Economics, Finance, and Operations | Malone Math

The Central Idea

Economics, finance, and operations can be viewed as systems of stocks, flows, constraints, incentives, and decisions. A general modeling structure is

\[ \boxed{ \text{state} +\text{flows} +\text{prices} +\text{constraints} +\text{uncertainty} +\text{decisions} \longrightarrow \text{outcomes} }. \]

The same mathematical tools may model a household budget, a firm's production plan, a bond, a bank balance sheet, an inventory system, a service queue, or a large supply network.

Stocks and Flows

A stock is measured at a point in time. A flow is measured per unit time.

Examples of stocks include cash, debt, inventory, capital, and outstanding orders. Examples of flows include revenue per month, interest per year, production per hour, and customer arrivals per minute.

A basic accounting identity is

\[ \boxed{ \frac{dS}{dt} =\text{inflow rate}-\text{outflow rate} }. \]

Discrete Stock-Flow Accounting

Over discrete periods,

\[ \boxed{ S_{t+1} =S_t+\text{inflows}_t-\text{outflows}_t }. \]

This structure appears in cash balances, inventories, debt balances, population accounts, and many operating systems.

Economic Quantities and Units

Dimensional consistency matters in economic models just as it does in physical models. A price might be measured in dollars per unit, a wage in dollars per hour, a production rate in units per day, and a return as a dimensionless rate per period.

Multiplying compatible rates gives meaningful quantities:

\[ \left(\frac{\text{dollars}}{\text{unit}}\right) \left(\frac{\text{units}}{\text{month}}\right) = \frac{\text{dollars}}{\text{month}}. \]

Revenue

If a firm sells quantity $Q$ at price $P$, total revenue is

\[ \boxed{R=PQ}. \]

If price depends on quantity, then

\[ \boxed{R(Q)=P(Q)Q}. \]

This distinction is important because increasing quantity can lower price in some markets.

Cost

A simple cost model separates fixed and variable costs:

\[ \boxed{ C(Q)=F+V(Q) }. \]

If variable cost is approximately constant per unit,

\[ \boxed{ C(Q)=F+cQ }. \]

Profit

Profit is revenue minus cost:

\[ \boxed{ \Pi(Q)=R(Q)-C(Q) }. \]

Profit models become optimization models when the decision variable $Q$ is chosen to maximize $\Pi$ subject to capacity, demand, financing, or regulatory constraints.

Break-Even Analysis

Break-even occurs when profit equals zero:

\[ R(Q)=C(Q). \]

With constant unit price $p$, fixed cost $F$, and variable unit cost $c$,

\[ pQ=F+cQ. \]

Therefore,

\[ \boxed{ Q_{\text{BE}}=\frac{F}{p-c} } \]

provided $p>c$.

Supply and Demand

A simple market model represents demand and supply as functions of price:

\[ Q_D=D(P), \qquad Q_S=S(P). \]

A competitive equilibrium in the simplified model occurs when

\[ \boxed{Q_D=Q_S}. \]

Linear Supply and Demand

A stylized linear demand curve may be

\[ Q_D=a-bP, \qquad b>0, \]

while supply may be

\[ Q_S=c+dP, \qquad d>0. \]

Setting them equal gives the equilibrium price

\[ \boxed{ P^*=\frac{a-c}{b+d} }. \]

Elasticity

Elasticity measures proportional responsiveness. Price elasticity of demand can be written

\[ \boxed{ \varepsilon = \frac{dQ}{dP}\frac{P}{Q} }. \]

Elasticity is dimensionless, which makes it useful for comparing responsiveness across markets with different units and scales.

Marginal Quantities

Derivatives describe marginal change:

\[ \boxed{ MC(Q)=C'(Q) } \]

for marginal cost, and

\[ \boxed{ MR(Q)=R'(Q) } \]

for marginal revenue.

Profit Maximization

Since

\[ \Pi(Q)=R(Q)-C(Q), \]

an interior optimum satisfies

\[ \Pi'(Q)=0. \]

Therefore,

\[ \boxed{MR(Q)=MC(Q)} \]

under the usual differentiability and interior-solution assumptions.

Growth Models

Economic quantities such as output, revenue, prices, or capital may be modeled with exponential or compound growth:

\[ \boxed{ X(t)=X_0e^{gt} }. \]

The parameter $g$ is a continuously compounded growth rate.

Discrete Compounding

If an amount $P$ grows at rate $r$ per period for $n$ periods,

\[ \boxed{ A=P(1+r)^n }. \]

This structure appears in interest, investment returns, inflation adjustments, debt accumulation, and growth projections.

Compound Interest with Multiple Compounding Periods

If a nominal annual rate $r$ is compounded $m$ times per year for $t$ years,

\[ \boxed{ A=P\left(1+\frac{r}{m}\right)^{mt} }. \]

Continuous Compounding

In the limit of increasingly frequent compounding,

\[ \boxed{ A=Pe^{rt} }. \]

Present Value

A future payment $F$ received $n$ periods from now has present value

\[ \boxed{ PV=\frac{F}{(1+r)^n} }. \]

Discounting converts cash flows occurring at different times into a common valuation date.

Net Present Value

For cash flows $CF_t$,

\[ \boxed{ NPV= \sum_{t=0}^{T} \frac{CF_t}{(1+r)^t} }. \]

A positive NPV means the discounted inflows exceed discounted outflows under the selected discount-rate assumptions.

Annuities

The present value of an ordinary annuity paying $C$ at the end of each period for $n$ periods is

\[ \boxed{ PV= C\frac{1-(1+r)^{-n}}{r} }. \]

This structure appears in loans, leases, pensions, and fixed payment streams.

Loan Amortization

If a loan of present value $P$ is repaid with $n$ equal end-of-period payments $M$ at periodic rate $r$, then

\[ P=M\frac{1-(1+r)^{-n}}{r}. \]

Solving for payment,

\[ \boxed{ M= P\frac{r}{1-(1+r)^{-n}} }. \]

Bond Valuation

A simplified fixed-coupon bond can be valued as the present value of coupon payments plus the present value of principal:

\[ \boxed{ P= \sum_{t=1}^{n} \frac{C}{(1+y)^t} + \frac{F}{(1+y)^n} }. \]

Here $y$ is the yield per period, $C$ the coupon payment, and $F$ the face value.

Returns

A simple one-period return is

\[ \boxed{ R=\frac{V_1-V_0+D}{V_0} }. \]

Here $V_0$ and $V_1$ are beginning and ending values, and $D$ represents cash distributions received during the period.

Log Returns

A continuously compounded or log return is

\[ \boxed{ r_{\log}=\ln\left(\frac{V_1}{V_0}\right) }. \]

Log returns add across consecutive periods, which can simplify time-series analysis.

Expected Value

If an uncertain payoff $X$ takes values $x_i$ with probabilities $p_i$,

\[ \boxed{ E[X]=\sum_i p_i x_i }. \]

Expected value summarizes the probability-weighted average outcome, but does not describe the spread or downside risk.

Variance and Standard Deviation

Variance measures dispersion around the expected value:

\[ \boxed{ \operatorname{Var}(X) =E[(X-E[X])^2] }. \]

Standard deviation is

\[ \boxed{ \sigma_X=\sqrt{\operatorname{Var}(X)} }. \]

Portfolio Expected Return

For portfolio weights $w_i$ and asset returns $R_i$,

\[ R_p=\sum_i w_iR_i. \]

Therefore,

\[ \boxed{ E[R_p] =\sum_i w_iE[R_i] }. \]

Portfolio Variance

For weight vector $\mathbf{w}$ and covariance matrix $\Sigma$,

\[ \boxed{ \sigma_p^2 =\mathbf{w}^{T}\Sigma\mathbf{w} }. \]

Covariance matters because portfolio risk depends not only on individual asset volatility but also on how returns move together.

Diversification

Combining imperfectly correlated assets can reduce portfolio variance without requiring every component to have low individual volatility.

Diversification is therefore a covariance effect, not merely a matter of owning many different assets.

Risk Measures

Different decisions require different definitions of risk. Possible measures include

variance or standard deviation;
probability of loss;
downside deviation;
quantiles of loss;
expected shortfall;
drawdown;
probability of insolvency or constraint violation.

No single risk statistic captures every operational or financial concern.

Scenario Analysis

Scenario analysis evaluates a model under alternative assumptions, such as higher interest rates, lower demand, supply disruption, cost inflation, slower collections, or increased capacity.

A scenario is conditional:

\[ \boxed{ \text{if assumptions }A\text{ occur, the model produces outcome }Y }. \]

This is different from claiming that $A$ will occur.

Stress Testing

Stress testing deliberately evaluates severe but plausible conditions. Examples include sudden demand collapse, interest-rate shocks, supplier failure, liquidity pressure, or major queue congestion.

Stress tests focus on resilience and constraint failure rather than average performance.

Monte Carlo Simulation

When several inputs are uncertain, Monte Carlo simulation repeatedly samples those inputs and evaluates the model:

\[ \boxed{ \theta^{(j)} \longrightarrow Y^{(j)} \qquad j=1,\dots,N }. \]

The resulting output distribution can estimate expected outcomes, tail risk, probabilities of loss, and confidence intervals.

Inventory as a Stock

Let $I_t$ be inventory at the start of period $t$, $Q_t$ replenishment, and $D_t$ demand. A basic inventory balance is

\[ \boxed{ I_{t+1}=I_t+Q_t-D_t }. \]

More realistic models may separately track backorders, lost sales, lead times, damaged stock, and multiple locations.

Reorder-Point Models

If average demand is $d$ units per unit time and deterministic lead time is $L$, a simple reorder point is

\[ \boxed{ R=dL }. \]

When demand or lead time is uncertain, safety stock is commonly added.

Safety Stock

A stylized service-level model may use

\[ \boxed{ SS=z\sigma_L }. \]

Here $\sigma_L$ is the standard deviation of demand during lead time and $z$ is a service-factor parameter chosen from the assumed probability model.

Economic Order Quantity

In the classical EOQ model, annual demand is $D$, fixed ordering cost is $S$, and holding cost per unit per year is $H$. Total relevant annual cost is approximated by

\[ C(Q) =\frac{DS}{Q} +\frac{HQ}{2}. \]

Minimizing gives

\[ \boxed{ Q^*=\sqrt{\frac{2DS}{H}} }. \]

Assumptions Behind EOQ

The classical EOQ formula assumes conditions such as steady demand, replenishment without uncertainty, constant costs, and no stockouts.

The formula is useful because it exposes the ordering-versus-holding tradeoff, not because every inventory system literally satisfies these assumptions.

Capacity

Capacity limits how much a system can process in a given period. A utilization ratio may be written

\[ \boxed{ u=\frac{\text{actual throughput}} {\text{available capacity}} }. \]

High average utilization can be efficient in deterministic systems but can create severe congestion when arrivals and service times are variable.

Queues

Queueing models represent systems in which customers, jobs, vehicles, data packets, or orders wait for service.

Important quantities include

arrival rate $\lambda$;
service rate $\mu$;
number in system $L$;
number waiting $L_q$;
time in system $W$;
waiting time $W_q$.

Little's Law

Under broad steady-state conditions,

\[ \boxed{ L=\lambda W }. \]

The average number of items in a stable system equals throughput rate times average time spent in the system.

Simple M/M/1 Queue

For a one-server queue with Poisson arrivals and exponentially distributed service times,

\[ \boxed{ \rho=\frac{\lambda}{\mu} } \]

is server utilization.

The steady-state model requires

\[ \boxed{\lambda<\mu}. \]

M/M/1 Average Time in System

Under the M/M/1 assumptions,

\[ \boxed{ W=\frac{1}{\mu-\lambda} }. \]

As $\lambda$ approaches $\mu$, expected time in system increases sharply.

Congestion Is Nonlinear

Queueing models demonstrate why a small increase in utilization near capacity can produce a large increase in delay.

This is an important operational lesson: maximizing utilization is not always equivalent to maximizing system performance.

Production Models

Production transforms inputs into outputs. A production function may be written

\[ \boxed{ Y=f(K,L,\ldots) }. \]

Here $K$ might represent capital and $L$ labor. Additional inputs may include materials, energy, technology, or organizational capacity.

Cobb-Douglas Production

A common stylized form is

\[ \boxed{ Y=AK^\alpha L^\beta }. \]

The parameter $A$ summarizes productivity, while $\alpha$ and $\beta$ determine output responsiveness to the two inputs under the model.

Returns to Scale

For the Cobb-Douglas model,

\[ f(cK,cL) =c^{\alpha+\beta}f(K,L). \]

Therefore,

$\alpha+\beta=1$: constant returns to scale;
$\alpha+\beta>1$: increasing returns to scale;
$\alpha+\beta<1$: decreasing returns to scale.

Linear Programming

Many operating decisions allocate scarce resources. A linear program has the form

\[ \boxed{ \text{maximize } \mathbf{c}^{T}\mathbf{x} } \]

subject to

\[ \boxed{ A\mathbf{x}\le\mathbf{b}, \qquad \mathbf{x}\ge0. } \]

The decision vector $\mathbf{x}$ may represent production quantities, shipments, staffing, or investment allocations.

Shadow Prices

In constrained optimization, a shadow price measures the marginal value of relaxing a binding constraint.

Operationally, this can estimate the value of one additional unit of scarce labor, machine time, storage, capital, or transport capacity.

Integer Decisions

Some decisions cannot be fractional. A warehouse is either opened or not, a vehicle is assigned or not, and a worker shift may require whole-person counts.

Integer programming adds constraints such as

\[ \boxed{x_i\in\{0,1\}} \]

\[ \boxed{x_i\in\mathbb{Z}}. \]

Scheduling

Scheduling models assign activities to times and resources while respecting precedence, capacity, and timing constraints.

Common objectives include minimizing completion time, lateness, overtime, or setup cost.

Forecasting

Operational planning frequently depends on forecasts of demand, sales, arrivals, prices, or workloads.

A simple moving average forecast is

\[ \boxed{ \hat y_{t+1} = \frac{1}{m} \sum_{j=0}^{m-1}y_{t-j} }. \]

Exponential Smoothing

Simple exponential smoothing updates a forecast using

\[ \boxed{ \hat y_{t+1} = \alpha y_t +(1-\alpha)\hat y_t }. \]

The parameter $0<\alpha<1$ determines how strongly the forecast responds to recent observations.

Forecast Error

If $y_t$ is observed and $\hat y_t$ predicted, forecast error is

\[ \boxed{ e_t=y_t-\hat y_t }. \]

Forecast evaluation should examine both average error and the distribution of errors.

Mean Absolute Error

One common forecast metric is

\[ \boxed{ MAE= \frac{1}{n} \sum_{t=1}^{n}|e_t| }. \]

Root Mean Square Error

Another is

\[ \boxed{ RMSE= \sqrt{ \frac{1}{n} \sum_{t=1}^{n}e_t^2 } }. \]

Squaring errors gives larger errors more influence on the metric.

Forecasts and Decisions Are Different

A forecast estimates what may occur. A decision model determines what action to take given forecasts, costs, constraints, and risk preferences.

Improving forecast accuracy does not automatically produce better decisions if the decision model is poorly specified.

Decision Trees

Decision trees organize sequential choices and uncertain outcomes. A chance node can be evaluated with expected value:

\[ \boxed{ EV=\sum_i p_iV_i }. \]

Decision nodes compare available actions using the selected objective.

Expected Utility

When decision makers care about risk as well as expected monetary value, a utility function $u(x)$ can represent preferences:

\[ \boxed{ E[u(X)] = \sum_i p_i u(x_i) }. \]

Different utility functions represent different attitudes toward risk.

Dynamic Decisions

Many operational and financial decisions are repeated through time. A state $x_t$ changes according to

\[ \boxed{ x_{t+1} =f(x_t,u_t,w_t) }. \]

Here $u_t$ is the decision and $w_t$ an uncertain disturbance.

Dynamic Programming

Dynamic programming solves sequential decisions by decomposing them into smaller state-dependent decisions. A finite-horizon Bellman recursion may be written

\[ \boxed{ V_t(x) = \max_u \left[ r(x,u) +E[V_{t+1}(x')] \right] }. \]

This framework appears in inventory control, investment, maintenance, scheduling, and resource allocation.

Cash-Flow Dynamics

Let $B_t$ be a cash balance, $R_t$ receipts, $E_t$ expenditures, and $F_t$ net financing flow. Then

\[ \boxed{ B_{t+1}=B_t+R_t-E_t+F_t }. \]

Liquidity models focus on whether the balance remains above required operating or covenant thresholds.

Debt Dynamics

A simple debt balance with interest rate $r$, new borrowing $B_t$, and repayment $P_t$ is

\[ \boxed{ D_{t+1} =(1+r)D_t+B_t-P_t }. \]

This stock-flow model makes clear that debt can rise even without new borrowing when interest exceeds repayment.

Working Capital

Operations and finance interact through receivables, inventory, payables, and cash. A simplified cash conversion cycle is

\[ \boxed{ CCC = DIO+DSO-DPO }. \]

Here DIO is days inventory outstanding, DSO days sales outstanding, and DPO days payables outstanding.

Economic Models Are Behavioral Models

Unlike many physical systems, economic outcomes can depend on expectations, incentives, strategic responses, institutions, and changing behavior.

A policy or operational rule may change the behavior being modeled, so parameters estimated under one regime may not remain stable under another.

Feedback

Economic systems contain feedback loops. For example,

\[ \text{price} \longrightarrow \text{demand} \longrightarrow \text{inventory} \longrightarrow \text{reordering} \longrightarrow \text{supply} \longrightarrow \text{price}. \]

Delays within feedback loops can create oscillation, overshoot, and instability.

The Bullwhip Effect

In supply chains, small changes in consumer demand can produce larger order fluctuations upstream when forecasting, batching, lead times, or local decision rules amplify the signal.

This is an example of system behavior generated by feedback and delay rather than by a large external shock.

Game-Theoretic Models

When outcomes depend on choices made by several strategic decision makers, a game-theoretic model may be appropriate.

Each participant has actions and payoffs, and an equilibrium describes a set of strategies consistent with the model's assumptions about incentives and information.

Agent-Based Economic Models

Agent-based models represent heterogeneous households, firms, traders, institutions, or consumers explicitly.

Such models can capture network effects, bounded rules, local interactions, and adaptation that aggregate equations may suppress.

System Dynamics Models

System dynamics represents stocks, flows, feedback loops, and delays using coupled difference or differential equations.

It is especially useful when operational behavior emerges from accumulation and delayed responses.

Parameter Estimation

Economic and operational models frequently estimate parameters from observed data. A regression model may be

\[ \boxed{ y_i =\beta_0+\beta_1x_{i1}+\cdots+\beta_px_{ip} +\varepsilon_i }. \]

Estimated coefficients describe relationships conditional on the model, variables, sampling process, and assumptions.

Prediction and Causation

A model can predict well without identifying causal effects. Conversely, a causal model may prioritize identification over raw predictive accuracy.

The modeling objective should distinguish whether the task is forecasting, explanation, policy evaluation, or decision optimization.

Endogeneity

In economic systems, an explanatory variable may be jointly determined with the outcome or correlated with omitted influences.

This can make an ordinary regression coefficient a poor estimate of causal effect even if the fitted relationship is statistically strong.

Model Calibration

Some economic and operational models are calibrated by selecting parameters so model-generated moments or trajectories match selected observations.

Calibration is not the same as validation. A calibrated model should still be evaluated against independent patterns relevant to its intended use.

Backtesting

Financial or forecasting models may be evaluated using historical periods not used for fitting.

Backtesting can reveal instability, but historical success does not guarantee future performance because market structure, policy, technology, and behavior can change.

Nonstationarity

A process is nonstationary when its statistical properties change over time. Economic and financial series often experience trends, regime shifts, structural breaks, or changing volatility.

Models that assume fixed relationships should therefore be tested for stability across periods.

Sensitivity Analysis

Sensitivity analysis asks how outputs change when assumptions or parameters change.

Important sensitivities may include

profit sensitivity to price or volume;
NPV sensitivity to discount rate;
liquidity sensitivity to collections delay;
inventory cost sensitivity to demand volatility;
queue delay sensitivity to utilization;
portfolio risk sensitivity to correlations;
capacity value sensitivity to demand growth.

Uncertainty Analysis

Uncertainty can arise from demand, costs, interest rates, returns, supply reliability, service times, policy, behavior, and model form.

Decision-grade models should distinguish what is measured, what is estimated, what is assumed, and what remains structurally uncertain.

Robust Decisions

A robust decision performs acceptably across a range of plausible conditions rather than being optimal only under one precise forecast.

Robust modeling is particularly valuable when uncertainty is large and the cost of failure is asymmetric.

Example: Break-Even Quantity

Suppose

\[ F=\$120{,}000, \qquad p=\$50, \qquad c=\$30. \]

Then

\[ Q_{\text{BE}} = \frac{120{,}000}{50-30} \] \[ =\boxed{6{,}000\text{ units}}. \]

Example: Present Value

A payment of $\$10{,}000$ received in three years at an annual discount rate of $6\%$ has present value

\[ PV= \frac{10{,}000}{(1.06)^3} \] \[ \boxed{PV\approx\$8{,}396.19}. \]

Example: Expected Payoff

A project has a $0.6$ probability of earning $\$100{,}000$ and a $0.4$ probability of losing $\$20{,}000$. Its expected payoff is

\[ E[X] =(0.6)(100{,}000) +(0.4)(-20{,}000) \] \[ \boxed{E[X]=\$52{,}000}. \]

The expected value does not show the probability or severity of downside by itself.

Example: EOQ

Suppose annual demand is $D=10{,}000$ units, ordering cost is $S=\$40$ per order, and holding cost is $H=\$2$ per unit per year. Then

\[ Q^* = \sqrt{ \frac{2(10{,}000)(40)}{2} } \] \[ = \sqrt{400{,}000} \] \[ \boxed{Q^*\approx632\text{ units}}. \]

Example: Little's Law

A process completes an average of $20$ jobs per hour, and a job spends an average of $0.75$ hour in the system. Then

\[ L=\lambda W =(20)(0.75) \] \[ \boxed{L=15\text{ jobs}}. \]

Example: M/M/1 Delay

Suppose arrivals occur at

\[ \lambda=8\text{ customers/hour} \]

and service capacity is

\[ \mu=10\text{ customers/hour}. \]

Then

\[ W=\frac{1}{10-8} =\boxed{0.5\text{ hour}}. \]

Average time in system is $30$ minutes under the M/M/1 assumptions.

Verification

Verification asks whether the model has been implemented and solved correctly. Useful checks include

stock-flow balances;
cash-flow reconciliation;
dimensional consistency;
constraint satisfaction;
comparison with analytical special cases;
simulation convergence;
independent recalculation of key outputs.

Validation

Validation asks whether the model is adequate for its intended decision.

A demand model may fit historical sales but fail under a new pricing regime. A queue model may reproduce average delay but miss peak congestion. A financial model may match expected returns while underestimating tail losses.

Validation should therefore focus on the outcomes and conditions that matter operationally.

Common Error: Mixing Stocks and Flows

A balance sheet amount measured in dollars cannot be added directly to a monthly cash-flow rate without accounting for the time interval.

Unit discipline prevents many accounting and operational modeling errors.

Common Error: Treating a Forecast as a Certainty

A point forecast hides uncertainty. Capacity, inventory, financing, and investment decisions should consider forecast error and alternative scenarios when those uncertainties are material.

Common Error: Optimizing the Wrong Objective

Maximizing revenue is not the same as maximizing profit. Maximizing utilization is not the same as minimizing delay. Minimizing unit cost is not necessarily the same as maximizing enterprise value.

The objective function must correspond to the actual decision.

Common Error: Ignoring Constraints

A mathematically optimal plan may be infeasible because of labor, capacity, capital, regulation, inventory, timing, or contractual constraints.

Constraints are part of the model, not an afterthought.

Common Error: Using Average Demand in a Congested System

Average demand can hide variability. Queues depend strongly on the distribution of arrivals and service times, particularly near capacity.

A deterministic average-flow model can therefore materially understate waiting and required buffer capacity.

Common Error: Assuming Historical Relationships Are Permanent

Economic and financial systems change through policy, technology, competition, institutions, expectations, and behavior.

Parameter stability should be tested rather than assumed.

Common Error: Reporting Expected Value Without Downside

Two decisions can have identical expected value and radically different loss profiles.

When downside matters, expected value should be supplemented with probabilities, quantiles, stress scenarios, or other relevant risk measures.

A Reliable Modeling Procedure

Define the economic, financial, or operational decision.
Identify stocks, flows, prices, resources, constraints, and time scales.
Separate measured data from assumptions and estimated parameters.
Write accounting identities before adding behavioral or optimization relationships.
Represent uncertainty in demand, prices, rates, returns, or service conditions when material.
Select an analytical, statistical, optimization, simulation, or hybrid model.
Verify balances, units, constraints, and numerical implementation.
Calibrate parameters using appropriate data without confusing fit with causation.
Validate against relevant out-of-sample behavior or independent benchmarks.
Perform sensitivity, scenario, stress, and uncertainty analysis before making resource commitments.

Practice 1: Break-Even Quantity

A product sells for $\$80$ per unit, has variable cost of $\$50$ per unit, and fixed cost of $\$150{,}000$. Find the break-even quantity.

Practice 2: Compound Growth

An investment of $\$5{,}000$ grows at $7\%$ per year for $5$ years with annual compounding. Find the ending value.

Practice 3: Present Value

Find the present value of $\$20{,}000$ received four years from now using a $5\%$ annual discount rate.

Practice 4: Expected Value

A decision has a $0.7$ probability of producing a gain of $\$60{,}000$ and a $0.3$ probability of producing a loss of $\$30{,}000$. Find the expected payoff.

Practice 5: Inventory Balance

Beginning inventory is $500$ units, replenishment is $300$ units, and demand is $620$ units. Find ending inventory using the basic stock-flow model.

Practice 6: Reorder Point

Average demand is $40$ units per day and lead time is $6$ days. Find the deterministic reorder point before adding safety stock.

Practice 7: EOQ

Annual demand is $D=12{,}000$ units, ordering cost is $S=\$25$ per order, and annual holding cost is $H=\$3$ per unit. Find the classical EOQ.

Practice 8: Little's Law

A service process handles $30$ customers per hour, and customers spend an average of $0.20$ hour in the system. Find the average number of customers in the system.

Practice 9: M/M/1 Queue

Customers arrive at $\lambda=9$ per hour and the server processes $\mu=12$ per hour. Find the utilization $\rho$ and the expected time in system $W$.

Practice 10: Choose the Better Model

A distribution center operates near full capacity. Managers use only average daily demand to estimate staffing, even though arrivals are highly variable and long queues form during peaks. Explain why an average-flow model may be inadequate and identify a better modeling direction.

Practice Answers

Practice 1

\[ Q_{\text{BE}} = \frac{150{,}000}{80-50} \] \[ =\boxed{5{,}000\text{ units}}. \]

Practice 2

\[ A=5000(1.07)^5 \] \[ \boxed{A\approx\$7{,}012.76}. \]

Practice 3

\[ PV= \frac{20{,}000}{(1.05)^4} \] \[ \boxed{PV\approx\$16{,}454.05}. \]

Practice 4

\[ E[X] =(0.7)(60{,}000) +(0.3)(-30{,}000) \] \[ =42{,}000-9{,}000 \] \[ \boxed{E[X]=\$33{,}000}. \]

Practice 5

\[ I_{t+1}=500+300-620 \] \[ \boxed{I_{t+1}=180\text{ units}}. \]

Practice 6

\[ R=dL =(40)(6) \] \[ \boxed{R=240\text{ units}}. \]

Practice 7

\[ Q^* = \sqrt{\frac{2DS}{H}} \] \[ = \sqrt{ \frac{2(12{,}000)(25)}{3} } \] \[ = \sqrt{200{,}000} \] \[ \boxed{Q^*\approx447\text{ units}}. \]

Practice 8

\[ L=\lambda W =(30)(0.20) \] \[ \boxed{L=6\text{ customers}}. \]

Practice 9

\[ \rho=\frac{\lambda}{\mu} =\frac{9}{12} =\boxed{0.75}. \]

The utilization is $75\%$.

\[ W=\frac{1}{\mu-\lambda} =\frac{1}{12-9} =\frac{1}{3}\text{ hour}. \] \[ \boxed{W=20\text{ minutes}}. \]

Practice 10

Average flow does not capture arrival variability or the nonlinear growth of waiting near capacity. A queueing model or discrete-event simulation should be used to represent the arrival distribution, service-time distribution, staffing levels, capacity, and peak-period congestion.

What to Remember

economic and operational models begin with stocks, flows, accounting identities, prices, and constraints;
revenue, cost, profit, supply, demand, and elasticity translate economic relationships into mathematical form;
discounting and compounding connect cash flows across time;
expected value must be separated from risk and downside exposure;
inventory models balance replenishment, demand, holding costs, and service requirements;
queueing models show that congestion rises sharply as uncertain systems approach capacity;
optimization allocates scarce resources subject to explicit constraints;
forecasting estimates future conditions, while decision models determine actions;
economic relationships may change when behavior, policy, technology, or institutions change;
verification, validation, sensitivity, stress testing, and uncertainty analysis are necessary before models support consequential decisions.

Where We Go Next

Economics, finance, and operations show how resources, incentives, uncertainty, and capacity interact across time. Many operational systems also depend on the physical movement of goods, vehicles, people, information, and services across connected networks.

The next lesson, Networks, Transportation, and Logistics, develops models of nodes and links, routing, flow, shortest paths, capacity, congestion, facility location, vehicle movement, supply networks, and logistical optimization.