The Central Idea
A static model describes relationships among quantities without explicitly modeling how those quantities evolve through time.
For example,
\[ Q=100-2P \]may describe the relationship between price \(P\) and quantity demanded \(Q\) under a given set of conditions.
A dynamic model describes how the state of a system changes.
For example,
\[ Q_{t+1}=Q_t+\Delta Q_t \]explicitly relates a future state to a current state.
The key distinction is whether change through time is part of the model itself.
A Snapshot versus a Process
A useful first distinction is:
- Static model: What relationships hold now or under a specified condition?
- Dynamic model: How does the system move from one state to another?
A static model is like a photograph.
A dynamic model is like a sequence showing how the scene changes.
What Makes a Model Static?
A static model does not explicitly represent the path of change over time.
It may describe:
- an equilibrium condition;
- a geometric relationship;
- a cost relationship at a chosen production level;
- a balance among quantities;
- a feasible region;
- an optimization problem at one planning stage.
Time may appear in the surrounding story without being a modeled variable.
Examples of Static Models
The area of a circle,
\[ A=\pi r^2, \]is static.
A cost function,
\[ C(q)=500+8q, \]can also be static when it describes cost at a given production quantity.
A resource constraint such as
\[ 3x+5y\leq 200 \]is static when it describes allowable combinations of \(x\) and \(y\) for one decision period.
Static Does Not Mean Constant
A static model can contain variables.
Consider
\[ y=3x+4. \]The value of \(y\) changes as \(x\) changes, but the model is still static if it does not describe how \(x\) or \(y\) evolves over time.
Static means that temporal evolution is not part of the model structure.
Equilibrium Models
Static models often describe equilibrium.
Suppose demand and supply are modeled by
\[ Q_d=100-2P \]and
\[ Q_s=10+P. \]Equilibrium requires
\[ Q_d=Q_s. \]Therefore,
\[ 100-2P=10+P. \]This determines the equilibrium price without describing how the market moves toward equilibrium.
Static Optimization
A one-period optimization model is often static.
Suppose profit is
\[ P(x,y)=40x+55y \]subject to
\[ 2x+3y\leq 180, \] \[ 4x+y\leq 160, \] \[ x\geq 0,\qquad y\geq 0. \]The model asks which feasible production plan is best under the current assumptions. It does not describe how production capacity changes over time.
What Makes a Model Dynamic?
A dynamic model contains an explicit rule for change.
The model identifies a current state and specifies how that state evolves.
A general discrete-time form is
\[ x_{t+1}=F(x_t,u_t), \]where \(x_t\) is the current state and \(u_t\) represents inputs or decisions.
A general continuous-time form is
\[ \frac{dx}{dt}=f(x,t). \]The State of a System
The state summarizes the information needed to describe the system at a particular time.
Examples include:
- inventory currently on hand;
- account balance;
- population size;
- position and velocity;
- temperature;
- number of customers in a queue.
Dynamic models specify how the state changes.
State Variables
A state variable records a quantity needed to determine future evolution.
For inventory,
\[ I_t \]may represent units on hand at time \(t\).
For motion, position and velocity may both be required:
\[ x(t),\qquad v(t). \]The correct state description depends on the mechanism.
Stocks and Flows
Dynamic modeling often distinguishes stocks from flows.
A stock is an accumulated quantity.
Examples include:
- money in an account;
- water in a reservoir;
- inventory in a warehouse;
- population in a region.
A flow changes a stock.
The Basic Stock-Flow Relationship
In discrete time,
\[ \boxed{ \text{new stock} = \text{old stock} + \text{inflow} - \text{outflow} }. \]For inventory,
\[ I_{t+1}=I_t+Q_t-D_t. \]For cash,
\[ B_{t+1}=B_t+C_t-O_t, \]where \(C_t\) is cash received and \(O_t\) is cash paid out.
Discrete-Time Models
A discrete-time model updates the system at separate time points.
Time might be measured in:
- days;
- months;
- years;
- production cycles;
- transactions;
- generations.
The model often uses subscripts such as
\[ x_0,x_1,x_2,\ldots. \]Recurrence Relations
A recurrence relation specifies a future value from previous values.
For constant additive growth,
\[ x_{t+1}=x_t+d. \]For constant proportional growth,
\[ x_{t+1}=(1+r)x_t. \]Recurrences make the update mechanism explicit.
Example: Savings Account
Suppose an account earns \(4\%\) per year and no deposits or withdrawals occur.
The dynamic model is
\[ A_{t+1}=1.04A_t. \]If
\[ A_0=1000, \]then
\[ A_1=1040, \] \[ A_2=1081.60, \]and so on.
Closed Form and Dynamic Form
The same process can sometimes be represented both dynamically and directly.
The recurrence
\[ A_{t+1}=1.04A_t \]with initial value \(A_0\) has the closed form
\[ A_t=A_0(1.04)^t. \]The closed form gives a direct value at time \(t\). The recurrence displays the step-by-step mechanism.
Continuous-Time Models
A continuous-time model treats time as continuously varying.
A common form is a differential equation:
\[ \frac{dx}{dt}=f(x,t). \]Rather than specifying the next step directly, the model specifies an instantaneous rate of change.
Example: Constant Velocity
If velocity is constant,
\[ \frac{dx}{dt}=v. \]With initial position
\[ x(0)=x_0, \]the solution is
\[ \boxed{x(t)=x_0+vt}. \]The differential equation expresses the dynamic mechanism. The solution gives the resulting trajectory.
Example: Exponential Growth
Continuous proportional growth is modeled by
\[ \frac{dP}{dt}=rP. \]The rate of change is proportional to the current population.
With
\[ P(0)=P_0, \]the solution is
\[ P(t)=P_0e^{rt}. \]Initial Conditions
A dynamic law often does not determine one unique trajectory until an initial condition is supplied.
If
\[ \frac{dx}{dt}=5, \]then many functions have derivative \(5\):
\[ x(t)=5t+C. \]If
\[ x(0)=20, \]then
\[ C=20 \]and
\[ x(t)=20+5t. \]Dynamic Models Need a Time Scale
A rate only makes sense relative to a time unit.
A growth rate of \(2\%\) per month is not the same as \(2\%\) per year.
Likewise,
\[ x_{t+1}=1.02x_t \]cannot be interpreted correctly until we know what one time step means.
Time resolution is part of the model.
Static Models Can Be Extracted from Dynamic Models
A dynamic model may have states that do not change.
Suppose
\[ x_{t+1}=F(x_t). \]An equilibrium \(x^*\) satisfies
\[ x^*=F(x^*). \]This equation is static even though it comes from a dynamic model.
It identifies a state that remains unchanged under the update rule.
Equilibrium in Continuous Time
If a dynamic model is
\[ \frac{dx}{dt}=f(x), \]equilibrium occurs where
\[ \frac{dx}{dt}=0. \]Therefore, equilibrium values satisfy
\[ \boxed{f(x)=0}. \]A dynamic model can therefore generate a static equilibrium condition.
Example: Tank Volume
Suppose water enters a tank at rate \(r_{\text{in}}\) and leaves at rate \(r_{\text{out}}\).
A continuous dynamic model is
\[ \frac{dV}{dt} = r_{\text{in}}-r_{\text{out}}. \]If
\[ r_{\text{in}}=r_{\text{out}}, \]then
\[ \frac{dV}{dt}=0. \]The tank volume is then at a static equilibrium.
Example: Inventory
A static inventory model might ask how much inventory is required to meet a known demand:
\[ I\geq D. \]A dynamic inventory model follows inventory through time:
\[ I_{t+1}=I_t+Q_t-D_t. \]The second model can represent accumulation, depletion, replenishment, and stockouts.
Example: Debt
A static debt calculation might find the interest due on a balance:
\[ I=rB. \]A dynamic debt model tracks the balance:
\[ B_{t+1}=(1+r)B_t-P_t. \]The payment \(P_t\) and interest accumulation determine the next state.
Example: Population
A static population model might compare population with capacity:
\[ P\leq K. \]A dynamic model might use
\[ P_{t+1}=P_t+rP_t \]or
\[ \frac{dP}{dt}=rP. \]These models describe how population changes rather than only its current level.
Example: Motion
A static geometric model might identify points satisfying
\[ x^2+y^2=25. \]A dynamic model of motion along a path requires position as a function of time:
\[ x=x(t),\qquad y=y(t). \]Velocity can then be modeled by
\[ \frac{dx}{dt},\qquad \frac{dy}{dt}. \]Feedback
Dynamic systems often contain feedback, meaning that the current state influences future change.
In
\[ P_{t+1}=(1+r)P_t, \]a larger current population produces a larger absolute increase.
The state affects its own future development.
Positive Feedback
Positive feedback reinforces change.
For example,
\[ \frac{dP}{dt}=rP,\qquad r>0, \]produces growth that accelerates in absolute magnitude as \(P\) increases.
More population produces more growth, which produces still more population.
Negative Feedback
Negative feedback counteracts deviations.
A simple adjustment model is
\[ \frac{dx}{dt}=k(T-x), \]where \(T\) is a target level and \(k>0\).
If \(x
If \(x>T\), the derivative is negative.
The system tends to move toward \(T\).
Lagged Effects
Dynamic models can represent delays.
For example,
\[ x_{t+1}=a x_t+b x_{t-1} \]allows the next state to depend on both the current and previous states.
Such lagged relationships are impossible to represent faithfully with a purely static snapshot.
Path Dependence
In some systems, the final outcome depends not only on current conditions but also on the path taken to reach them.
Dynamic models are useful when:
- past states affect current possibilities;
- accumulation matters;
- switching costs matter;
- delays matter;
- history changes future behavior.
A static model may miss these effects.
Time-Varying Parameters
Dynamic models can allow parameters themselves to change through time.
Instead of
\[ x_{t+1}=(1+r)x_t, \]we may write
\[ x_{t+1}=(1+r_t)x_t. \]The growth factor now depends on the period.
This adds flexibility but also requires more information.
Dynamic Constraints
Constraints may also change through time.
For example,
\[ 0\leq I_t\leq K_t \]allows warehouse capacity \(K_t\) to vary by period.
A dynamic model may therefore combine state-update equations with time-dependent constraints.
Dynamic Models Can Be Deterministic or Stochastic
Static versus dynamic is a different distinction from deterministic versus stochastic.
A deterministic dynamic model might be
\[ x_{t+1}=1.05x_t. \]A stochastic dynamic model might be
\[ x_{t+1}=1.05x_t+\varepsilon_{t+1}. \]Both evolve through time. Only the second explicitly includes randomness.
Four Possible Combinations
The two distinctions create four broad possibilities:
- static and deterministic;
- static and stochastic;
- dynamic and deterministic;
- dynamic and stochastic.
These categories describe different structural features of a model.
Static and Deterministic Example
\[ C(q)=500+8q. \]The model gives one cost for each \(q\) and does not track change through time.
Static and Stochastic Example
\[ Y=\beta_0+\beta_1x+\varepsilon. \]The model represents random variation in \(Y\), but it need not describe a time-evolving process.
Dynamic and Deterministic Example
\[ P_{t+1}=1.03P_t. \]The state changes through time, and the same current state always gives the same next state.
Dynamic and Stochastic Example
\[ P_{t+1}=1.03P_t+\varepsilon_{t+1}. \]The state changes through time and includes random variation.
When a Static Model Is Enough
A static model may be sufficient when:
- the question concerns one period or one condition;
- the path of adjustment is irrelevant;
- equilibrium is the main object of interest;
- time effects are negligible;
- the model is intended as a first approximation.
Do not add dynamics merely because time exists in the real world.
When a Dynamic Model Is Necessary
A dynamic model becomes important when:
- stocks accumulate or deplete;
- growth or decay matters;
- feedback matters;
- delays matter;
- past states influence future states;
- the path matters, not just the endpoint;
- policies or decisions are repeated over time.
Common Error: Calling Every Function of Time Dynamic
The appearance of a time symbol does not automatically make a model meaningfully dynamic.
For example,
\[ C(t)=100+5t \]may simply assign cost to a time value.
A stronger dynamic interpretation appears when the model specifies how the current state generates future states or rates of change.
Common Error: Ignoring the Initial State
A rule of change may be incomplete without a starting condition.
The recurrence
\[ x_{t+1}=2x_t \]does not determine a unique sequence until \(x_0\) is known.
Different initial states produce different trajectories.
Common Error: Mixing Stocks and Flows
A stock is measured at a point in time. A flow is measured per unit of time.
If inventory is measured in units and sales are measured in units per day, then the time interval must be included when updating inventory.
For a one-day step,
\[ I_{t+1}=I_t-D_t \]is appropriate when \(D_t\) means units sold during that day.
Common Error: Using the Wrong Time Step
A monthly update rule should not be applied once per day without conversion.
If
\[ x_{t+1}=1.02x_t \]means \(2\%\) growth per month, then applying it \(365\) times per year changes the model dramatically.
The meaning of one update step must be explicit.
Common Error: Assuming Equilibrium Explains the Path
Knowing the equilibrium state does not tell us how quickly the system approaches it, whether it oscillates, or whether it is stable.
Those questions require a dynamic model.
A static equilibrium condition describes a destination, not necessarily the journey.
Common Error: Adding Dynamics Without a Mechanism
A dynamic model should explain why the state changes.
Writing
\[ x_{t+1}=1.7x_t+4 \]is not automatically meaningful.
The coefficients and update structure should come from a mechanism, data, calibration, or a justified approximation.
A Reliable Procedure
- State the modeling question.
- Ask whether the path through time matters.
- Identify the state variables if the system evolves.
- Identify inflows, outflows, growth, decay, or other mechanisms of change.
- Choose a discrete or continuous time representation.
- Specify the time unit or update interval.
- Write the state-update rule or rate equation.
- Supply initial conditions.
- Add constraints, parameters, and inputs.
- Check units and dimensional consistency.
- Examine equilibrium states when useful.
- Compare the model's behavior with the real system.
Practice 1: Static or Dynamic?
Classify
\[ C(q)=200+12q. \]Assume the formula describes cost at a chosen production quantity and does not track production through time.
Practice 2: Inventory Update
Inventory begins a day at \(500\) units. During the day, \(120\) units arrive and \(160\) units are sold.
Write the state update and find the ending inventory.
Practice 3: Recurrence
A population grows by \(4\%\) each year.
Write a discrete-time dynamic model.
Practice 4: Initial Condition
A quantity changes according to
\[ \frac{dx}{dt}=3 \]and begins at
\[ x(0)=10. \]Find \(x(t)\).
Practice 5: Equilibrium
A discrete system follows
\[ x_{t+1}=0.8x_t+10. \]Find the equilibrium \(x^*\).
Practice 6: Stock or Flow?
Classify each quantity as a stock or a flow:
- water currently in a reservoir;
- liters entering the reservoir per minute.
Practice 7: Static or Dynamic Question?
Which requires a dynamic model?
- What production plan maximizes profit this month?
- How does inventory change over the next twelve months?
Practice 8: Continuous Growth
Write a continuous-time model in which population growth is proportional to current population.
Practice 9: Identify the Error
A model uses a \(5\%\) monthly growth factor but applies the update every day.
Explain the problem.
Practice 10: Deterministic or Stochastic?
Compare
\[ x_{t+1}=1.02x_t \]with
\[ x_{t+1}=1.02x_t+\varepsilon_{t+1}. \]Both are dynamic. Which one is stochastic?
Practice Answers
Practice 1
The model is static because it describes a relationship between cost and production quantity without modeling temporal evolution.
Practice 2
\[ I_{\text{new}} = I_{\text{old}} + \text{arrivals} - \text{sales}. \] \[ I_{\text{new}}=500+120-160. \] \[ \boxed{I_{\text{new}}=460}. \]Practice 3
\[ \boxed{P_{t+1}=1.04P_t}. \]Practice 4
Integrating gives
\[ x(t)=3t+C. \]Since \(x(0)=10\),
\[ C=10. \]Therefore,
\[ \boxed{x(t)=10+3t}. \]Practice 5
At equilibrium,
\[ x^*=0.8x^*+10. \]Therefore,
\[ 0.2x^*=10 \]and
\[ \boxed{x^*=50}. \]Practice 6
- Water currently in the reservoir is a stock.
- Liters entering per minute is a flow.
Practice 7
The second question requires a dynamic model because it asks how inventory evolves over multiple periods.
Practice 8
\[ \boxed{\frac{dP}{dt}=rP}. \]Practice 9
The update frequency does not match the parameter's time scale. A monthly growth factor cannot be applied daily without converting the rate or redefining the model.
Practice 10
The model
\[ \boxed{x_{t+1}=1.02x_t+\varepsilon_{t+1}} \]is stochastic because it explicitly includes a random term.
What to Remember
- A static model describes relationships without explicitly modeling temporal evolution.
- A dynamic model describes how a system changes from one state to another.
- Static does not mean constant; static models can contain variables.
- Dynamic models require a rule of change.
- State variables summarize the current condition of the system.
- Stocks accumulate; flows change stocks.
- Discrete models use updates such as \(x_{t+1}=F(x_t)\).
- Continuous models often use differential equations.
- Initial conditions determine which trajectory a dynamic law follows.
- Equilibrium is a static condition that can be extracted from a dynamic model.
- Static/dynamic and deterministic/stochastic are separate classifications.
- The time scale must match the model's rates and update rules.
The central modeling question is: Do we need only the relationship among quantities, or do we need the mechanism by which the system changes?
Where We Go Next
Once the structure and time behavior of a model have been selected, the next task is connecting the model to evidence.
Parameters may need to be estimated from observations, and the resulting model must be checked against data or known behavior.
The next lesson, Calibration and Validation, develops the distinction between fitting a model and testing whether it is useful beyond the information used to construct it.