Title: When Money Dies The Dynamics of Speculative Hyperinflations
Domain: economics

Question: Standard economic models of hyperinflation show the real value of money approaching zero asymptotically over an infinite time horizon. Your task is to design a continuous-time, micro-founded monetary model that demonstrates a more radical and counter-intuitive outcome: the value of money becoming exactly zero in a finite amount of time. Derive the analytical expression for this finite 'time of death' (T) and, most importantly, establish the necessary and sufficient condition on agents' preferences for this phenomenon to be possible.

Background: This problem is situated in the field of monetary economics, specifically concerning the dynamics of hyperinflation. Classic models, such as Cagan's adaptive expectations model, explain hyperinflation as a phenomenon driven by explosive growth in the money supply. These models typically predict that the real value of money will approach zero only asymptotically, i.e., over an infinite time horizon. However, this query explores a more radical possibility: a complete collapse in money's value in finite time, even with a constant money supply. This can occur in a speculative equilibrium, where the expectation of a future collapse becomes a self-fulfilling prophecy. To model this, one must move beyond representative-agent models and use a search-theoretic framework (like the Shi-Trejos-Wright model). In these models, money is essential for trade due to frictions like the lack of a double coincidence of wants. The use of continuous time is also crucial, as it allows the model to bypass the backward induction arguments that often rule out such collapses in discrete-time settings.

Definitions: Value of Money (y_t): The quantity of a real consumption good that one unit of indivisible money can purchase at time t. It is the central dynamic variable. | Agent's Utility (u(y)): A twice-differentiable, strictly increasing, and concave utility function from consuming y units of the good, with u(0) = 0. Its properties near y=0 are critical. | Key Parameters: ρ is the agents' rate of time preference. θ is the effective velocity of money, determined by the rate of meetings (α), the probability of a desirable match (σ), and the stock of money (M). θ = ασ(1 – M).

Methodology: Begin by setting up the continuous-time Bellman equations for agents holding money (V_1,t) and agents not holding money (V_0,t). The difference, y_t = V_1,t - V_0,t, represents the seller's surplus, which equals the value of money. | Derive the law of motion for y_t. This will result in an autonomous Bernoulli ordinary differential equation (ODE) of the form: ẏ_t = f(y_t). | To find the time T for y_t to go from an initial value y_0 to 0, separate variables in the ODE and express T as a definite integral. | The core of the problem is to analyze the convergence of this integral. The condition for T to be finite depends entirely on whether the integrand 1/f(y) is integrable in the neighborhood of y=0. This, in turn, depends on the limiting behavior of the utility function u(y) as y approaches zero.

Answer: 1. Model Formulation: A Continuous-Time Search Model | We analyze a continuous-time Shi-Trejos-Wright (STW) model. A unit measure of agents are specialized, discounting the future at a rate ρ > 0. Agents meet randomly at a Poisson rate α, and a fraction σ of meetings are "single-coincidence." Fiat money is essential for trade. Money is indivisible, and agents hold at most one unit. M is the fraction of agents holding money. | Let V_1,t be the lifetime expected utility of an agent with money (a buyer) and V_0,t be the utility of an agent without money (a seller) at time t. The value of money, y_t, is the amount of goods exchanged for it, determined by a take-it-or-leave-it offer making the seller just indifferent: y_t = V_1,t - V_0,t. | The Bellman equations for the agents are: | ρV_1,t = ασ(1-M)[u(y_t) - y_t] + d(V_1,t)/dt (1a) | ρV_0,t = ασM[y_t] + d(V_0,t)/dt (1b) | In a stationary equilibrium, the time derivative terms are zero. | 2. The Law of Motion for the Value of Money | To analyze out-of-steady-state dynamics (speculative hyperinflation), we consider a path where y_t evolves over time, so ẏ_t ≠ 0. Since y_t = V_1,t - V_0,t, we have ẏ_t = d(V_1,t)/dt - d(V_0,t)/dt. | Subtracting (1b) from (1a): | ρ(V_1,t - V_0,t) = ασ(1-M)[u(y_t) - y_t] - ασM[y_t] + (d(V_1,t)/dt - d(V_0,t)/dt) | ρy_t = ασ(1-M)u(y_t) - ασ(1-M)y_t - ασMy_t + ẏ_t | Let θ = ασ(1 – M) be the velocity of money. Rearranging gives the fundamental law of motion, an autonomous Bernoulli ordinary differential equation (ODE): | ẏ_t = (ρ + θ)y_t - θu(y_t) (2) | A speculative hyperinflation is an equilibrium path that is a solution to this ODE, starting from y_0 > 0 and converging to y=0. | 3. Derivation of the Finite Time of Death (T) | The time T it takes for money to become valueless can be found by solving this ODE. Separating variables gives: | dt = dy / [(ρ + θ)y - θu(y)] | To find the total time T for the value of money to fall from an initial state y_0 (at t=0) to zero (at t=T), we integrate both sides: | ∫(from 0 to T) dt = ∫(from y_0 to 0) dy / [(ρ + θ)y - θu(y)] | T = - ∫(from 0 to y_0) dy / [(ρ + θ)y - θu(y)] | Since the denominator is negative for small y, the expression for T is positive: | T = ∫(from 0 to y_0) dy / [θu(y) - (ρ + θ)y] (3) | 4. The Necessary and Sufficient Condition for a Finite T | The time T is finite if and only if the integral in equation (3) converges. The convergence of this improper integral depends on the behavior of the integrand 1 / [θu(y) - (ρ + θ)y] as y → 0+. | As y → 0+, the term (ρ + θ)y becomes negligible compared to θu(y). Thus, the convergence of the integral is determined by the convergence of: | ∫(from 0 to y_0) dy / [θu(y)] | This integral converges if and only if u(y) grows "faster" than y near the origin. More formally, T is finite if and only if ∫(from 0 to ε) [1/u(y)] dy is finite for some ε > 0. | This is the necessary and sufficient condition. A necessary (but not sufficient) part of this condition is that u'(0) = +∞. If u'(0) were finite, u(y) would behave like y * u'(0) near the origin, and ∫ 1/y dy diverges, making T infinite. | 5. Example: CRRA Preferences | Let's assume a Constant Relative Risk Aversion (CRRA) utility function: u(y) = y^(1-η) / (1-η) for 0 < η < 1. Here, u'(y) = y^(-η), so u'(0) = +∞. | Checking the integrability condition: | ∫ 1/u(y) dy = ∫ (1-η)/y^(1-η) dy = (1-η) ∫ y^(η-1) dy = (1-η) * [y^η / η]. | Evaluating this from 0 to ε gives (1-η)ε^η / η, which is finite. Therefore, the condition is met, and T must be finite. | We can solve for T explicitly. The ODE becomes ẏ_t = (ρ + θ)y_t - (θ/(1-η))y_t^(1-η). This is a Bernoulli equation. Let x_t = y_t^η. Then ẋ_t = ηy_t^(η-1)ẏ_t. Substituting gives a linear ODE for x_t: | ẋ_t = η(ρ + θ)x_t - ηθ/(1-η) | Solving this linear ODE for a path from x_0 to 0 and transforming back to y_t yields the closed-form solution for the time of death: | T = [1 / (η(ρ+θ))] * ln[1 - ((ρ+θ)(1-η)/θ) * y_0^η]^(-1) | This expression is finite for any y_0 within the basin of attraction of the non-monetary equilibrium.

Checklist: 1. Correctly formulate the continuous-time Bellman equations for V_1,t and V_0,t based on the specified search-theoretic environment.  
2. Accurately derive the autonomous Bernoulli ODE that governs the dynamics of the value of money, y_t: ẏ_t = (ρ + θ)y_t - θu(y_t).  
3. Derive the general formula for T, the time for money to become valueless, by expressing it as a definite integral with respect to y.  
4. State and rigorously prove that T is finite if and only if the integral ∫(from 0 to y_0) [1/u(y)] dy converges.  
5. Translate this mathematical condition into a precise economic statement about agents' preferences: a finite lifespan for money requires that the marginal utility of consumption grows infinitely fast as consumption approaches zero, in a specific, integrable way.  
6. Demonstrate the result with a concrete example, by solving for T under the assumption of CRRA utility, u(y) = y^(1-η), and showing it is finite for 0 < η < 1.  
7. Correctly interpret the economic intuition: for agents to hold money as its value plummets towards zero at an infinite rate, the liquidity return (driven by u(y)) must also become infinite to compensate them.