δr}^{cb}).
- Macroprudential authority loss L_mp (abnormal credit and LTV instrument variability):
  - L_mp = 1/(Ω_z^{mp} σ_z^2 + Ω_{ΔLTV}^{mp} σ_{ΔLTV}^2) (Ω_z^{mp} + Ω_{ΔLTV}^{mp}).
- Tax authority loss L_tp (housing affordability and tax-rate variability):
  - L_tp = 1/(Ω_u^{tp} σ_u^2 + Ω_{ΔTAX}^{tp} σ_{ΔTAX}^2) (Ω_u^{tp} + Ω_{ΔTAX}^{tp}).
- Overall economy loss:
  - L = Ψ_cb L_cb + (1−Ψ_cb)( Ψ_mp L_mp + (1−Ψ_mp) L_tp ).
- Baseline loss-function weights:
  - Ω_π^{cb} = 1
  - Ω_y^{cb} = 0.5
  - Ω_{Δr}^{cb} = 0.1
  - Ω_z^{mp} = 1
  - Ω_{ΔLTV}^{mp} = 0.1
  - Ω_u^{tp} = 1
  - Ω_{ΔTAX}^{tp} = 0.1

### Policy scenarios examined
- Six optimal-policy configurations:
  - Monetary Policy: r
  - LTV Policy: r, LTV
  - Tax Policy: r, TAX
  - Tax Policy (savers): r, TAX (savers)
  - ALL Policies: r, LTV, TAX
  - ALL Policies (savers tax): r, LTV, TAX (savers)
- Three objective-weighting specifications:
  - Baseline: Ψ_cb = 0.5, Ψ_mp = 0.5
  - Debt Focus: Ψ_cb = 0.5, Ψ_mp = 0.9
  - Affordability Focus: Ψ_cb = 0.5, Ψ_mp = 0.1

### Main quantitative results
- More instruments improve outcomes:
  - Monetary policy alone yields higher losses than when augmented with LTV and/or tax policy.
  - All policies in operation produce the largest gains.
- Macroprudential policy (LTV) outperforms property-transfer tax:
  - LTV reduces overall loss more than property-transfer tax across specifications.
  - This holds whether the tax targets all homebuyers or only savers (non-collateral-constrained buyers).
- Policy ranking depends on objective focus:
  - Baseline: all instruments operating gives slightly better outcomes than when tax policy is not operating.
  - Debt-focus with all policies: a tax on savers is better than a tax on all homebuyers.
  - Affordability-focus with all policies: tax on all homebuyers is very similar to taxing savers alone.
- Instrument volatility considerations:
  - Property-tax-rate volatility is very high relative to monetary policy and LTV policy.
  - Baseline specification: asymptotic standard deviation of the property tax rate ≈ 45 percent.
  - Implication: a 99 percent confidence interval around typical dynamics would require the tax rate to swing by around 150 percentage points around its steady-state level—entailing negative tax rates in downturns and rates above 100 percent in expansions.
  - Tax-rate volatility is higher when taxes are targeted at savers alone and is particularly large when the share of savers ω is low.
  - Tax-rate volatility tends to decline as the share of savers increases, but targeting a narrower set of homebuyers always leads to greater volatility than targeting all homebuyers.

### Policy implications and recommendations (from model)
- Multiple instruments can better achieve multiple objectives (macroeconomic stability, financial stability, housing affordability).
- Between an LTV limit and a property-transfer tax, an LTV limit is preferable in this framework because it more effectively reduces overall loss.
- Property-transfer taxes can induce excessive volatility in tax rates; this problem worsens when taxes are narrowly targeted (e.g., savers or nonresidents).
- Targeting property-transfer taxes at a broader set of homebuyers is more effective and less volatile than targeting savers/nonresidents alone.
- Given nonresident homebuyers represent only a small fraction of existing housing stock in Vancouver and Toronto, taxes targeting nonresidents may have limited effectiveness.

### Limitations and avenues for future research
- Model is stylized and does not explicitly model systemic risk; systemic-risk distortions that motivate macroprudential policy are not directly included.
- Housing-affordability impacts on consumer welfare are not micro-foundationed; objectives use ad-hoc loss functions.
- Future extensions:
  - Analyze optimal policy under noncooperative policymaker behavior (authorities focusing on their own objectives).
  - Explicitly model nonresident homebuyer behavior rather than treating them as domestic savers.

*Source: wp18237 (sections II–VI, model description, estimation, and policy experiments).*

### 1. Calibrated Parameter ................................................................................................

### 1. Calibrated Parameter

### Background: housing market risks in Canada
- Housing market imbalances are prominent in Canada and are a key source of systemic risk.
- Rapidly-rising house prices are usually coupled with rising household indebtedness.
- High household debt raises the vulnerability of financial institutions to sharp corrections in house prices, and the size of exposure and interconnectedness of financial institutions make this risk systemic, not merely idiosyncratic.
- Macroprudential measures are typically used to mitigate these risks, contain the build-up of vulnerabilities over time, and enhance the resilience of the financial sector.

### Housing affordability concerns
- Rapid rise in housing prices has led to worsening housing affordability, posing a major problem to some of Canada’s most dynamic metropolitan regions.
- Canada’s overall affordability indices are not yet among the worst globally, but the Toronto and Vancouver regions are becoming severely unaffordable.
- Footnote reference: 14th Annual Demographia International Housing Affordability Survey (third quarter, 2017).

### Recent policy actions in Canada
- Federal macroprudential measures introduced over the past ten years targeted the high LTV ratio, government-backed insured mortgage market, and reducing government exposure to the housing sector (see Arvai, Krznar, and Ustyugova, 2017).
- In early 2018, the Office of the Superintendent of Financial Institutions (OSFI) tightened underwriting requirements for low-ratio mortgages to address rising risk in the uninsured mortgage market. Low-ratio mortgages are now subject to:
  - (i) a stress test for mortgage interest rates;
  - (ii) Loan-to-Value (LTV) measures and limits that reflect housing market risks, to be updated as housing markets and the economic environment evolve; and
  - (iii) restrictions on combining mortgages with other lending products (e.g. co-lending arrangements) that could circumvent LTV limits (see IMF 2018).
- Provincial property-tax measures deployed by British Columbia (BC) and Ontario to curb speculation and improve affordability:
  - A 15 percent nonresident property-transfer tax was introduced for the Toronto and Vancouver areas between 2016–17.
  - The BC government later increased the property-transfer tax on nonresident homebuyers to 20 percent and expanded its geographic coverage.

### Research question and methodological approach
- Central question: Which policy—macroprudential policy or property-tax policy—best satisfies the overall objectives of policymakers?
- The paper develops a simple Dynamic-Stochastic-General-Equilibrium (DSGE) framework to assess the effectiveness of a specific macroprudential policy—an LTV limit—against property-tax measures.
- The model is estimated using Bayesian methods and Canadian data.
- Operational objectives are specified for:
  - the central bank,
  - the macroprudential authority, and
  - the property-tax authority (assumed to be provincial governments).
- Optimal policy experiments are conducted to assess the overall performance of each policy given the specified objectives.

### Structure and scope of the analysis
- The paper proceeds as follows:
  - Section II outlines the DSGE model used in the analysis.
  - Sections II and III describe details about the estimation of the model.
  - Section IV outlines details of the policy experiments.
  - Results from the policy experiments are discussed subsequently.
- Figures listed in the source (titles preserved as provided):
  - Figure 1. Model Fit: Estimated Standard Deviations
  - Figure 2. Optimal Policy Frontiers
  - Figure 3. Instrument Volatility
  - Figure 4. Tax Volatility and the Share of Savers

*Source: wp18237 - 1. Calibrated Parameter*

### section V and section VI concludes with a discussion of the results and avenues for future

### wp18237 - section V and section VI concludes with a discussion of the results and avenues for future

### Model structure and key mechanisms
- Two household types: 휔 borrowers and (1- 휔) savers; borrowers discount the future faster (impatient), savers are patient.
- Borrowers face credit constraints tied to housing collateral; housing is in fixed supply and traded between household types.
- Two production sectors: housing (H) and non-housing consumer goods (C). Intermediate-good firms are monopolistically competitive; final consumer goods are tradable domestically and internationally.
- Three policy instruments: an interest rate, a loan-to-value (LTV) ratio, and a tax rate applied to house purchases.
- Government runs a balanced budget each period using lump-sum transfers.

### Households — preferences, constraints, and first-order conditions
- Borrowers:
  - Maximize expected utility: E0 ∑(βb)^t [1/(1+σ) (X_t^b)^(1−σ) − 1/(1+φ) (N_C,t^b)^(1+φ) − 1/(1+φ) (N_H,t^b)^(1+φ)] (equation (1)).
  - Welfare-relevant consumption index: X_t^b ≡ (Ĉ_t^b)^(1−γ ε_t) (H_t^b)^(γ ε_t), with Ĉ_t^b = C_t^b − h C_{t−1}^b.
  - Faces housing preference shock ε_t = exp(ϵ_t).
  - Budget constraint (nominal bonds only; no international borrowing): C_t^b + Q_t (1+τ) I_t^b ε_t^τ − B_t^b = −R_{t−1} B_{t−1}^b / Π_{C,t} + W_{C,t}^b / P_{C,t} + W_{H,t}^b / P_{C,t} + T_t^b (equation (2)).
  - Housing investment: I_t^b ≡ H_t^b − (1−δ) H_{t−1}^b; δ is housing depreciation rate.
  - Property-transfer tax shock: ε_t^τ ≡ exp(TAX_t).
  - Borrowing constraint: R_t B_t^b ≤ (1−χ)(1−δ) E_t[ Q_{t+1} H_t^b / Π_{C,t+1} ] ε_t^{LTV} (equation (3)); (1−χ) is steady-state LTV; ε_t^{LTV} ≡ exp(LTV_t).
  - First-order conditions:
    - Labor-leisure tradeoff: W_{j,t}^b / P_{C,t} = (X_t^b)^σ (N_{j,t}^b)^φ (Ĉ_t^b)^ε_t (1−γ ε_t) (H_t^b)^φ, j = C,H (equation (4)).
    - Marginal utility of non-housing consumption vs shadow value of housing services (equation (5)).
    - Consumption Euler adjusted for borrowing constraint: R_t ψ_t = 1 − β_b E_t[ ... ] (equation (6)); ψ_t is marginal value of borrowing (Lagrange multiplier), and ψ_t = 0 recovers standard New Keynesian Euler equation.
- Savers:
  - Maximize expected utility: E0 ∑(βs)^t [1/(1+σ) (X_t^s)^(1−σ) − 1/(1+φ) ... ] (equation (7)).
  - Budget constraint includes foreign bond holdings B_t^{s,F} and nominal exchange rate Z_t (equation (8)).
  - First-order conditions mirror borrowers' when ψ_t = 0.
  - Additional intertemporal saving condition (equation (9)): 1 = β_s E_t[ ... Z_{t+1}/Z_t R_t / Π_{C,t+1} ].

### Tradable goods sector, prices, and international linkages
- Non-housing consumption index aggregates domestic and foreign consumption: C_t ≡ [ (1−α)^{1/η} C_{D,t}^{(η−1)/η} + α^{1/η} C_{F,t}^{(η−1)/η} ]^{η/(η−1)}.
- Sub-aggregates: C_{D,t}, C_{F,t} integrate differentiated goods with elasticities ε, ζ, and η; α is degree of openness.
- Consumer price index: P_t ≡ [ (1−α) P_{C,D,t}^{1−η} + α P_{C,F,t}^{1−η} ]^{η/(η−1)} (equation (10)).
- Terms of trade S_t and real exchange rate ℛ_t defined as: S_t = Z_t P_{C,F,t}^* / P_{C,D,t} = P_{C,F,t} / P_{C,D,t} (equation (11)); ℛ_t = S_t P_t^* / P_t (equation (12)).
- International risk sharing condition for savers (complete markets): ℛ_t = (X_t^s / X_t^{s,*})^{−σ} (Ĉ_t^s ε_t / Ĉ_t^{s,*} ε_t^*)^γ (H_t^s ε_t / H_t^{s,*} ε_t^*)^γ (equation (13)).

### Firms, price setting, and inflation dynamics
- Retailers combine domestic intermediate goods with CES production; wholesale uses Cobb–Douglas: Y_{j,t}(k) = A_{j,t} N_{j,t}(k).
- Price-setting: Calvo variant with fraction (1−θ_j) adjusting and fraction ι_j forward-looking; ι_j measures backward-looking pricing.
- New Keynesian Phillips curves (log-linearized):
  - Consumption-sector inflation: (1+β_s ι_C) π̂_{C,t} = β_s E_t π̂_{C,t+1} + ι_C π̂_{C,t−1} + κ_C mĉ_{C,t−1} + ϵ_t μ_C (equation (14)).
  - Housing-sector inflation: (1+β_s ι_H) π̂_{H,t} = β_s E_t π̂_{H,t+1} + ι_H π̂_{H,t−1} + κ_H mĉ_{H,t−1} + ϵ_t μ_H (equation (15)).
- κ_j = (1−θ_j)(1−β_s θ_j)/θ_j is slope of Phillips curve; ϵ_t μ_j are sector-specific cost-push shocks.

### Equilibrium conditions and aggregates
- Government purchases fraction f_t of output in each sector financed by lump-sum taxation; define government expenditure shock g_t as log(1−f_t) = exp(−g_t).
- Linearized market clearing:
  - ŷ_{C,t} = (1−α) ĉ_t + α ĉ_t^* + α(ζ+η(1−α)) ŝ_t + g_t (equation (18)).
  - ŷ_{H,t} = î_t + g_t (equation (19)).
- Aggregates:
  - C_t = ω C_t^b + (1−ω) C_t^s; H_t = ω H_t^b + (1−ω) H_t^s.
  - N_t = ω N_t^b + (1−ω) N_t^s; B_t = ω B_t^b + (1−ω) B_t^s.
- Aggregate real output satisfies P_{D,t} Y_t = P_{C,D,t} Y_{C,t} + P_{H,t} Y_{H,t}; aggregate output price index is weighted average of domestic consumption and housing prices.

### Monetary policy specification
- Taylor-type rule (log-linearized): r̂_t = ρ_r r̂_{t−1} + (1−ρ_r)(φ_π π̂_{C,t} + φ_y Δŷ_t) + ν_t^r (equation (20)).
- Parameters: φ_π (inflation reaction), φ_y (output reaction), ρ_r (interest-rate inertia), ν_t^r monetary policy shock.

### Exogenous processes (shocks) and estimation setup
- Exogenous AR(1) processes (log-linearized) driving model dynamics:
  - a_{C,t} = ρ_{aC} a_{C,t−1} + ν_{aC,t} (21)
  - a_{H,t} = ρ_{aH} a_{H,t−1} + ν_{aH,t} (22)
  - ĉ_t^*̂ = ρ_{c*} ĉ_{t−1}^*̂ + ν_{c*,t} (23)
  - ϵ_t^μ_C = ρ_{μC} ϵ_{t−1}^μ_C + ν_{μC,t} (24)
  - ϵ_t^μ_H = ρ_{μH} ϵ_{t−1}^μ_H + ν_{μH,t} (25)
  - ϵ_t = ρ ϵ_{t−1} + ν_t (26)
  - g_t = ρ_G g_{t−1} + ν_{g,t} (27)
  - ν_t^r = ν_t^r (28)
- All ν_{t,i} ~ N(0, σ_i^2); with the exception of ν_t^r, shocks are AR(1).
- For estimation, LTV and tax policy shocks ε_t^τ and ε_t^{LTV} are set to zero (inactive over sample).

### Data and estimation approach
- Quarterly data: 1993–2017.
- Measurement variables: real GDP per capita, real consumption per capita, real residential investment per capita, employment per capita, headline CPI inflation, house price inflation, overnight policy interest rate.
- Variables expressed as log deviations from steady states; steady states via Hodrick-Prescott filter for real variables and sample averages for nominal variables.
- Estimation: Bayesian methods (An and Schofheide, 2007) with Metropolis-Hastings draws: 500,000 draws, first half discarded for convergence.
- Calibrated steady-state parameters (Table 1):
  - β_s = 0.99
  - β_b = 0.98
  - δ = 0.01
  - (1−χ) LTV steady state = 0.60
  - τ property tax steady state = 0.05
  - α degree of openness = 0.30
  - γ share of housing in utility = 0.30
- Estimated parameters (posterior means and [10%, 90%] intervals) include (selected from Table 2):
  - σ = 1.05 [0.97, 1.12]
  - ψ = 1.97 [1.81, 2.13]
  - h = 0.69 [0.67, 0.71]
  - ω = 0.61 [0.59, 0.63]
  - ζ = 1.13 [1.06, 1.21]
  - η = 1.09 [1.01, 1.17]
  - ε = 4.05 [3.89, 4.21]
  - ρ_r = 0.76 [0.72, 0.79]
  - φ_π = 2.10 [1.83, 2.36]
  - φ_y = 0.21 [0.13, 0.29]
  - θ_C = 0.71 [0.67, 0.74]; θ_H = 0.50 [0.47, 0.54]
  - ι_C = 0.80 [0.72, 0.89]; ι_H = 0.79 [0.71, 0.88]
  - Persistence and shock standard deviations reported for technology, cost-push, preference, foreign, and government shocks (see Table 2 for exact values and intervals).

### Model fit and simulation evidence
- The estimated model replicates the volatility in historical data reasonably well.
- Simulated standard deviations from the model are comparable to those from historical data estimated via a VAR (4 lags) simulated 1000 times with bootstrapping.
- The model fit figure compares standard deviations for Output, Consumption, Residential Investment, Consumer Price Inflation, House Price Inflation, and Interest Rate; uncertainty bands reflect parameter and shock uncertainty.
- The policy experiments conducted in the analysis use policymakers setting instruments to minimize variances of key variables; model performance in matching historical volatility supports this exercise.

### Policy objectives and loss functions
- Central bank loss (minimize inflation and output variability and interest-rate volatility):
  - L_cb = 1/(Ω_π^{cb} σ_π^2 + Ω_y^{cb} σ_y^2 + Ω_{Δr}^{cb} σ_{Δr}^2) (Ω_π^{cb} + Ω_y^{cb} + Ω_{Δr}^{cb}) (equation (29)).
  - Weights Ω_π^{cb}, Ω_y^{cb}, Ω_{Δr}^{cb} ≥ 0; Ω_{Δr}^{cb} included to avoid excessive policy-rate volatility.
- Macroprudential authority loss (reacts to "abnormal" credit measured by debt-to-GDP ratio z and variability of LTV instrument):
  - L_mp = 1/(Ω_z^{mp} σ_z^2 + Ω_{ΔLTV}^{mp} σ_{ΔLTV}^2) (Ω_z^{mp} + Ω_{ΔLTV}^{mp}) (equation (30)).
  - Ω_z^{mp}, Ω_{ΔLTV}^{mp} ≥ 0; symmetric loss implies dislike of both high and low leverage.
- Tax authority loss (housing affordability measured by house-price-to-income ratio u and variability of property-transfer tax TAX):
  - L_tp = 1/(Ω_u^{tp} σ_u^2 + Ω_{ΔTAX}^{tp} σ_{ΔTAX}^2) (Ω_u^{tp} + Ω_{ΔTAX}^{tp}) (equation (31)).
  - Ω_u^{tp}, Ω_{ΔTAX}^{tp} ≥ 0; TAX can apply to all home purchases or only to savers (non-collateral constrained buyers).
- Nonresident homebuyers are modeled as part of savers (not collateral-constrained), capturing cash purchases and limited access to mortgages.
- Overall economy loss is weighted average of authorities:
  - L = Ψ_cb L_cb + (1−Ψ_cb)( Ψ_mp L_mp + (1−Ψ_mp) L_tp ) (equation (32)), with weights Ψ_cb, Ψ_mp between 0 and 1.

*Source: Staff estimates and model description from the supplied content of wp18237 (sections II–IV, data, estimation, and policy objectives).*

### 1. Thus, the total loss of the economy depends on the weights associated with the objectives

### wp18237 - 1. Thus, the total loss of the economy depends on the weights associated with the objectives

### Optimal policy: framework and assumptions
- Authorities select policy rules that minimize the intertemporal version of the total loss function (4) using optimal policy rules (Woodford, 2003).
- Optimal policy rules determine the paths of:
  - the policy interest rate (r),
  - the LTV ratio (LTV),
  - the property tax rate (TAX).
- In the optimal-rule case, the three policy instruments are functions of all model variables (and Lagrange multipliers).
- Assumptions for the baseline analysis:
  - Weights in individual loss functions are fixed at standard values from the monetary policy and macroprudential literature.
  - The standard deviation of monetary policy shocks is set to zero.
  - All other parameters are set to their estimated values.
- Baseline weights in the loss functions (equations 30, 31, and 32) are:
  - Ω휋푐푏 = 1
  - Ω푦푐푏 = 0.5
  - ΩΔ푟푐푏 = 0.1
  - Ω푧푚푝 = 1
  - ΩΔ퐿푇푉푚푝 = 0.1
  - Ω푢푡푝 = 1
  - ΩΔ푇퐴푋푡푝 = 0.1

### Policy scenarios and weightings
- Six different optimal-policy configurations are examined depending on which policy instruments operate:
  - Monetary Policy: r
  - LTV Policy: r, LTV
  - Tax Policy: r, TAX
  - Tax Policy (savers): r, TAX (savers)
  - ALL Policies: r, LTV, TAX
  - ALL Policies (savers tax): r, LTV, TAX (savers)
- When an instrument is not in operation (LTV or TAX), the policymaker responsible for that instrument is “passive” and relies on other authorities to adjust their instruments to minimize total loss including objectives of all policymakers.
- Three specifications for how policymakers’ objectives are weighted in overall loss:
  - Baseline: Ψ푐푏 = 0.5, Ψ푚푝 = 0.5
  - Debt Focus: Ψ푐푏 = 0.5, Ψ푚푝 = 0.9
  - Affordability Focus: Ψ푐푏 = 0.5, Ψ푚푝 = 0.1

### Main results and comparative findings
- Having more policy instruments leads to better outcomes:
  - Monetary policy alone leads to higher losses than when monetary policy is augmented with LTV and/or tax policy.
  - Having all policies in operation leads to even greater gains.
- Macroprudential policy (LTV) is better than tax policy:
  - LTV policy reduces overall loss by more than tax policy.
  - This holds regardless of whether the tax is applied to all homebuyers or only to savers that are not subject to collateral constraints.
- The ‘best’ policy depends on the focus of objectives:
  - Baseline specification: applying all policy instruments leads to slightly better outcomes than when tax policy is not operating.
  - Debt-focus specification with all policies operating: a tax on savers is better than a tax on all homebuyers.
  - Affordability-focus specification with all policies operating: a tax on all homebuyers is very similar to imposing a tax on savers alone.
  - Overall: the ‘best’ policy from a loss-minimization perspective depends on the focus of policymakers’ objectives.
- Instrument volatility should be considered when evaluating policy viability over the business cycle.

### Instrument volatility and tax-rate dynamics
- Tax rate volatility is very high relative to monetary policy and/or LTV policy:
  - Baseline specification: the asymptotic standard deviation of the property tax rate is about 45 percent.
  - Implication: a 99 percent confidence interval around typical dynamics would require the tax rate to swing by around 150 percentage points around its steady state level—requiring negative tax rates in downturns and rates above 100 percent in expansions.
- Tax rate volatility is higher when taxes are targeted at savers alone, irrespective of policymakers’ objective-focus.
- Volatility of property-tax rates as the share of savers increases:
  - Property-tax policy targeted at savers alone leads to greater volatility than a tax that targets all homebuyers.
  - Volatility is particularly high when taxes are targeted at savers and the share of savers in the economy is low.
  - Tax-rate volatility tends to decline when the share of savers increases, but targeting a narrower set of homebuyers always leads to greater volatility than targeting all homebuyers.
- Figures referenced (instrument volatility, tax volatility by share of savers) support these quantitative volatility conclusions (source: Staff estimates).

### Summary and policy implications
- Multiple policy instruments can yield better outcomes when policymakers face multiple objectives (macroeconomic stability, financial stability, housing affordability).
- Between an LTV limit and a property-transfer tax, policymakers should choose an LTV limit because it can be more effective in achieving desired objectives in this framework.
- Deploying multiple policy instruments can lead to excessive volatility in property-tax rates; this problem is exacerbated when taxes are targeted at a narrower base of homebuyers.
- Targeting property-transfer taxes at a broader set of homebuyers is more effective than targeting savers (or nonresident homebuyers) alone:
  - Targeting savers alone requires greater swings in tax rates to achieve desired objectives.
  - If nonresident homebuyers are largely not collateral constrained, taxing all homebuyers (residents and nonresidents) would likely be more effective than targeting nonresidents alone.
- Empirical note: recent data show nonresident homeowners represent only a small fraction of the existing housing stock in Vancouver and Toronto, which could limit the effectiveness of property transfer taxes on nonresidents.

### Limitations and avenues for future research
- Model limitations:
  - The model is stylized and simple; it does not explicitly include systemic risk—the key distortion macroprudential policy should address—reflecting difficulties in rigorously modeling systemic risk.
  - The impact of housing affordability on consumer welfare is not modeled; analysis uses ad-hoc objective functions rather than micro-founded, welfare-based measures.
- Assumptions to relax in future work:
  - Policymaker cooperation: current experiments assume cooperation to achieve overall objectives; future work could assess optimal policy when policymakers do not cooperate and focus only on their own objectives.
  - Nonresident behavior: future models could explicitly model nonresident homebuyer behavior instead of assuming they behave like domestic savers.

*Source: wp18237 - 1. Thus, the total loss of the economy depends on the weights associated with the objectives*

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_Source: https://www.imf.org/-/media/files/publications/wp/2018/wp18237.pdf_