Unit 4: Mathematical Economics

Difference and Differential Equations with Economic Applications

Mathematical economics uses mathematical tools to represent economic theories and analyze economic problems. Among these tools, difference and differential equations play a crucial role in understanding dynamic economic systems, where variables change over time. Difference equations are used when time is discrete, such as in yearly economic data, while differential equations are employed when time is continuous, like in instantaneous rates of change.

Difference Equations

A difference equation is an equation that relates successive terms of a sequence. In economics, these equations are used to model how economic variables evolve over discrete time periods. The general form of a first-order linear difference equation is:

yt+1 = a yt + b

where:

• yt+1 is the value of the variable at time t+1.
• yt is the value of the variable at time t.
• a is a constant representing the multiplier or the effect of the previous period's value on the current period's value.
• b is a constant representing an autonomous or exogenous factor.

Types of Difference Equations

Difference equations can be classified based on their order and linearity. The order of a difference equation is determined by the highest difference between time indices. A first-order equation involves terms like yt+1 and yt, while a second-order equation might involve yt+2, yt+1, and yt.

Linear difference equations have variables raised to the power of one and no products of variables. Non-linear equations involve powers greater than one or products of variables.

Solving First-Order Linear Difference Equations

The general solution to a first-order linear difference equation yt+1 = a yt + b consists of two parts: the complementary solution (or homogeneous solution) and a particular solution.

1. Complementary Solution: This is the solution to the homogeneous equation yt+1 = a yt. The solution is yt(c) = A at, where A is a constant determined by initial conditions.

2. Particular Solution: This is any solution to the non-homogeneous equation. If b is a constant, a common particular solution is yt(p) = b / (1 - a), provided a ≠ 1. This is the equilibrium value.

The general solution is yt = yt(c) + yt(p) = A at + b / (1 - a).

To find A, we use the initial condition, say y0. y0 = A a0 + b / (1 - a) = A + b / (1 - a) So, A = y0 - b / (1 - a).

The final solution is: yt = (y0 - b / (1 - a)) at + b / (1 - a)

Stability Conditions for First-Order Linear Difference Equations

The behavior of the solution as t approaches infinity depends on the value of a.

• If |a| < 1, the term at approaches 0 as t → ∞. The system is stable and converges to the equilibrium y* = b / (1 - a).
• If |a| > 1, the term at grows infinitely large. The system is unstable.
• If a = 1, the equation becomes yt+1 = yt + b, which is an arithmetic progression. If b ≠ 0, it diverges. If b = 0, it is constant.
• If a = -1, the system oscillates between two values: y0 and y0 - b.

Economic Applications of Difference Equations

Difference equations are widely used in macroeconomics and finance.

1. The Cobweb Model: This model explains price fluctuations in agricultural markets. Producers decide output for the next period based on the current period's price, which is determined by the previous period's supply. Let Pt be the price in period t and Qt be the quantity supplied in period t. Assume demand is Qd,t = α - β Pt and supply is Qs,t = γ + δ Pt-1. In equilibrium, Qd,t = Qs,t. α - β Pt = γ + δ Pt-1 Pt = (α - γ) / β - (δ / β) Pt-1 This is a first-order linear difference equation of the form Pt = a Pt-1 + b, where a = -δ / β and b = (α - γ) / β. The market is stable if |a| = |-δ / β| < 1, i.e., δ / β < 1. This means the responsiveness of supply to price changes is less than the responsiveness of demand to price changes.

2. Multiplier-Accelerator Model: This model explains business cycles. The multiplier effect describes how an initial change in investment leads to a larger change in national income. The accelerator principle states that investment depends on the change in income. Let Yt be income in period t, It be investment, and Ct be consumption. Yt = Ct + It Ct = c Yt-1 (where c is the marginal propensity to consume) It = v (Yt-1 - Yt-2) (where v is the accelerator coefficient) Substituting these into the income equation: Yt = c Yt-1 + v (Yt-1 - Yt-2) Yt = (c + v) Yt-1 - v Yt-2 This is a second-order linear homogeneous difference equation. Its stability and the pattern of fluctuations depend on the roots of the characteristic equation r2 - (c + v) r + v = 0.

3. Inventory Models: Firms adjust their inventory levels based on demand and production costs. These adjustments can be modeled using difference equations to find optimal inventory policies.

Differential Equations

A differential equation is an equation that relates a function with its derivatives. In economics, differential equations are used to model continuous-time dynamic systems. They are essential for analyzing instantaneous rates of change and market adjustments that occur smoothly over time. The general form of a first-order linear differential equation is:

dy/dt = a y + b

where:

• dy/dt is the derivative of the variable y with respect to time t, representing the instantaneous rate of change.
• y is the variable.
• a and b are constants.

Types of Differential Equations

Similar to difference equations, differential equations can be classified by their order (highest derivative) and linearity.

Solving First-Order Linear Differential Equations

The general solution to dy/dt = a y + b can be found using separation of variables or by recognizing it as a process of change proportional to the deviation from equilibrium.

1. Separation of Variables: If a y + b ≠ 0, we can write: dy / (a y + b) = dt Integrate both sides: (1/a) ln|a y + b| = t + C1 ln|a y + b| = a t + a C1 |a y + b| = eat + aC1 = eaC1 eat a y + b = ± eaC1 eat Let C = ± eaC1. a y + b = C eat a y = C eat - b y(t) = (C/a) eat - b/a Let A = C/a. y(t) = A eat - b/a

2. Using Equilibrium Concept: The equilibrium value y* occurs when dy/dt = 0. 0 = a y* + b, so y* = -b/a. Let z(t) = y(t) - y*. Then dz/dt = dy/dt. The equation becomes dz/dt = a (y(t) - y*) = a z(t). The solution is z(t) = z0 eat, where z0 = y0 - y*. Substituting back: y(t) - y* = (y0 - y*) eat y(t) = y* + (y0 - y*) eat y(t) = -b/a + (y0 + b/a) eat This is equivalent to the previous solution if we adjust the constant.

Stability Conditions for First-Order Linear Differential Equations

The stability of the system depends on the sign of the coefficient a in the exponent eat.

• If a < 0, then eat → 0 as t → ∞. The system is stable and converges to the equilibrium y* = -b/a.
• If a > 0, then eat → ∞ as t → ∞. The system is unstable.
• If a = 0, the equation is dy/dt = b, which means y(t) = b t + y0, a linear growth path.

Economic Applications of Differential Equations

Differential equations are fundamental in continuous-time economic models.

1. Solow Growth Model (Continuous Time): This model describes capital accumulation in an economy. Let k(t) be capital per worker at time t. The change in capital per worker is given by: dk/dt = s f(k) - (δ + n) k where:

• s is the savings rate.
• f(k) is the production function per worker (e.g., kα).
• δ is the depreciation rate.
• n is the population growth rate.

The equilibrium level of capital per worker k* is found where dk/dt = 0: s f(k*) = (δ + n) k* The stability of the model depends on the shape of the f(k) function and the parameters. If the function g(k) = s f(k) - (δ + n) k is such that g'(k*) < 0, the equilibrium is stable.

2. Optimal Control Theory (Dynamic Optimization): Many economic problems involve optimizing an objective function over time, such as maximizing lifetime utility or minimizing costs. Differential equations are used to describe the evolution of state variables and solve these optimization problems. For example, in an optimal consumption problem, the rate of change of wealth or capital is determined by savings decisions, which are influenced by interest rates and consumption choices.

3. Asset Pricing Models: In finance, the Black-Scholes model for option pricing is based on a partial differential equation (the Black-Scholes equation). More generally, the instantaneous rate of change of an asset's price is often modeled as a stochastic differential equation (e.g., Geometric Brownian Motion). dS/S = μ dt + σ dW where S is the asset price, μ is the drift (expected rate of return), σ is volatility, and dW is a Wiener process (Brownian motion).

4. Market Adjustment Speed: Consider a market where price P adjusts to excess demand D(P) - S(P). The speed of adjustment can be modeled as: dP/dt = λ [D(P) - S(P)] where λ > 0 is the speed of adjustment parameter. The equilibrium price P* is where D(P*) = S(P*). The stability depends on the sign of dP/dP evaluated at equilibrium, which is λ [D'(P*) - S'(P*)]. For stability, this must be negative.

Second-Order Difference and Differential Equations

Higher-order equations are used to model more complex dynamics involving more lags or rates of change.

Second-Order Linear Difference Equations

The general form is yt+2 + a1 yt+1 + a0 yt = b. The solution involves finding the roots of the characteristic equation: r2 + a1 r + a0 = 0. The nature of the roots (real and distinct, real and repeated, complex) determines the stability and behavior (e.g., oscillations, explosive growth).

Example: The multiplier-accelerator model Yt = (c + v) Yt-1 - v Yt-2 leads to the characteristic equation r2 - (c + v) r + v = 0. The roots determine whether the economy experiences stable growth, cyclical fluctuations, or explosive cycles.

Second-Order Linear Differential Equations

The general form is a y'' + b y' + c y = f(t), where y' = dy/dt and y'' = d2y/dt2. The homogeneous equation a y'' + b y' + c y = 0 has a characteristic equation ar2 + br + c = 0. The roots determine the behavior:

• Real distinct roots: Exponential decay or growth.
• Real repeated roots: Exponential decay or growth with linear term.
• Complex conjugate roots: Damped or explosive oscillations.

Example: In physics, the motion of a damped harmonic oscillator is described by a second-order linear differential equation. In economics, such equations can model phenomena with cyclical behavior, like inventory cycles or business cycles, where the rate of change depends on the current level and the previous rate of change.

Comparison: Difference vs. Differential Equations

Feature	Difference Equations	Differential Equations
Time Variable	Discrete (t = 0, 1, 2, ...)	Continuous (t ∈ [0, ∞))
Core Relationship	Relates values at successive time points (e.g., yt+1, yt)	Relates a function to its derivatives (e.g., dy/dt, y)
Typical Economic Use	Macroeconomic models with annual data, Cobweb model, discrete investment adjustments.	Continuous growth models, optimal control, asset pricing, instantaneous market adjustments.
Solution Form	Involves powers (e.g., at)	Involves exponentials (e.g., eat)
Stability Condition (First-order linear)	|a| < 1 for convergence (yt+1 = a yt + b)	a < 0 for convergence (dy/dt = a y + b)

Understanding the distinction and applications of both difference and differential equations is vital for students of mathematical economics. They provide the framework for analyzing how economies evolve and respond to shocks over time, whether in discrete steps or continuous flows.