## _wp16138appendix

## Source details

**Canonical URL:** [_wp16138appendix](https://www.imf.org/-/media/websites/imf/imported-publications/external/pubs/ft/wp/2016/_wp16138appendix.pdf)

## Other formats

- [Markdown version](/-/media/websites/imf/imported-publications/external/pubs/ft/wp/2016/_wp16138appendix.pdf.md)
- [Structured JSON version](/-/media/websites/imf/imported-publications/external/pubs/ft/wp/2016/_wp16138appendix.pdf.json)

---

### A.1 The SOE-DEFK Model — Households, Entrepreneurs, Workers, Firms, Final Goods, Government, Aggregation
- Households (A.1.1)
  - Continuum of identical households; each household contains members j ∈ [0,1]; entrepreneurs j ∈ [0,ζ) and workers j ∈ [ζ,1].
  - Intra-household asset split equal across members at beginning of period; no reallocation after role realization → liquidity constraints.
  - Utility: Et Σ_{s=t}^∞ β^{s−t} [ C_{s}^{1−γ}/(1−γ) + (1/(1+φ)) ∫_0^1 H_s(j)^{1+φ} dj ] (equation (A.1)); parameters: β, γ, φ.
  - Instruments and returns:
    - New capital price p^I_t ≡ P^I_t / P^C_t; rental r^k_t; depreciation δ; market value q_t. Return on new capital: r^k_{t+1} + (1−δ) q_{t+1} / p^I_t.
    - Entrepreneurs issue equity N^I_t; private equity purchase N^O_t at price q_t; return on equity: [r^k_{t+1} + (1−δ) q_{t+1}] / q_t.
    - Net equity N_t ≡ N^O_t + K_t − N^I_t.
    - Domestic government bond L_t (gross nominal rate R_t); foreign bond L^*_t (gross nominal rate R^*_t) with actual domestic yield R^*_t (1 − ζ(·)).
  - Risk premium: ζ( s_t L^*_t / Y_t ) = ξ [ exp( s_t L^*_t / Y_t ) − 1 ] with s_t ≡ S_t P^*_C_t / P^C_t.
  - Timing: households receive dividends D_t and D^K_t and lump-sum taxes τ_t; intertemporal budget constraint given by (A.2).
  - Inflation measures: Π^C_t ≡ P^C_t / P^C_{t−1}; Π^{C*}_t ≡ P^{C*}_t / P^{C*}_{t−1}.

- Entrepreneurs (A.1.2)
  - Investment: q_t > p^I_t in steady state and post-shock equilibria → return on new capital exceeds return on equity and real return on government bonds.
  - Liquidity constraints:
    - No short positions on government bonds: L_t(j) ≥ 0 and L^*_t(j) ≥ 0 for j ∈ [0,ζ).
    - Borrowing constraint: can issue equity up to fraction θ ∈ (0,1) of new investment.
    - Resaleability constraint: can sell at most ω_t ∈ (0,1) fraction of net equity holdings each period.
  - Net equity law of motion per entrepreneur: N_t(j) = (1−ω_t)(1−δ) N_{t−1}(j) + (1−θ) I_t(j).
  - Entrepreneurs set C_t(j) = 0 for j ∈ [0,ζ).
  - Aggregate investment (A.3):
    - I_t = { [ r^k_t + (1−δ) q_t ] ω_t N_{t−1} + R_{t−1} Π^C_t L_{t−1} + R^*_{t−1} [1−ζ(·)] Π^{C*}_t s_t L^*_{t−1} + D_t + D^K_t − τ_t } / p^I_t − θ q_t
    - Emphasis: aggregate investment depends on abundance of households' liquidity; contrasts with standard DSGE without liquidity frictions.

- Workers and Wage-setting (A.1.3)
  - Workers' FOCs derived from household problem after entrepreneurs' decisions.
  - Euler equations (A.4)–(A.6) for domestic bonds, private equity, and foreign bonds:
    - Domestic bonds (A.4): C_t^{−γ} = β E_t [ C_{t+1}^{−γ} ( R_t / Π^C_{t+1} + { (q_{t+1} − p^I_{t+1}) / (p^I_{t+1} − θ q_{t+1}) } R_t / Π^C_{t+1} ) ]
    - Private equity (A.5): C_t^{−γ} = β E_t [ C_{t+1}^{−γ} ( [r^k_{t+1} + (1−δ) q_{t+1}]/q_t + { (q_{t+1} − p^I_{t+1}) / (p^I_{t+1} − θ q_{t+1}) } [r^k_{t+1} + (1−δ) q_{t+1}] ω_{t+1} / q_t ) ]
    - Foreign bonds (A.6): C_t^{−γ} = β E_t [ C_{t+1}^{−γ} ( R^*_t [1−ζ(·)] s_{t+1} / Π^{C*}_{t+1} s_t + { (q_{t+1} − p^I_{t+1}) / (p^I_{t+1} − θ q_{t+1}) } R^*_t [1−ζ(·)] s_{t+1} / Π^{C*}_{t+1} s_t ) ]
  - Liquidity premium: the {·} terms capture liquidity value for holders of assets when future investment opportunities are scarce (non-zero κ ≡ {·}).
  - Uncovered interest parity (A.7): R_t / E_t(Π^C_{t+1}) = E_t [ R^*_t [1−ζ(·)] Π^{C*}_{t+1} s_{t+1} / s_t ].
  - New Keynesian wage setting with differentiated workers, labour aggregator (A.8) and demand H_t(j) (A.9). Staggered wage resetting prob. (1−ξ_w); optimal wage condition (A.10); aggregate real wage dynamics (A.11).

- Intermediate-goods firms (A.1.4)
  - Monopolistic competition; production Y_t(i) = A_t K_t(i)^α H_t(i)^{1−α}.
  - Real marginal cost (A.12): mc_t = mc_t(i) = 1/A_t [ w_t^{1−α} ( r^k_t )^{α} ]^{1−α}.
  - Calvo price stickiness: prob. of not resetting price 1−ξ_p; separate home and export price-setting problems (A.13) and (A.14).
  - Demand functions: Y^H_t(i) = [ P^H_t(i) / P^H_t ]^{−(1+η_f)/η_f} Y^H_t and X_t(i) = [ P^{H*}_t(i) / P^{H*}_t ]^{−(1+η_f)/η_f} X_t.
  - Optimal price-setting conditions (A.15) and (A.16); price-index evolutions (A.17) and (A.18).

- Final-goods firms (A.1.5)
  - Private consumption aggregator (A.19): C_t = [ (1−ω)^{1/ζ} C^H_t^{(ζ−1)/ζ} + ω^{1/ζ} C^F_t^{(ζ−1)/ζ} ]^{ζ/(ζ−1)} with ω inverse home bias, ζ elasticity of substitution.
  - CES allocations and price index P^C_t (A.20)–(A.22).
  - Investment goods: I_t [1 + S(·)] CES between I^H_t and I^F_t; adjustment cost S( I_t / I ) = (κ/2) [ I_t / I − 1 ]^2 with κ > 0; capital-goods FOC p^I_t = 1 + S(·) + S'(·) I_t / I (A.24).
  - Public consumption uses only home-produced goods; aggregate domestic demand Y^H_t(i) = C^H_t(i) + I^H_t(i) + G_t(i).

- Government policies (A.1.6)
  - Government budget constraint (A.29): G_t + R_{t−1} Π^C_t L_{t−1} = τ_t + L_t.
  - Fiscal rule (A.30): τ_t − τ = ρ_τ [ (R_{t−1} Π^C_t L_{t−1} − R Π^C L) ] with ρ_τ > 0; ρ_τ low → fiscal expansion financed mainly by bond issuance.
  - Monetary policy Taylor rule (A.31): R_t = max( R · (Π^C_t)^{ϕ_π} ( Y_t / Y )^{ϕ_y} ; 1 ) with ϕ_π > 1, ϕ_y ∈ (0,1), zero lower bound R_t ≥ 1.

- Aggregation and resource constraints (A.1.7)
  - Labour and capital market clearing: H_t = ∫_0^1 H_t(i) di; K_{t−1} = ∫_0^1 K_{t−1}(i) di.
  - Optimal K/H ratio (A.32): K_{t−1} / H_t = (1−α) w_t / r^k_t.
  - Aggregate production (A.33) and ownership K_t = N_t (A.34).
  - Capital accumulation (A.35): K_t = (1−δ) K_{t−1} + I_t.
  - Resource constraint (A.36): Y_t = C^H_t + I^H_t + G_t + X_t with related expressions (A.37)–(A.39).
  - Consolidated aggregate investment expression (A.40):
    - I_t = { [ r^k_t + (1−δ) q_t ] ω_t N_{t−1} + r_{t−1} L_{t−1} + r^*_{t−1} [1−ζ(·)] s_t L^*_{t−1} + p^H_t Y^H_t + s_t p^{H*}_t X_t − w_t H_t − r^k_t K_{t−1} + p^I_t I_t − [1 + S(·)] I_t − τ_t } / p^I_t − θ q_t
    - Real interest rates: r_t ≡ R_t / E_t(Π^C_{t+1}); r^*_t ≡ R^*_t / E_t(Π^{C*}_{t+1}).

### Rest of the World (Section 2)
- Modeled as one large economy with continuum of monopolistically competitive firms i ∈ [0;1]; price-resetting Calvo probability 1−θp.
- Export price-setting problem (A.43): Et ∑_{s=t} β^{s−t} (C̃'_s/C̃'_t) [ eP_Ft(i) S_s − MC'_s ] M_s(i); world nominal marginal cost MC'_t and world variables Y'_t, R'_t, π'_t, MC'_t taken as exogenous by SOE.
- Small open economy exports (A.44): X_t = ψ (P_H'_t / P_C'_t)^{−η} Y'_t — exports increase with world aggregate output Y'_t.

### Steady State (Section 2)
- Steady-state assumptions:
  - No inflation: π_C = π_C' = 1.
  - R = r and R' = r' with uncovered interest parity implying r = r'.
  - Normalisations: p_H = p_H' = p_F = 1 and s = 1.
  - Home foreign bond holdings L' = 0 ⇒ sL'/Y = 0.
  - C_H = (1−ω) C and C_F = ω C; capital adjustment cost S(1) = S'(1) = 0 → I_H = (1−ω) I and I_F = ω I.
  - Zero net exports at steady state: X − M = 0 ⇒ Y = C + I + G (A.45).
- Key steady-state conditions:
  - p_I = 1 (A.24).
  - Euler reductions:
    - 1/β = r [ 1 + φ (q−1)/(1−θq) ]  (A.46)
    - 1/β = r_k + (1−δ) q [ 1 + φ (q−1)/(1−θq) ] − φ (q−1) (1−δ) (1−ϵ)/(1−θq)  (A.47)
  - Real wage markup relation (A.48) and optimal K/H (A.49): K_H = (1−α) w / r_k.
  - Marginal cost and price-setting (A.50)–(A.51): mc = 1/(1+μ_f) = 1/A [ w^{1−α} r_k^{α} ]; Y/K = (1+μ_f) r_k.
  - Government budget: τ = (r−1) L + G (A.52).
  - Aggregate investment at steady state (A.53):
    - I = { [ r_k + (1−δ) q ϵ ] K + r L + Y − w H − r_k K − τ } / (1−θq)
  - Steady-state identity (A.54) linking δ, θ_q, φ, q, μ_f, L/Y, G/K and r_k.

### The Log-linear Model (selected linearised equations)
- Log-deviation notation: x̂_t ≡ ln(x_t / x).
- Representative linearised equations (equation references preserved):
  - Euler for domestic bonds (A.55): −χ Ĉ_t = −χ E_t[Ĉ_{t+1}] + b̂R_t − E_t[π̂_{t+1}] + φ_r {1−θ_q}/(1−θ_q)^2 q E_t[ q̂_{t+1} ] − φ_r {1−θ_q}/(1−θ_q)^2 q E_t[ p̂_{I,t+1} ].
  - Euler for equity (A.56): q̂_t − χ Ĉ_t = −χ E_t[Ĉ_{t+1}] + (φ/r_k) q [ 1 + φ (q−1)/(1−θ_q) ] E_t[ r̂_{k,t+1} ] + ... (full expression A.56).
  - Uncovered interest parity (A.57): b̂R_t − E_t[π̂_{t+1}] = b̂R'_t − E_t[π̂'_{t+1}] + E_t[ŝ_{t+1}] − ŝ_t − ψ b̂F_t.
  - Wage-setting (A.58) and real-wage evolution (A.59).
  - Marginal cost (A.60): ĉmc_t = (1−α) b̂w_t + α b̂r_{k,t} − b̂A_t.
  - Phillips curves for home goods, exports, imports (A.61)–(A.63).
  - Demand and price index conditions (A.64)–(A.66).
  - Investment and capital relations including p̂_I,t = λ_I î_t (A.67) and capital law of motion (A.76).
  - Government budget and fiscal rule linearised (A.70)–(A.71).
  - Monetary policy (A.72): b̂R_t = max[ ρ ( π̄ b̂π_t + γ_Y b̂Y_t ); −log(R̄) ].
  - Net foreign assets evolution (A.81) and aggregate investment linear form (A.82).
  - World economy linearised equations (A.86)–(A.91).

### Calibration (parameters and steady-state values)
- Structural parameters (Table 2):
  - β 0.989
  - ϵ 1.1515
  - δ 0.025
  - α 0.36
  - λ 2.4
  - ϕ 1.9697
  - ω 0.3368
  - η 1
  - μ_f 0.11
  - κ_w 0.11
  - θ_p 0.7191
  - θ_w 0.7923
- Liquidity-constraint parameters:
  - φ 0.05
  - θ_q 0.19
  - ϵ 0.19
  - Notes: Entrepreneurs can sell up to 57% (= 1−(1−0.19)^4) of equity in a year. Investment opportunity φ = 0.05 per quarter → 19% (= 1−(1−0.05)^4) per year.
- Policy parameters:
  - ρ_π 1.5
  - γ_Y 0.125
  - ρ_G 0.80
  - τ̄ 0.1
- Calibration choices and sources:
  - λ = 2.4 (Groth (2008)); α = 0.36; δ = 0.025.
  - DSGE-standard parameters from Cogley et al. (2011): ϵ = 1.1515; ϕ = 1.9697; θ_p = 0.7191; θ_w = 0.7923; markups 0.11.
  - Openness: η = 1; ω = 0.3368 (posterior mode from Cogley et al. (2011)).
  - Monetary policy: ρ_π = 1.5; γ_Y = 0.125 (quarterly).
  - Fiscal rule τ̄ = 0.1; ρ_G = 0.8.

- Selected steady-state values of endogenous variables (Table 3):
  - Consumption-to-GDP ratio C/Y 0.58
  - Investment-to-GDP ratio I/Y 0.22
  - Government spending share G/Y 0.20
  - Import-to-GDP ratio M/Y 0.27
  - Export-to-GDP ratio X/Y 0.27
  - Quarterly GDP Y 3.06
  - Quarterly labour H 0.90
  - Capital stock K 26.79
  - Public debt-to-GDP ratio L=4Y 0.40
  - Tax-to-GDP ratio τ/Y 0.21
  - Aggregate real wage w 1.95
  - Capital rent r_k 3.70%
  - Investment cost of new capital p_I 1
  - Market price of equity q 1.04
  - Real marginal cost mc 0.90
  - Real interest rate (annualised) 4 (R−1) 3.43%
- Additional steady-state calibrations:
  - Public debt-to-GDP ratio L=4Y = 40% to match UK data (42% on average) for 1992-2007.
  - Government consumption share G/Y set to 20% (UK average 1992-2007).
  - Steady-state inflation π = 1.

*Source: Appendix Section 1, "The Fiscal Multiplier in a Small Open Economy: The Role of Liquidity Frictions" (A.1 The SOE-DEFK Model) — Online Appendix by Jasmin Sin.*

### Section 1

### The Fiscal Multiplier in a Small Open Economy: The Role of Liquidity Frictions — Appendix Section 1 (A.1 The SOE-DEFK Model)

### Households (A.1.1)
- Economy: continuum of identical households; each household consists of a continuum of members j ∈ [0,1].
- Role differentiation: members receive i.i.d. opportunity to invest in new capital; those who receive it (j ∈ [0,ζ)) are "entrepreneurs", others (j ∈ [ζ,1]) are "workers".
- Intra-household asset allocation: household assets divided equally among members at beginning of each period; no reallocation after realization of roles → liquidity constraints if members need extra funds.
- Household utility (aggregate consumption Ct used jointly):
  - Utility: Et Σ_{s=t}^∞ β^{s−t} [ C_{s}^{1−γ}/(1−γ) + (1/(1+φ)) ∫_0^1 H_s(j)^{1+φ} dj ] (equation (A.1))
  - Parameters: β = discount factor; γ = coefficient of relative risk aversion; φ = inverse Frisch elasticity of labour supply.
- Choices each period: non-durable consumption, saving in bonds or equity, and (for entrepreneurs) investment in capital.
- Asset/investment instruments:
  - New capital cost p^I_t ≡ P^I_t / P^C_t per unit; rental r^k_t; depreciation δ; market value q_t. Return on new capital: r^k_{t+1} + (1−δ) q_{t+1} / p^I_t.
  - Entrepreneurs borrow by issuing equity N^I_t.
  - Private equity purchase N^O_t at market price q_t; return on equity: [r^k_{t+1} + (1−δ) q_{t+1}] / q_t.
  - Net equity: N_t ≡ N^O_t + K_t − N^I_t.
  - Domestic government bond L_t (gross nominal rate R_t) unit face value.
  - International government bond L^*_t (gross nominal rate R^*_t) denominated in foreign currency; actual nominal yield in home country R^*_t (1 − ζ(·)) where ζ(·) is a risk premium function.
- Risk premium: ζ(·) depends on net foreign asset position via ζ( s_t L^*_t / Y_t ) = ξ [ exp( s_t L^*_t / Y_t ) − 1 ], where ξ is risk premium parameter; s_t ≡ S_t P^*_C_t / P^C_t is real exchange rate; S_t nominal exchange rate.
- Timing and dividends: households receive dividends D_t and D^K_t; lump-sum taxes τ_t. Intertemporal budget constraint given by equation (A.2) (preserves full algebraic form).
- Inflation definitions: Π^C_t ≡ P^C_t / P^C_{t−1}; Π^{C*}_t ≡ P^{C*}_t / P^{C*}_{t−1}.

### Entrepreneurs (A.1.2)
- Investment incentives: q_t > p^I_t in steady state and post-shock equilibria → return on new capital exceeds return on equity and real return on government bonds.
- Liquidity constraints:
  - Entrepreneurs cannot take short positions on government bonds: L_t(j) ≥ 0 and L^*_t(j) ≥ 0 for j ∈ [0,ζ).
  - Borrowing constraint: can issue equity up to θ ∈ (0,1) fraction of new investment.
  - Resaleability constraint: can sell at most ω_t ∈ (0,1) fraction of net equity holdings each period.
- Net equity law of motion (per entrepreneur): N_t(j) = (1−ω_t)(1−δ) N_{t−1}(j) + (1−θ) I_t(j) (text statement).
- Consumption: entrepreneurs set C_t(j) = 0 for j ∈ [0,ζ).
- Aggregate investment expression (A.3):
  - I_t = { [ r^k_t + (1−δ) q_t ] ω_t N_{t−1} + R_{t−1} Π^C_t L_{t−1} + R^*_{t−1} [1−ζ(·)] Π^{C*}_t s_t L^*_{t−1} + D_t + D^K_t − τ_t } / p^I_t − θ q_t
  - Emphasis: aggregate investment depends on the abundance of households' liquidity; contrasts with standard DSGE without liquidity frictions.

### Workers (A.1.3)
- Workers' problem derived from household as whole after entrepreneurs' decision.
- Euler equations (first-order conditions):
  - Domestic bonds (A.4): C_t^{−γ} = β E_t [ C_{t+1}^{−γ} ( R_t / Π^C_{t+1} + { (q_{t+1} − p^I_{t+1}) / (p^I_{t+1} − θ q_{t+1}) } R_t / Π^C_{t+1} ) ]
  - Private equity (A.5): C_t^{−γ} = β E_t [ C_{t+1}^{−γ} ( [r^k_{t+1} + (1−δ) q_{t+1}]/q_t + { (q_{t+1} − p^I_{t+1}) / (p^I_{t+1} − θ q_{t+1}) } [r^k_{t+1} + (1−δ) q_{t+1}] ω_{t+1} / q_t ) ]
  - Foreign bonds (A.6): C_t^{−γ} = β E_t [ C_{t+1}^{−γ} ( R^*_t [1−ζ(·)] s_{t+1} / Π^{C*}_{t+1} s_t + { (q_{t+1} − p^I_{t+1}) / (p^I_{t+1} − θ q_{t+1}) } R^*_t [1−ζ(·)] s_{t+1} / Π^{C*}_{t+1} s_t ) ]
- Liquidity premium interpretation: holders of bonds or equity obtain extra liquidity units for future investment opportunities; {·} terms capture liquidity value when investment opportunities are scarce (non-zero κ ≡ {·}).
- Uncovered interest parity (A.7) follows from (A.4) and (A.6):
  - R_t / E_t(Π^C_{t+1}) = E_t [ R^*_t [1−ζ(·)] Π^{C*}_{t+1} s_{t+1} / s_t ].
- Wage-setting: New Keynesian wage setting with differentiated workers supplied via employment agencies.
  - Labour aggregator (A.8) with parameter υ = 1/(1−ι) modification as in DEFK.
  - Demand for type-j labour (A.9): H_t(j) = [ W_t(j) / W_t ]^{−(1+η_w)/η_w} H_t, with η_w > 0; W_t aggregate nominal wage index defined in text.
  - Staggered wage reset probability: (1−ξ_w) reset each period; optimal wage-setting Euler-like condition (A.10).
  - Aggregate real wage dynamics (A.11): w_t^{−1/η_w} = (1−ξ_w) e w_t^{−1/η_w} + ξ_w [ w_{t−1} / Π^C_t ]^{−1/η_w}.

### Intermediate-Goods Firms (A.1.4)
- Continuum of monopolistically competitive firms i ∈ [0,1].
- Production: Y_t(i) = A_t K_t(i)^α H_t(i)^{1−α}, where α ∈ [0,1] capital share.
- Real marginal cost (A.12): mc_t = mc_t(i) = 1/A_t [ w_t^{1−α} ( r^k_t )^{α} ]^{1−α} (preserves algebraic relation as in text).
- Calvo price stickiness: probability of not resetting price is 1−ξ_p; firms charge different prices at home and abroad.
- Domestic price-resetting problem (A.13): firms choose eP^H_t(i) to maximize discounted expected profits with markup and marginal cost; analogous foreign-market problem (A.14) with export price eP^{H*}_t(i) and exports X_t(i).
- Demand functions:
  - Domestic demand: Y^H_t(i) = [ P^H_t(i) / P^H_t ]^{−(1+η_f)/η_f} Y^H_t.
  - Export demand: X_t(i) = [ P^{H*}_t(i) / P^{H*}_t ]^{−(1+η_f)/η_f} X_t.
- Optimal price-setting conditions (A.15) and (A.16) for home and foreign markets (preserve full functional forms).
- Price index evolution:
  - Domestic (A.17): P^H_t evolution as function of reset price and past index.
  - Export (A.18): P^{H*}_t evolution similarly.

### Final-Goods Firms (A.1.5)
- Final goods: private consumption, investment goods, public consumption.
- Private consumption aggregator (A.19):
  - C_t = [ (1−ω)^{1/ζ} C^H_t^{(ζ−1)/ζ} + ω^{1/ζ} C^F_t^{(ζ−1)/ζ} ]^{ζ/(ζ−1)}.
  - C^H_t and C^F_t are CES aggregators of home-produced and imported intermediate goods.
  - ω = inverse of home bias (trade openness index), ζ > 0 elasticity of substitution between home and foreign goods.
- Demands for intermediate goods by private consumption firms (A.20).
- Optimal allocations between home and foreign goods (A.21) with aggregate price index P^C_t given by (A.22).
- Investment goods production (A.23):
  - I_t [1 + S(·)] = CES aggregator between I^H_t and I^F_t; adjustment cost S(·) quadratic in I_t/I (steady state I).
  - S( I_t / I ) = (κ/2) [ I_t / I − 1 ]^2 with κ > 0; S(1)=S'(1)=0 and S''(1)>0.
- Capital-goods firms' first-order condition (A.24): p^I_t = 1 + S(·) + S'(·) I_t / I.
- Demands for intermediate goods in investment production (A.25) and optimal allocations (A.26)–(A.27).
- Public consumption goods: government consumes only home-produced goods; demand for intermediate good i for public consumption (A.28).
- Aggregate home market demand for intermediate good i: Y^H_t(i) = C^H_t(i) + I^H_t(i) + G_t(i).
- Imports demand: M_t(i*) = C^F_t(i*) + I^F_t(i*).

### Government Policies (A.1.6)
- Government budget constraint (A.29): G_t + R_{t−1} Π^C_t L_{t−1} = τ_t + L_t.
- Fiscal rule for taxes (A.30): τ_t − τ = ρ_τ [ (R_{t−1} Π^C_t L_{t−1} − R Π^C L) ] with ρ_τ > 0 and steady-state values without time subscript.
  - Interpretation: ρ_τ low to reflect slow tax adjustment relative to bond issuance; fiscal expansion financed mainly by bond issuance.
- Monetary policy: Taylor rule (A.31):
  - R_t = max( R · (Π^C_t)^{ϕ_π} ( Y_t / Y )^{ϕ_y} ; 1 )
  - Parameters: ϕ_π > 1; ϕ_y ∈ (0,1). Zero lower bound binding at R_t ≥ 1.

### Aggregation and Resource Constraint (A.1.7)
- Market clearing for labour and capital: H_t = ∫_0^1 H_t(i) di; K_{t−1} = ∫_0^1 K_{t−1}(i) di.
- Optimal capital-labour ratio (A.32): K_{t−1} / H_t = (1−α) w_t / r^k_t.
- Aggregate production (A.33): A_t K_{t−1}^α H_t^{1−α} = ∫_0^1 Y_t(i) di.
- Ownership: all capital owned by households via private equity (A.34): K_t = N_t.
- Capital accumulation (A.35): K_t = (1−δ) K_{t−1} + I_t.
- Resource constraint (A.36): Y_t = C^H_t + I^H_t + G_t + X_t.
  - Aggregate domestic demand for home-produced goods (A.37): Y^H_t = C^H_t + I^H_t + G_t.
  - Aggregate imports (A.38): M_t = C^F_t + I^F_t.
- Net foreign assets evolution (A.39): s_t L^*_t = R^*_{t−1} [1−ζ(·)] Π^{C*}_t s_t L^*_{t−1} + s_t p^{H*}_t X_t − p^F_t M_t.
- Dividend distribution: profits distributed to households; substituting D_t and D^K_t into investment expression yields consolidated I_t expression (A.40) (preserves full algebraic form):
  - I_t = { [ r^k_t + (1−δ) q_t ] ω_t N_{t−1} + r_{t−1} L_{t−1} + r^*_{t−1} [1−ζ(·)] s_t L^*_{t−1} + p^H_t Y^H_t + s_t p^{H*}_t X_t − w_t H_t − r^k_t K_{t−1} + p^I_t I_t − [1 + S(·)] I_t − τ_t } / p^I_t − θ q_t
  - Real interest rates: r_t ≡ R_t / E_t(Π^C_{t+1}); r^*_t ≡ R^*_t / E_t(Π^{C*}_{t+1}).

*Source: Appendix Section 1, "The Fiscal Multiplier in a Small Open Economy: The Role of Liquidity Frictions" (A.1 The SOE-DEFK Model) — Online Appendix by Jasmin Sin.*

### Section 2

### _wp16138appendix - Section 2

### Rest of the World
- The rest of the world is modelled as one large economy populated with a continuum of monopolistically competitive firms i ∈ [0;1]. Firms reset prices each period with Calvo probability 1−θp.
- Firms resetting export prices choose the nominal price eP_Ft(i) in domestic currency to maximize expected discounted profits:
  - Et ∑_{s=t} β^{s−t} (C̃'_s/C̃'_t) [ eP_Ft(i) S_s − MC'_s ] M_s(i)  (A.43)
  - MC'_t is the world nominal marginal cost; world variables Y'_t, R'_t, π'_t and MC'_t are taken as exogenous by the small open economy.
- Small open economy exports X_t increase with world aggregate output Y'_t:
  - X_t = ψ (P_H'_t / P_C'_t)^{−η} Y'_t  (A.44)

### Steady State
- Assumptions at steady state:
  - No inflation: π_C = π_C' = 1
  - R = r and R' = r' with uncovered interest parity implying r = r'
  - Normalisations: p_H = p_H' = p_F = 1 and s = 1
  - Home country foreign bond holdings L' = 0 so sL'/Y = 0
  - C_H = (1−ω) C and C_F = ω C
  - Capital adjustment cost S(1) = S'(1) = 0 → I_H = (1−ω) I and I_F = ω I
  - Zero net exports at steady state: X − M = 0 → Resource constraint: Y = C + I + G  (A.45)
- First-order and steady-state conditions:
  - p_I = 1 (from capital-goods producers FOC A.24)
  - Euler reductions:
    - 1/β = r [ 1 + φ (q−1)/(1−θq) ]  (A.46)
    - 1/β = r_k + (1−δ) q [ 1 + φ (q−1)/(1−θq) ] − φ (q−1) (1−δ) (1−ϵ)/(1−θq)  (A.47)
  - Real wage as markup over MRS:
    - w = (1+κ_w) [ 1 / (1−χ_H v_C − ϕ) ]  (A.48)  (expression as in source)
  - Optimal capital-labour ratio:
    - K_H = (1−α) w / r_k  (A.49)
  - Marginal cost and price setting:
    - mc = 1/(1+μ_f) = 1/A [ w^{1−α} r_k^{α} ]  (A.50)  (as implied by A.15 and A.12)
    - Y/K = (1+μ_f) r_k  (A.51)
  - Government budget constraint:
    - τ = (r−1) L + G  (A.52)
  - Aggregate investment at steady state (A.53) and rewritten (A.54):
    - I = { [ r_k + (1−δ) q ϵ ] K + r L + Y − w H − r_k K − τ } / (1−θq)  (A.53)
    - δ = [ δ θ_q + φ (1−δ) ϵ ] q = { ... } expression yields:
      - δ − [ δ θ_q + φ (1−δ) ϵ ] q = { 1 + (1+μ_f) (L/Y) + μ_f } r_k − φ (G/K)  all over (1−θq)  (A.54)

### The Log-linear Model
- Log-deviation notation: x̂_t ≡ ln(x_t / x)
- Key linearised equations (selected):
  - Euler for domestic bonds:
    - −χ Ĉ_t = −χ E_t[Ĉ_{t+1}] + b̂R_t − E_t[π̂_{t+1}] + φ_r {1−θ_q}/(1−θ_q)^2 q E_t[ q̂_{t+1} ] − φ_r {1−θ_q}/(1−θ_q)^2 q E_t[ p̂_{I,t+1} ]  (A.55)
  - Euler for equity:
    - q̂_t − χ Ĉ_t = −χ E_t[Ĉ_{t+1}] + (φ/r_k) q [ 1 + φ (q−1)/(1−θ_q) ] E_t[ r̂_{k,t+1} ] + ... (full expression A.56)
  - Uncovered interest parity (log-linearised):
    - b̂R_t − E_t[π̂_{t+1}] = b̂R'_t − E_t[π̂'_{t+1}] + E_t[ŝ_{t+1}] − ŝ_t − ψ b̂F_t  (A.57)
  - Wage-setting:
    - [ ζ (1+v)/(1+κ_w) ] b̂ew_t − v(1−ξ_w)/(1+κ_w) b̂w_t = (1−ξ_w)[ −χ Ĉ_t + v b̂H_t ] + ξ_w [ ζ (1+v)/(1+κ_w) ] E_t[ b̂ew_{t+1} + π̂_{t+1} ]  (A.58)
  - Real wage evolution:
    - b̂w_t = (1−ξ_w) b̂ew_t + ξ_w [ b̂w_{t−1} − π̂_t ]  (A.59)
  - Marginal cost:
    - ĉmc_t = (1−α) b̂w_t + α b̂r_{k,t} − b̂A_t  (A.60)
  - Phillips curves:
    - Home goods: π̂_Ht = −(1−ξ_p)/(1−ξ_p θ_p) [ θ_p π̂_t − ĉmc_t − p̂_Ht ] + β E_t[ π̂_H{t+1} ]  (A.61)
    - Exports: π̂_H't = −(1−ξ_p)/(1−ξ_p θ_p) [ θ_p π̂_t − ĉmc_t − ŝ_t − p̂_H't ] + β E_t[ π̂_H'{t+1} ]  (A.62)
    - Imports: π̂_Ft = −(1−ξ_p)/(1−ξ_p θ_p) [ θ_p π̂_t − ĉmc'_t + ŝ_t − p̂_Ft ] + β E_t[ π̂_F{t+1} ]  (A.63)
  - Under law of one price (sensitivity analysis): P_Ft = S_t P_C'_t and P_H'_t = P_H_t S_t (replace A.62 and A.63)
  - Demand for home and imported consumption goods:
    - Ĉ_Ht = −η p̂_Ht + Ĉ_t  (A.64)
    - Ĉ_Ft = −η p̂_Ft + Ĉ_t  (A.65)
  - Consumption price index condition:
    - (1−ω) p̂_Ht + ω p̂_Ft = 0  (A.66)
  - Capital-goods producers condition:
    - p̂_I,t = λ_I î_t  (A.67)
  - Investment decomposition:
    - Î_Ht = −η p̂_Ht + Î_t  (A.68)
    - Î_Ft = −η p̂_Ft + Î_t  (A.69)
  - Government budget (log-linear):
    - τ̂_t = r L̄ [ b̂R_{t−1} + b̂L_{t−1} − E_t[π̂_{t}] ] − L̄ b̂L_t + Ḡ b̂G_t  (A.70)
  - Fiscal rule:
    - τ̂_t = τ̄ [ r L̄ ( b̂R_{t−1} + b̂L_{t−1} − E_t[π̂_t] ) ]  (A.71)
  - Monetary policy (Taylor) rule:
    - b̂R_t = max[ ρ ( π̄ b̂π_t + γ_Y b̂Y_t ); −log(R̄) ]  (A.72)
  - Optimal capital-labour ratio log-linear:
    - b̂K_{t−1} = b̂w_t − b̂r_{k,t} + b̂H_t  (A.73)
  - Aggregate production:
    - b̂Y_t = b̂A_t + b̂K_{t−1} + (1−α) b̂H_t  (A.74)
  - Capital ownership:
    - b̂K_t = b̂N_t  (A.75)
  - Capital law of motion:
    - b̂K_t = (1−δ) b̂K_{t−1} + δ b̂I_t  (A.76)
  - Resource constraint:
    - b̂Y_t = (1−ω)(C/Y) b̂C_Ht + (1−ω)(I/Y) b̂I_Ht + (G/Y) b̂G_t + (X/Y) b̂X_t  (A.77)
  - Home demand for domestically produced goods:
    - (Y−X) b̂Y_Ht = (1−ω) C b̂C_Ht + (1−ω) I b̂I_Ht + G b̂G_t  (A.78)
  - Aggregate imports demand:
    - (C+I) ĉM_t = C b̂C_Ft + I b̂I_Ft  (A.79)
  - Exports aggregate:
    - b̂X_t = −η p̂_H't + b̂Y'_t  (A.80)
  - Net foreign assets evolution:
    - b̂F_t = R' b̂F_{t−1} + (X/Y) [ ŝ_t + p̂_H't + b̂X_t ] − (M/Y) [ p̂_Ft + ĉM_t ]  (A.81)
  - Aggregate investment function (A.82) — full log-linear form retained in source.
  - Relative prices definitions:
    - π̂_Ht = π̂_Ct + p̂_Ht − p̂_H{t−1}  (A.83)
    - π̂_Ft = π̂_Ct + p̂_Ft − p̂_F{t−1}  (A.84)
    - π̂_H't = π̂_C't + p̂_H't − p̂_H'{t−1}  (A.85)
- World economy linearised:
  - Euler: −χ b̂Y'_t = −χ E_t[b̂Y'_{t+1}] + b̂R'_t − E_t[π̂'_{t+1}]  (A.86)
  - Phillips (world): π̂'_t = −(1−ξ_p)/(1−ξ_p θ_p) θ_p cmc'_t + β E_t[π̂'_{t+1}]  (A.87)
  - World marginal cost relation:
    - cmc'_t = (φ + ω) b̂Y'_t  (A.88)
  - World Taylor rule:
    - b̂R'_t = ρ π̂'_t + γ_Y b̂Y'_t  (A.89)
  - Fisher equations:
    - r̂_t = b̂R_t − E_t[π̂_{t+1}]  (A.90)
    - r̂'_t = b̂R'_t − E_t[π̂'_{t+1}]  (A.91)

### Calibration
- Structural parameters (Table 2):
  - β 0.989   Discount factor
  - ϵ 1.1515  Relative risk aversion
  - δ 0.025   Depreciation rate
  - α 0.36    Capital share
  - λ 2.4     Capital goods adjustment cost parameter
  - ϕ 1.9697  Inverse Frisch elasticity of labour supply
  - ω 0.3368  Degree of trade openness
  - η 1       Elasticity of substitution between home and foreign goods
  - μ_f 0.11  Price mark-up
  - κ_w 0.11  Wage mark-up
  - θ_p 0.7191 Price Calvo probability
  - θ_w 0.7923 Wage Calvo probability
- Parameters related to liquidity constraints:
  - φ 0.05    Probability of investment opportunity
  - θ_q 0.19  Borrowing constraint at steady state
  - ϵ 0.19    Equity resaleability constraint at steady state
- Policy parameters:
  - ρ_π 1.5   Taylor rule coefficient on inflation
  - γ_Y 0.125 Taylor rule coefficient on output
  - ρ_G 0.80  Persistence of government spending
  - τ̄ 0.1     Fiscal rule parameter
- Calibration notes:
  - Borrowing constraint θ_q and resaleability constraint ϵ set to 0.19 (slightly higher than DEFK's 0.185) to reflect higher UK steady-state real interest rate (~3.5% on 3-month T-bills). Entrepreneurs can sell up to 57% (= 1−(1−0.19)^4) of equity in a year.
  - Investment opportunity φ = 0.05 per quarter → 19% (= 1−(1−0.05)^4) per year.
  - Capital adjustment cost λ set to 2.4 (Groth (2008)); α = 0.36; δ = 0.025.
  - DSGE-standard parameters drawn from Cogley et al. (2011): ϵ = 1.1515; ϕ = 1.9697; θ_p = 0.7191; θ_w = 0.7923; markups 0.11.
  - Openness parameters: η = 1; ω = 0.3368 (posterior mode from Cogley et al. (2011)).
  - Monetary policy: ρ_π = 1.5; γ_Y = 0.125 (quarterly output coefficient because annual 0.5/4 = 0.125).
  - Fiscal rule parameter τ̄ = 0.1; government spending persistence ρ_G = 0.8.
- Steady-state values of endogenous variables (Table 3):
  - Consumption-to-GDP ratio C/Y 0.58
  - Investment-to-GDP ratio I/Y 0.22
  - Government spending share G/Y 0.20
  - Import-to-GDP ratio M/Y 0.27
  - Export-to-GDP ratio X/Y 0.27
  - Quarterly GDP Y 3.06
  - Quarterly labour H 0.90
  - Capital stock K 26.79
  - Public debt-to-GDP ratio L=4Y 0.40
  - Tax-to-GDP ratio τ/Y 0.21
  - Aggregate real wage w 1.95
  - Capital rent r_k 3.70%
  - Investment cost of new capital p_I 1
  - Market price of equity q 1.04
  - Real marginal cost mc 0.90
  - Real interest rate (annualised) 4 (R−1) 3.43%
- Additional steady-state calibrations:
  - Public debt-to-GDP ratio L=4Y = 40% to match UK data (42% on average) for 1992-2007.
  - Government consumption share G/Y set to 20% (UK average 1992-2007).
  - Inflation zero at steady state: π = 1.

*Italic: Source: _wp16138appendix - Section 2*

---


_Source: https://www.imf.org/-/media/websites/imf/imported-publications/external/pubs/ft/wp/2016/_wp16138appendix.pdf_