## 3. Three period example ## Source details **Canonical URL:** [3. Three period example](https://www.imf.org/-/media/websites/imf/imported-full-text-pdf/external/pubs/ft/wp/2013/_wp1385.pdf) ## Other formats - [Markdown version](/-/media/websites/imf/imported-full-text-pdf/external/pubs/ft/wp/2013/_wp1385.pdf.md) - [Structured JSON version](/-/media/websites/imf/imported-full-text-pdf/external/pubs/ft/wp/2013/_wp1385.pdf.json) --- ### I. Introduction — core intuition and implications - Market liquidity affects trade activity, marginal utilities and state prices; asset prices are higher in liquidity-constrained states of nature. - States with ‘tighter’ liquidity constraints receive greater weight under the risk-neutral measure; correlation between monetary transaction costs (linked to the short-term interest rate) and asset payoffs generates risk-premia in asset prices. - The liquidity-induced premium exists even in absence of aggregate uncertainty (aggregate endowment not state-dependent) and for any additively separable utility function. - Funding transaction costs affect individuals’ marginal utilities though not the aggregated representative agent’s utility — providing a potential explanation for the Term Premium Puzzle. - Liquidity is modeled as: - supply of money by the Central Bank (or banking system); - easiness with which commodities or assets are traded (captured via cash-in-advance constraints and liquidity parameters; e.g., a share λ of receipts can be used immediately). - Intermediate results required for main propositions: - nominal determinacy: cash-in-advance + outside money generate money demand and ensure nominal determinacy; - higher monetary costs worsen consequences of market imperfections even in pure exchange economies (wedge between buying and selling prices proxies market inefficiencies). - Implication: yield curve contains a term premium even in absence of aggregate uncertainty; monetary aggregates provide information beyond interest rates on activity, inflation and asset prices. ### II. The baseline model — structure and timing - Infinite-horizon exchange economy with money; real uncertainty hedged (complete AD securities), monetary transaction costs un-insurable. - Two agent types, α and β, maximize - U_i(c_i) ≡ E_0[∑_{t=0}^∞ δ^t u_{it}(c_{it})], i∈{α,β}, with (u_{it})′>0 and (u_{it})′′<0, and initially δ = 1. - Trading roles alternate by period parity: - at t = 2z, agent β holds positive endowments and agent α endowment is null; - at t = 2z+1, agent α is rich and β has no endowment. - Timing: commodity and money markets meet in sub-periods; agents cannot use receipts of current sales to purchase until end of period — they borrow in the short-term money market and repay when receipts arrive. - Central Bank provides money in all money markets: ∀s∈F_t, t≥0 money supply M_{t,s} and short-term interest rate r_{t,s}; interest rates inversely proportional to M. - Arrow-Debreu securities are nominal, pay 1 unit of account in a specific state, zero in others; all AD securities in zero net supply. Budget constraint relations (representative equations as indexed in the text): - Agent α: - ∀t,∀k∈F_t, p_{t,k} q^α_{t,k} = m^α_t + h^α_k + ∑_{s∈F_{t+1}} θ_s h^α_s (φ^α_{t,k}) - ∀t,∀s∈F_{t+1}, h^α_s + ∑_{l∈F_{t+2}} θ_l h^α_l = μ^α_s/(1 + r_s) + m^α_{t+1} (Ψ^α_{t+1,s}) - μ^α_s = p_s q^α_s (χ^α_{t+1,s}) - Agent β (mirror): - ∀t,∀k∈F_t, h^β_{t,k} + ∑_{s∈F_{t+1}} θ_s h^β_s = m^β_t + μ^β_{t,k}/(1 + r_{t,k}) (Ψ^β_{t,k}) - μ^β_k = p_k q^β_k (χ^β_{t,k}) - ∀t,∀s∈F_{t+1}, p_s q^β_s = m^β_{s+1} + h^β_s + ∑_{l∈F_{t+2}} θ_l h^β_l (φ^β_{t+1,s}) ### III. Monetary equilibrium — liquidity effect and non-neutrality results - Outside money ensures positive nominal interest rates; in the simple version r = m/M (following Dubey and Geanakoplos). - Positive interest rates generate nominal determinacy and money non-neutrality. Key lemmas and propositions (preserve exact statements): - Lemma 1: Value of Money - In an interior equilibrium, ∀t,∀k∈F_t, r_k M_k = m^α_t + m^β_t - Lemma 2: Quantity Theory of Money - ∀t = 2z, ∀k∈F_t, p_k q^β_k = p_k q^α_k = m^α_t + m^β_t + M_k and ∀s∈F_{t+1}, p_s q^α_s = p_s q^β_s = m^α_{t+1} + m^β_{t+1} + M_s - Proposition 1: Non-Neutrality of Money - In an interior equilibrium, without aggregate uncertainty, states with higher interest rates are those where trade is lower (i.e. r_s > r_{s′} ⇒ q^α_s < q^α_{s′}, ∀s,s′∈F_{t+1}, s ≠ s′). - Proposition 2: Asset Prices - Assume ∀t = 2z, ∀x∈F_{t+1}, and ∀q>0, RRA_β(q) = −u″_β(q) q / u′_β(q) ≥ 1. - Then ∀s,s′∈F_{t+1}, s ≠ s′, r_{s′} > r_s ⇒ θ_{s′} π_{s′} > θ_s π_s. - Restated in risk-neutral probabilities ˆπ_s = θ_s / ∑_{j∈F_{t+1}} θ_j: - ∀s ≠ s′, r_{s′} > r_s ⇐⇒ ˆπ_{s′} π_{s′} > ˆπ_s π_s. Key implication: - Even absent aggregate real uncertainty, states with higher interest rates receive higher weights in state prices — a liquidity-driven term premium. Results hold locally (Proposition 5 in Appendix 1) and under CRRA (Example 1). ### III. Example 1: CRRA closed-form relations (exact expressions) - If α and β have CRRA utility with common risk-aversion ρ, the model yields closed-form relations (equations as printed): - q^α_s / e^α_s q^α_{s′} / e^α_{s′} = ( (π^α_{s′} / π^β_{s′}) (1 + r_{s′} / 1 + r_k) )^{1/ρ} + e^β_k q^β_k^{−1} ( (π^α_s / π^β_s) (1 + r_s / 1 + r_k) )^{1/ρ} + e^β_k q^β_k^{−1} (equation (10) as printed) - θ_s / θ_{s′} = (π^β_s / π^β_{s′}) ( (e^α_{s′} / e^α_s) )^{ρ−1} (M_{s′} + m^α_{t+1} + m^β_{t+1}) / (M_s + m^α_{t+1} + m^β_{t+1}) × ( ( (π^α_s / π^β_s) (1 + r_s / 1 + r_k) )^{1/ρ} + e^β_k q^β_k^{−1} ( (π^α_{s′} / π^β_{s′}) (1 + r_s / 1 + r_k) )^{1/ρ} + e^β_k q^β_k^{−1} )^{ρ−1} (equation (11) as printed) - Interpretation: trade proportion increases as interest rate decreases; state prices depend on subjective probabilities, aggregate endowments, relative money supplies (nominal payoffs), and trade effect via r_s. ### IV. Liquidity and the term structure — three-period illustration and liquidity-risk term premium - Setup: one-commodity, three-period economy with logarithmic utility; only third period has S possible money supplies. - F_0 = {0}, F_1 = {1}, F_2 = {1...S}; subjective probabilities equal 1/S for each state. - Example 2 (closed-form solution): - r_0 = m_0 / M_0 - r_1 = m_1 / M_1 - ∀s, r_s = m_2 / M_s - θ_1 = 2 (M_0 + m_0) − ̄M over M_1 + m_1 [printed as: θ_1 = 2(M_0 + m_0) − ̄M / M_1 + m_1] - θ_s = (M_1 + m_1) / S(m_2 + M_s) (2 − ̄M/(M_0 + m_0)) [printed as: θ_s = M_1 + m_1 / S(m_2 + M_s) (2 − ̄M/(M_0 + m_0))] - ̄M is an exogenous inherited liability for agent α. - Note: model requires one agent to be initially indebted so first-period bond market is active for reasonable stable values of money supply. - No-arbitrage and bond-rate implications: - First inter-period bond price 0P_1 exceeds θ_1 / (1 + r_1) to prevent arbitrage by liquidity-constrained agents; hence 0r_1 is lower than combined rate ̄r_1 + r_1 where ̄r_1 = 1/θ_1 − 1. - Second-period bond price 1P_2 = ∑_{s∈F_2} θ_s / (1 + r_s) = 1/(1 + ̄r_2) ∑_{s∈F_2} ˆπ_s / (1 + r_s), where 1/(1 + ̄r_2) = ∑_{s∈F_2} θ_s. - Approximation: 1f_2 ≈ ̄r_2 + ∑_{s∈F_2} ˆπ_s (r_s) r_s (printed as: 1f_2 ≈ ̄r_2 + ∑_{s∈F_2} ˆπ_s (r_s) r_s) - Core message: since ˆπ_s is increasing in r_s, forward interest rate ̄r_2 + E_{ˆπ}[r_f] is above expected future spot rate ̄r_2 + E_0[r_f]; uncertainty in Central Bank money supply generates a liquidity-risk term premium. - Larger variance in expected spot rates (liquidity risk) generates a higher term premium — stability of monetary policy matters for long-term interest rates. ### V. Extensions (brief) and Appendix 2 — generalized cash-in-advance and liquidity parameters - General cash-in-advance constraint form: - p1 x1 + p2 x2 ≤ p1 Λ1 e1 + p2 Λ2 e2 - Interpretation of Λ and λ parameters: - If good 1 is illiquid and cannot serve for transactions, Λ1 = 0. - If good 2 is fiat money (liquid but nonconsumable), Λ2 = 1 and x2 = 0. - Parameter λ captures illiquidity of assets: only a fraction λs of assets’ payoff hβs can be used to pay for commodities. - Economic intuition: - Different commodities are not equally liquid; not all receipts from sales can be contemporaneously used for other purchases. - Financial intermediation requirements (proxied by interest rates) matter for trade. - If some Λ parameters < 1, money (or credit) demand is positive to bridge the gap between expenditures and receipts. - Agent budget sets and constraints (two-period stylized): - Agent α maximization: - max qα0, (qαs, hαs, μαs)_{s∈F} uα(qα0) + ∑_{s∈F} παs uα(eαs − qαs) (equation (12)) - p0 qα0 = ∑_{s∈F} θs hαs (φα0) (equation (13)) - ∀s∈F: hαs = Λs ps qαs + μαs / (1 + rs) (Ψαs) (equation (14)) - μαs = (1 − Λs) ps qαs (χαs) (equation (15)) - Agent β maximization: - max qβ0, (qβs, hβs, μβ0)_{s∈F} uβ(eβ0 − qβ0) + ∑_{s∈F} πβs uβ(qβs) (equation (16)) - ∑_{s∈F} θs hβs = Λ0 p0 qβ0 + μβ0 / (1 + r0) + mβ0 (φβ0) (equation (17)) - μβ0 = (1 − Λ0) p0 qβ0 (χβ0) (equation (18)) - ∀s∈F: ps qβs = μβs / (1 + rs) + λs hβs + mβs (Ψβs) (equation (19)) - μβs = (1 − λs) hβs (χβs) (equation (20)) Lemmas and comparative statics with liquidity parameters (exact relations preserved): - Lemma 3: Short-Term Interest Rates - ∀s∈F∪{0} rs = mβs / Ms - Lemma 4: Quantity theory of money and Liquidity - p0 q0 = M0 + mβ0 / (1 − Λ0) - ∀s∈F ps q s = M s + mβs / (1 − Λs) + (1 − λs)(Λs + (1 − Λs)/(1 + rs)) - Proposition 3: Non-Neutrality of Money and Liquidity - If ∀s∈F eαs = eα, then, ∀s ≠ s′, - rs (2 − λs − Λs) > rs′ (2 − λs′ − Λs′) =⇒ q s < q s′ - Proposition 4: Asset Prices and Liquidity - Assume ∀q, RRA(q) ≥ 1. Ceteris paribus, - rs > rs′ =⇒ θs > θs′ - λs > λs′ =⇒ θs < θs′ - Λs > Λs′ =⇒ θs < θs′ - Interpretation: liquidity parameters λs and Λs magnify financing costs, affect effective money quantity and asset prices, and are necessary (along with monetary aggregates) to assess markets’ liquidity needs and monetary transmission. Multiple-commodity extension (selected results preserved): - Lemma 4 (multiple commodities): - ∀s ∈ F_t ∪ {0}, r_s = (m^α_s + m^β_s) / M_s. - ∀s ∈ F_t ∪ {0}, ∑_{l ∈ L^α_s(−)} p_{s,l} q^α_{s,l} + ∑_{l ∈ L^β_s(−)} p_{s,l} q^β_{s,l} = ∑_{l ∈ L^β_s(+)} p_{s,l} q^β_{s,l} + ∑_{l ∈ L^α_s(+)} p_{s,l} q^α_{s,l} = M_s + m^α_s + m^β_s. - Proposition 6 (Non-Neutrality with multiple commodities): - Under stated directional-trade assumptions, ∀s ≠ s′, r_s > r_{s′} ⇐⇒ q^α_{s,L} < q^α_{s′,L}. - Proposition 7 (Asset Prices, multiple commodities): - Under stated assumptions (RRA ≥ 1, same sets L^i_s(−)), r_s > r_{s′} ⇒ θ_s / π_s > θ_{s′} / π_{s′}. ### VI. Conclusion — core findings and policy-relevant implications - Core findings: - In states with high interest rates, market imperfections that lead to unsold commodities or assets lower trade and affect state prices. - Risk-neutral probabilities are higher in states with higher interest rates and, in general, with low liquidity. - The result arises from interaction of monetary technology with the exchange economy and is absent from a general equilibrium without money. - Risk in the supply of money and in the extent of market imperfections determines risk in trade values. - Mechanism for upward-sloping term structure: - Two reinforcing phenomena: - Futures spot interest rates are affected by liquidity/market frictions. - Risk-neutral probabilities are modified by liquidity risk. - The interaction pushes long-term rates above the historical average of future spot rates, even without aggregate real risk. - Greater uncertainty in future spot rates implies higher long-term rates. - Policy implications (exact policy-relevant claims preserved): - Stability of monetary policy is required to maintain flat yield curves. - Changes in the term structure cannot be systematically related only to standard fundamentals (inflation, inflation risk, macroeconomic volatility, risk-aversion, fiscal policy). - Interpretations must include development of financial markets and liquidity risks. - Risk in market functioning and asset liquidity may generate term premia larger than those explained by non-monetary models. - Monetary aggregates are important, alongside short-term rates, to assess prospects for trade activity, inflation, and asset prices. *Source: _wp1385 - 3. Three period example (IMF working paper PDF excerpt, appendices and references).* ### References ............................................................................................................. ### _wp1385 - References ### References and Appendices - References ................................................................................................................................21 - Appendix 1 ...............................................................................................................................23 - Appendix 2 ...............................................................................................................................29 - Appendix 3 ...............................................................................................................................33 ### Figures - Figure 1. Time and uncertainty structure of the model .........................................................................8 - Figure 2. Timing of commodity, asset and money markets ..................................................................8 *Source: https://www.imf.org/-/media/websites/imf/imported-full-text-pdf/external/pubs/ft/wp/2013/_wp1385.pdf* ### 3. Three period example ................................................................................................ ### _wp1385 - 3. Three period example ......................................................................................................... ### I. INTRODUCTION — core intuition and implications - Market liquidity affects trade activity, marginal utilities and state prices; asset prices are higher in liquidity-constrained states of nature. - States with ‘tighter’ liquidity constraints receive greater weight under the risk-neutral measure; correlation between monetary transaction costs (linked to the short-term interest rate) and asset payoffs generates risk-premia in asset prices. - The liquidity-induced premium exists even in absence of aggregate uncertainty (aggregate endowment not state-dependent) and for any additively separable utility function. - Funding transaction costs affect individuals’ marginal utilities though not the aggregated representative agent’s utility — providing a potential explanation for the Term Premium Puzzle. - Liquidity in this paper refers to two concepts: - supply of money by the Central Bank (or banking system); - easiness with which commodities or assets are traded (captured via cash-in-advance constraints and liquidity parameters; e.g., a share λ of receipts can be used immediately). - Two intermediate results required for the main propositions: - nominal determinacy: cash-in-advance + outside money generate money demand and ensure nominal determinacy; - higher monetary costs worsen consequences of market imperfections even in pure exchange economies (wedge between buying and selling prices proxies market inefficiencies). - Implication: yield curve contains a term premium even in absence of aggregate uncertainty; monetary aggregates provide information beyond interest rates on activity, inflation and asset prices. ### II. THE BASELINE MODEL — structure and timing - Infinite-horizon exchange economy with money; real uncertainty hedged (complete AD securities), monetary transaction costs un-insurable. - Two agent types, α and β, maximize - U_i(c_i) ≡ E_0[∑_{t=0}^∞ δ^t u_{it}(c_{it})], i∈{α,β}, with (u_{it})′>0 and (u_{it})′′<0, and initially δ = 1. - Agents consume and trade a single commodity (extension to multiple commodities addressed later). - Endowments: e^i_{t,s} and monetary endowments m^i_t (monetary endowments non-stochastic). - Trading roles alternate by period parity: - at t = 2z, agent β holds positive endowments and agent α endowment is null; - at t = 2z+1, agent α is rich and β has no endowment. - Periods are defined so that the cash-in-advance constraint binds (periods correspond to intervals where buying/selling roles change). - Timing: commodity and money markets meet in sub-periods; agents cannot use receipts of current sales to purchase until end of period — they borrow in the short-term money market and repay when receipts arrive. - Central Bank provides money in all money markets: ∀s∈F_t, t≥0 money supply M_{t,s} and short-term interest rate r_{t,s}; interest rates inversely proportional to M. - Arrow-Debreu securities are nominal, pay 1 unit of account in a specific state, zero in others; all AD securities in zero net supply. ### II.B–C. Budget constraints (key equations) - Agent α (notation: subscripts k for states where α buys at t, subscripts s for states at t+1 where she sells): - purchases of goods = outside money + asset paying off + asset sales: - ∀t,∀k∈F_t, p_{t,k} q^α_{t,k} = m^α_t + h^α_k + ∑_{s∈F_{t+1}} θ_s h^α_s (φ^α_{t,k}) - servicing AD + asset purchases = short-term borrowing + outside money: - ∀t,∀s∈F_{t+1}, h^α_s + ∑_{l∈F_{t+2}} θ_l h^α_l = μ^α_s/(1 + r_s) + m^α_{t+1} (Ψ^α_{t+1,s}) - short-term debt repayment = income from selling goods: - μ^α_s = p_s q^α_s (χ^α_{t+1,s}) - Agent β (mirror roles): - servicing AD + asset purchases = outside money + short-term borrowing: - ∀t,∀k∈F_t, h^β_{t,k} + ∑_{s∈F_{t+1}} θ_s h^β_s = m^β_t + μ^β_{t,k}/(1 + r_{t,k}) (Ψ^β_{t,k}) - short-term debt repayment = income from selling goods: - μ^β_k = p_k q^β_k (χ^β_{t,k}) - purchases of goods = outside money + servicing asset + asset purchases: - ∀t,∀s∈F_{t+1}, p_s q^β_s = m^β_{s+1} + h^β_s + ∑_{l∈F_{t+2}} θ_l h^β_l (φ^β_{t+1,s}) ### III. MONETARY EQUILIBRIUM — liquidity effect and non-neutrality results - Outside money ensures positive nominal interest rates; following Dubey and Geanakoplos, a one-period relation r = m/M obtains in the simple version. - Positive interest rates generate nominal determinacy and money non-neutrality. - Lemma 1: Value of Money - In an interior equilibrium, ∀t,∀k∈F_t, r_k M_k = m^α_t + m^β_t - Lemma 2: Quantity Theory of Money - ∀t = 2z, ∀k∈F_t, p_k q^β_k = p_k q^α_k = m^α_t + m^β_t + M_k and ∀s∈F_{t+1}, p_s q^α_s = p_s q^β_s = m^α_{t+1} + m^β_{t+1} + M_s - Proposition 1: Non-Neutrality of Money - In an interior equilibrium, without aggregate uncertainty, states with higher interest rates are those where trade is lower (i.e. r_s > r_{s′} ⇒ q^α_s < q^α_{s′}, ∀s,s′∈F_{t+1}, s ≠ s′). - Proposition 2: Asset Prices - Assume ∀t = 2z, ∀x∈F_{t+1}, and ∀q>0, RRA_β(q) = −u″_β(q) q / u′_β(q) ≥ 1. - Then ∀s,s′∈F_{t+1}, s ≠ s′, r_{s′} > r_s ⇒ θ_{s′} π_{s′} > θ_s π_s. - Restated in risk-neutral probabilities ˆπ_s = θ_s / ∑_{j∈F_{t+1}} θ_j: - ∀s ≠ s′, r_{s′} > r_s ⇐⇒ ˆπ_{s′} π_{s′} > ˆπ_s π_s. - Key implication: even absent aggregate real uncertainty, states with higher interest rates receive higher weights in state prices — a liquidity-driven term premium. The result holds locally when subjective probabilities or endowments differ infinitesimally (Proposition 5 in Appendix 1) and can be extended under CRRA assumptions (Example 1). ### III. Example 1: CRRA closed-form relations (exact expressions) - If α and β have CRRA utility with common risk-aversion ρ: - q^α_s / e^α_s q^α_{s′} / e^α_{s′} = ( (π^α_{s′} / π^β_{s′}) (1 + r_{s′} / 1 + r_k) )^{1/ρ} + e^β_k q^β_k^{−1} ( (π^α_s / π^β_s) (1 + r_s / 1 + r_k) )^{1/ρ} + e^β_k q^β_k^{−1} (equation (10) as printed) - θ_s / θ_{s′} = (π^β_s / π^β_{s′}) ( (e^α_{s′} / e^α_s) )^{ρ−1} (M_{s′} + m^α_{t+1} + m^β_{t+1}) / (M_s + m^α_{t+1} + m^β_{t+1}) × ( ( (π^α_s / π^β_s) (1 + r_s / 1 + r_k) )^{1/ρ} + e^β_k q^β_k^{−1} ( (π^α_{s′} / π^β_{s′}) (1 + r_s / 1 + r_k) )^{1/ρ} + e^β_k q^β_k^{−1} )^{ρ−1} (equation (11) as printed) - Interpretation: trade proportion increases as interest rate decreases; state prices depend on subjective probabilities, aggregate endowments, relative money supplies (nominal payoffs), and trade effect via r_s. ### IV. LIQUIDITY AND THE TERM STRUCTURE — three-period illustration and liquidity-risk term premium - Model applied to a one-commodity, three-period economy with logarithmic utility; only third period has S possible money supplies. - Structure: F_0 = {0}, F_1 = {1}, F_2 = {1...S}; subjective probabilities equal 1/S for each state. - Example 2: Three-Period Example — closed-form solution (exact expressions) - r_0 = m_0 / M_0 - r_1 = m_1 / M_1 - ∀s, r_s = m_2 / M_s - θ_1 = 2 (M_0 + m_0) − \bar{M} over M_1 + m_1 [printed as: θ_1 = 2(M_0 + m_0) − ̄M / M_1 + m_1] - θ_s = (M_1 + m_1) / S(m_2 + M_s) (2 − ̄M/(M_0 + m_0)) [printed as: θ_s = M_1 + m_1 / S(m_2 + M_s) (2 − ̄M/(M_0 + m_0))] - ̄M is an exogenous inherited liability for agent α. - Note: model requires one agent to be initially indebted so first-period bond market is active for reasonable stable values of money supply. - No-arbitrage and implications for bond rates: - First inter-period bond price 0P_1 exceeds θ_1 / (1 + r_1) to prevent arbitrage by liquidity-constrained agents; hence 0r_1 is lower than combined rate ̄r_1 + r_1 where ̄r_1 = 1/θ_1 − 1. - Second-period bond price 1P_2 = ∑_{s∈F_2} θ_s / (1 + r_s) = 1/(1 + ̄r_2) ∑_{s∈F_2} ˆπ_s / (1 + r_s), where 1/(1 + ̄r_2) = ∑_{s∈F_2} θ_s. - Approximation: 1f_2 ≈ ̄r_2 + ∑_{s∈F_2} ˆπ_s (r_s) r_s (printed as: 1f_2 ≈ ̄r_2 + ∑_{s∈F_2} ˆπ_s (r_s) r_s) - Core message: since ˆπ_s is increasing in r_s, forward interest rate ̄r_2 + E_{ˆπ}[r_f] is above expected future spot rate ̄r_2 + E_0[r_f]; uncertainty in Central Bank money supply generates a liquidity-risk term premium. - Larger variance in expected spot rates (liquidity risk) generates a higher term premium — stability of monetary policy matters for long-term interest rates. ### V. EXTENSIONS (brief) - Multiple commodities extension feasible for additively separable utility; main result holds if the direction of trade of at least one good is state-independent (sufficient: one good always bought in all states and its price decreases when money supply decreases). - General point: transaction costs and liquidity parameters generate additional un-insurable risks that affect marginal utilities and asset prices beyond production/consumption risk. *Source: _wp1385 - 3. Three period example ................................................................................................ (IMF working paper PDF excerpt).* ### Appendix 2. ### Appendix 2. ### B. An Extension of Cash-In-Advance Constraints - General form of the cash-in-advance budget constraint: - p1 x1 + p2 x2 ≤ p1 Λ1 e1 + p2 Λ2 e2 - Interpretation of liquidity parameters Λ1 and Λ2: - If good 1 is illiquid and cannot serve for transactions, Λ1 = 0. - If good 2 is fiat money (liquid but nonconsumable), Λ2 = 1 and x2 = 0. - Economic intuition: - Different commodities are not equally liquid; not all receipts from sales can be contemporaneously used for other purchases. - Financial intermediation requirements (proxied by interest rates) matter for trade. - If some Λ parameters < 1, money (or credit) demand is positive to bridge the gap between expenditures and receipts. - Extensions introduced: - Follow Grandmont and Younes (1972) in defining liquidity of commodities. - Add parameter λ capturing illiquidity of assets. ### C. Budget Set for Agent α (two-period model) - Agent α maximisation problem: - max qα0, (qαs, hαs, μαs)_{s∈F} uα(qα0) + ∑_{s∈F} παs uα(eαs − qαs) (equation (12)) - Constraints: - p0 qα0 = ∑_{s∈F} θs hαs (φα0) (equation (13)) - ∀s∈F: hαs = Λs ps qαs + μαs / (1 + rs) (Ψαs) (equation (14)) - μαs = (1 − Λs) ps qαs (χαs) (equation (15)) - Interpretation: - A share 1 − Λs of commodity q_s sales receipts cannot be used to clear asset payments due to β. - As a result, α must borrow from the Central Bank μαs/(1 + rs). ### D. Budget Set for Agent β - Agent β maximisation problem: - max qβ0, (qβs, hβs, μβ0)_{s∈F} uβ(eβ0 − qβ0) + ∑_{s∈F} πβs uβ(qβs) (equation (16)) - Constraints: - ∑_{s∈F} θs hβs = Λ0 p0 qβ0 + μβ0 / (1 + r0) + mβ0 (φβ0) (equation (17)) - μβ0 = (1 − Λ0) p0 qβ0 (χβ0) (equation (18)) - ∀s∈F: ps qβs = μβs / (1 + rs) + λs hβs + mβs (Ψβs) (equation (19)) - μβs = (1 − λs) hβs (χβs) (equation (20)) - Interpretation of λs: - Only a fraction λs of the assets payoff hβs can be used to pay for the commodity in states (cost ps qβs). - Illiquidity of assets implies borrowing needs to the Central Bank μβs/(1 + rs). ### Lemmas on liquidity, money, and interest rates - Lemma 3: Short-Term Interest Rates - ∀s∈F∪{0} rs = mβs / Ms - Lemma 4: Quantity theory of money and Liquidity - p0 q0 = M0 + mβ0 / (1 − Λ0) - ∀s∈F ps q s = M s + mβs / (1 − Λs) + (1 − λs)(Λs + (1 − Λs)/(1 + rs)) - Interpretation: - Lemma 4 mirrors the canonical Quantity Theory of Money MV = PQ with V implicitly set to one. - Parameters λ and Λ capture limits on instantaneous liquidation of assets/commodities and thus magnify financing costs. - The intrinsic liquidity of assets/commodities affects effective money quantity and asset prices but not short-term interest rates. ### Propositions on non-neutrality of money and asset prices - Proposition 3: Non-Neutrality of Money and Liquidity - If ∀s∈F eαs = eα, then, ∀s ≠ s′, - rs (2 − λs − Λs) > rs′ (2 − λs′ − Λs′) =⇒ q s < q s′ - Proposition 4: Asset Prices and Liquidity - Assume ∀q, RRA(q) ≥ 1. Ceteris paribus, - rs > rs′ =⇒ θs > θs′ - λs > λs′ =⇒ θs < θs′ - Λs > Λs′ =⇒ θs < θs′ - Implications: - Short-term interest rate paths alone do not determine future trade and asset prices. - Monetary aggregates and liquidity parameters (λs and Λs) are necessary to assess markets’ liquidity needs and the transmission of monetary policy. ### VI. Conclusion — core findings and policy-relevant implications - Core findings: - In states with high interest rates, market imperfections that lead to unsold commodities or assets lower trade and affect state prices. - Risk-neutral probabilities are higher in states with higher interest rates and, in general, with low liquidity. - The result arises from interaction of monetary technology with the exchange economy and is absent from a general equilibrium without money. - Risk in the supply of money and in the extent of market imperfections determines risk in trade values. - Mechanism for upward-sloping term structure: - Two reinforcing phenomena: - Futures spot interest rates are affected by liquidity/market frictions. - Risk-neutral probabilities are modified by liquidity risk. - The interaction pushes long-term rates above the historical average of future spot rates, even without aggregate real risk. - Greater uncertainty in future spot rates implies higher long-term rates. - Policy implications: - Stability of monetary policy is required to maintain flat yield curves. - Changes in the term structure cannot be systematically related only to standard fundamentals (inflation, inflation risk, macroeconomic volatility, risk-aversion, fiscal policy). - Interpretations must include development of financial markets and liquidity risks. - Risk in market functioning and asset liquidity may generate term premia larger than those explained by non-monetary models. - Monetary aggregates are important, alongside short-term rates, to assess prospects for trade activity, inflation, and asset prices. *Appendix 2. (from the provided PDF content)* ### REFERENCES ### _wp1385 - REFERENCES ### References - Aiyagari, S.R., 1994, “Uninsured Idiosyncratic Risk and Aggregate Saving”, Quarterly Journal of Economics, Vol. 109(3), pp. 659–684. - Backus, D., A. Gregory and S. Zin, 1989, “Risk Premiums in the Term Structure: Evidence from Artificial Economies,” Journal of Monetary Economics, Vol. 24(3), pp. 371–399. - Bansal, R., and W.J. Coleman II, 1996, “A Monetary Explanation of the Equity Premium, Term Premium and Risk-Free Rate Puzzles,” Journal of Political Economy, Vol. 104(6), pp. 1135–1171. - Bloize, G., J.H. Dreze and H. Polemarchakis, 2005, “Monetary Economy over Infinite Horizon,” Economic Theory, Vol. 25(1), pp. 51–74. - Breeden, D., 1979, “An Intertemporal Asset Pricing Model with Stochastic Consumption and Investment Opportunities, Journal of Financial Economics, Vol. 7, pp. 265–296. - Chabi-Yo, F., D. Leisen and E. Renault (2007), “Implications of Asymmetry Risk for Portfolio Analysis and Asset Pricing,” Bank of Canada Working Paper 2007–47. - Clower, B.W., 1967, “A Reconsideration of the Microeconomic Foundations of Monetary Theory,” Western Economic Journal, Vol. 6, pp. 1–8. - Dubey, P., and J. Geanakoplos, 1992, “The Value of Money in a Finite-Horizon Economy: a Role for Banks,” in P. Dasgupta, D. Gale et al. (eds), Economic Analysis of Market and Games, Cambridge MA: MIT Press. - Dubey, P., and J. Geanakoplos, 2003, “Monetary Equilibrium with Missing Markets,” Journal of Mathematical Economics, Vol. 39, pp. 585–618. - Dubey, P., and J. Geanakoplos, 2006, “Determinacy with Nominal Assets and Outside Money,” Economic Theory, Vol. 27, pp. 79–106. - Elul, R., 1997, “Financial Innovation, Precautionary Saving and the Risk-free Rate,” Journal of Mathematical Eonomics, Vol. 27, pp. 113–131 - Fan, M., 2006, “Heterogeneous Beliefs, the Term Structure and Time-Varying Risk Premia,” Annals of Finance, Vol. 2, pp. 259–285. - Fisher, I., 1911, The Purchasing Power of Money, New York: MacMillan. - Grandmont, J.M., and Y. Younes, 1972, “On the Role of Money and the Existence of a Monetary Equilibrium,” Review of Economic Studies, Vol. 39, pp. 355–372. - Grossman, S., A. Melino and R. Shiller, 1987, “Estimating the Continuous Time Consumption Based Asset Pricing Model,” Journal of Business and Economic Statistics, Vol. 5(3), pp. 315–327. - Gurley, J.G., and E.S. Shaw, 1960, Money in a Theory of Finance, Washington DC: The Brookings Institution. - Lagos, R., and R. Wright, 2005, “A Unified Framework for Monetary Theory and Policy Analysis,” Journal of Political Economy, Vol. 113(3), pp. 463–484. - Lagos, R., 2010, “Asset Prices and liquidity in an exchange economy,” Journal of Monetary Economics, Vol. 57, pp. 913–930. - Lucas, R., 1978, “Asset Prices in an Exchange Economy,” Econometrica, Vol. 46(6), pp. 1429–1445. - Lucas, R., and N. Stokey, 1983, “Optimal Fiscal and Monetary Policy in an Economy Without Capital,” Journal of Monetary Economics, Vol. 12, pp. 55–93. - Lucas, R., and N. Stokey, 1987, “Money and Interest in a Cash-in-Advance Economy,” Econometrica, Vol. 55(3), pp. 491–513. - Lucas, R., 1990, “Liquidity and Interest Rates,” Journal of Economic Theory, Vol. 50(2), pp. 237–264. - Radner, R., 1972 “Existence of Equilibrium of Plans, Prices, and Price Expectations in a Sequence of Markets,” Econometrica, Vol. 40(2), pp. 289–303. - Shubik, M., and D.P. Tsomocos, 1992, “A Strategic Market Game with a Mutual Bank with Fractional Reserves and Redemption in Gold (A Continuum of Traders),” Journal of Economics, Vol. 55(2), pp. 123–150. - Shubik, M., and C. Wilson, 1977, “The Optimal Bankruptcy Rule in a Trading Economy Using Fiat Money,” Journal of Economics, Vol. 37 (3-4), pp. 337–354. - Svensson, L.E.O., 1985, “Money and Asset Prices in a Cash-in-Advance Economy,” Journal of Political Economy, Vol. 93 (5), pp. 919–944. - Tsomocos, D.P., 2008, “Generic Determinacy and Money Non-Neutrality of International Monetary Equilibria,” Journal of Mathematical Economics, Vol. 44(7–8), pp. 866887. - Weil, P., 1992, “Equilibrium Asset Prices with Undiversifiable Labor Income Risk,” Journal of Economic Dynamics and Control, Vol. 16, pp. 769–790. - Woodford, M., 1996, “Control of the Public Debt: A Requirement for Price Stability?” NBER Working Paper No. 5684, Cambridge, MA: National Bureau of Economic Research. ### Appendix 1 — Proofs, Lemmas, Propositions (selected results) - Lemma 1: Money carrying over is dominated — no agent would carry money from t to t+1 because she could reduce money demand, save interest payments, and sell less of her endowment to repay loans. Proof applies symmetrically for periods t = 2z and t = 2z+1. - Lemma 2: For t = 2z, relations (21) and (22) imply μ^β_{t,k}/(1+r_{t,k}) = M_{t,k}, and with μ^β_{t,k}/(1+r_{t,k}) = M_{t,k} the stated result follows; same for t = 2z+1. - Proposition 1: From first-order conditions of agents α and β, equations (23)–(26) yield: - If ∀s e^α_s = e, define f(q^α_s) = u′_α(e−q^α_s)/u′_β(q^α_s) which is increasing in q^α_s. - Therefore, if r_s > r_{s′}, then q^α_s < q^α_{s′}. - Proposition 2: Using (24), quantity theory, and assuming relative risk aversion > 1 (so u′_β(q)q is decreasing), deduce: - θ_s is lower when r_s is lower. - Since r_s is an inverse function of M_s, ∀s≠s′, r_{s′} > r_s ⇐⇒ θ_{s′} π_{s′} > θ_s π_s. - Proposition 3 (Non-Neutrality of Money and Liquidity): Under e^α_s = e^α and small intra-period interest rates (1/(1+r_s) ≈ 1), define f(q^α_s) = u′_α(e−q^α_s)/u′_β(q^α_s) (increasing in q^α_s). Then: - r_s (2−λ_s − Λ_s) > r_{s′} (2−λ_{s′} − Λ_{s′}) ⇒ q_s < q_{s′}. - Proof uses first-order conditions (34)–(36) and approximation 1/((Λ_s + (1−Λ_s)/(1+r_s))(λ_s + (1−λ_s)/(1+r_s))) ≈ 1 + r_s (2−Λ_s − λ_s). - Proposition 4 (Asset Prices and Liquidity): Assume ∀q, RRA(q) ≥ 1. Ceteris paribus: - r_s > r_{s′} ⇒ θ_s > θ_{s′} - λ_s > λ_{s′} ⇒ θ_s < θ_{s′} - Λ_s > Λ_{s′} ⇒ θ_s < θ_{s′} - Proof rearranges β’s first-order conditions (35), expresses θ_s/θ_{s′} via u′_β(q)q and L_s factors, and shows ∂L_s/∂M_s < 0, ∂L_s/∂λ_s < 0, ∂L_s/∂Λ_s < 0, yielding stated comparative statics. ### Example 2 — Three-Period Example (closed-form solution) Assumptions: - Three periods with F_0 = {0}, F_1 = {1}, F_2 = {1...S}. - u_α = u_β = ln. - π^α_s = π^β_s = 1/S, ∀s ∈ F_2. - ¯M is an exogenous inherited liability for agent α. Solution (stated): - r_0 = m_0 / M_0 - r_1 = m_1 / M_1 - ∀s, r_s = m_2 / M_s - θ_1 = 2(M_0 + m_0) − ¯M / (M_1 + m_1) - θ_s = M_1 + m_1 / [S(m_2 + M_s)(2 − ¯M/(M_0 + m_0))] Notes on derivation: - p_0 q_0 = M_0 + m_0, p_1 q_1 = M_1 + m_1, ∀s p_s q_s = M_s + m_2. - From β’s first-order conditions, θ_s h_s = θ_{s′} h_{s′} for s,s′ ∈ F_2; uses market clearing and interior equilibrium conditions to solve for θ_s and θ_1 via (28) and (29). Additional remark: - Footnote: “We need one agent to be in debt initially so that the first period bond market be active for reasonnable stable values of the money supply.” ### Appendix 2 — Multiple Commodities: Non-Neutrality and Asset Prices - Setup: Two-period restriction to simplify; goods indexed by l ∈ L = {1,...,L}; L^i_s(+) goods agent i sells in state s; L^i_s(−) goods agent i buys. - Lemma 4 (Short-term interest rate, multiple commodities): - ∀s ∈ F_t ∪ {0}, r_s = (m^α_s + m^β_s) / M_s. - Lemma 4 (Quantity theory of money, multiple commodities): - ∀s ∈ F_t ∪ {0}, ∑_{l ∈ L^α_s(−)} p_{s,l} q^α_{s,l} + ∑_{l ∈ L^β_s(−)} p_{s,l} q^β_{s,l} = ∑_{l ∈ L^β_s(+)} p_{s,l} q^β_{s,l} + ∑_{l ∈ L^α_s(+)} p_{s,l} q^α_{s,l} = M_s + m^α_s + m^β_s. - Proposition 6 (Non-Neutrality of Money With Multiple Commodities): - If for any states s ∈ F_t ∪ {0} a good (s,L) is bought by β and e^α_{s,L} > 0, e^β_{s,L} = 0, ∀s ∈ F_t (L sold by α in all states), then in absence of aggregate uncertainty: - ∀s ≠ s′, r_s > r_{s′} ⇐⇒ q^α_{s,L} < q^α_{s′,L}. - Proposition 7 (Asset Prices with Multiple Commodities): - Assume for all i ∈ {α,β}, ∀s,s′ ∈ F_t, s ≠ s′, L^i_s(−) = L^i_{s′}(−); assume further that RRA ≥ 1 and no aggregate uncertainty. Then: - r_s > r_{s′} ⇒ θ_s / π_s > θ_{s′} / π_{s′}. - Proof uses β’s first-order conditions, market clearing, quantity theory, and existence of a good g with p_{s,g} q_{i,s,g} ≤ p_{s′,g} q_{i,s′,g} to conclude θ_s/π_s > θ_{s′}/π_{s′}. ### Appendix 3 — Additional Lemmas and Propositions - Lemma 3 (Short-Term Interest Rates): ∀s ∈ F ∪ {0}, r_s = m^β_s / M_s. - Derivation: Money market equilibrium and asset market equilibrium conditions yield (33) and subsequent identities implying r_0 = m^β_0 / M_0 and, more generally, M_s(1 + r_s) = μ^α_s + μ^β_s = M_s + m^β_s, hence r_s = m^β_s / M_s. - Lemma 4 (Quantity theory of money and Liquidity, alternate form): - p_0 q_0 = (M_0 + m^β_0) / (1−Λ_0) - ∀s ∈ F, p_s q_s = (M_s + m^β_s) / [1−Λ_s + (1−λ_s)(Λ_s + (1−Λ_s)/(1+r_s))] - Proposition 3 (alternate statement and proof sketch reproduced): - If ∀s ∈ F, e^α_s = e^α, then ∀s ≠ s′, - r_s (2 − λ_s − Λ_s) > r_{s′} (2 − λ_{s′} − Λ_{s′}) ⇒ q_s < q_{s′}. - Uses first-order conditions (34)–(36), approximation for small intra-period interest rates, and monotonicity of f(q) = u′_α(e−q)/u′_β(q). - Proposition 4 (Asset Prices and Liquidity, alternate development): - Presents θ_s/θ_{s′} expression via u′_β(q) q, p q, and liquidity-related factor L_s; shows ∂L_s/∂M_s < 0, ∂L_s/∂λ_s < 0, ∂L_s/∂Λ_s < 0. Concludes that higher M_s (lower r_s), higher λ_s, or higher Λ_s is associated with lower θ_s, consistent with earlier propositions. *Content derived from the PDF chapter "_wp1385 - REFERENCES" and accompanying appendices.* --- _Source: https://www.imf.org/-/media/websites/imf/imported-full-text-pdf/external/pubs/ft/wp/2013/_wp1385.pdf_