Demand, Supply, and Surplus – Key Concepts and Calculations

Definitions

Demand – The relationship between the price of a good and the quantity that consumers are willing and able to purchase at that price, usually expressed as a function $Q_d=f_d(p)$ or $p=f_d(Q_d)$.

Supply – The relationship between the price of a good and the quantity that producers are willing and able to offer for sale, expressed as $Q_s=f_s(p)$ or $p=f_s(Q_s)$.

Market equilibrium – The point at which the quantity demanded equals the quantity supplied, $Q_d=Q_s$, resulting in a unique equilibrium price $p^{\ast }$ and quantity $Q^{\ast }$.

Consumer surplus (CS) – The monetary benefit consumers receive when they pay a price lower than the maximum price they are willing to pay. Graphically, it is the area between the demand curve and the equilibrium price, above the price line.

Producer surplus (PS) – The monetary benefit producers receive when they sell at a price higher than the minimum price they are willing to accept. Graphically, it is the area between the equilibrium price and the supply curve, below the price line.

Total surplus (TS) – The sum of consumer and producer surplus, $TS=CS+PS$, representing the overall welfare generated by the market.

Demand Function

A linear demand function can be written as
$$p=a-bQ,$$
where $a$ is the intercept (maximum price consumers are willing to pay when quantity is zero) and $b$ is the slope (rate at which price decreases as quantity increases).

Example:
$p=30-2Q$

• Intercept $a=30$
• Slope $b=2$

The demand curve is downward sloping, reflecting the law of demand.

Supply Function

A linear supply function is typically expressed as
$$p=c+dQ,$$
where $c$ is the intercept (minimum price producers require when quantity is zero) and $d$ is the slope (rate at which price increases as quantity increases).

Example:
$p=5+Q$

• Intercept $c=5$
• Slope $d=1$

The supply curve is upward sloping, reflecting the law of supply.

Market Equilibrium

To find equilibrium, set demand equal to supply:
$$a-bQ=c+dQ.$$
Solve for $Q^{\ast }$:
$$Q^{\ast }=\frac{a-c}{b+d}.$$
Substitute $Q^{\ast }$ back into either function to find $p^{\ast }$.

Graphical Representation

The blue shaded area represents consumer surplus; the red shaded area represents producer surplus.

Consumer Surplus

Mathematically,
$$CS={\int }_0^{Q^{\ast }}(p_d(Q)-p^{\ast })dQ.$$
For a linear demand $p=a-bQ$, the integral simplifies to
$$CS=\frac{1}{2}(a-p^{\ast })Q^{\ast }.$$

Producer Surplus

Similarly,
$$PS={\int }_0^{Q^{\ast }}(p^{\ast }-p_s(Q))dQ.$$
For a linear supply $p=c+dQ$,
$$PS=\frac{1}{2}(p^{\ast }-c)Q^{\ast }.$$

Total Surplus

$$TS=CS+PS=\frac{1}{2}(a-c)Q^{\ast }.$$

The total surplus is the area of the triangle formed by the demand and supply curves between $Q=0$ and $Q=Q^{\ast }$.

Worked Example: Bakery

A local bakery sells loaves of bread.
Demand: $p=30-2Q$
Supply: $p=5+Q$

1. Equilibrium
   $$30-2Q=5+Q\Rightarrow 25=3Q\Rightarrow Q^{\ast }=\frac{25}{3}\approx \mathrm{8.33.}$$
   $$p^{\ast }=5+Q^{\ast }=5+\frac{25}{3}=\frac{40}{3}\approx \mathrm{13.33.}$$

2. Consumer Surplus
   $$CS=\frac{1}{2}(30-13.33)(8.33)\approx \frac{1}{2}(16.67)(8.33)\approx \mathrm{69.44.}$$

3. Producer Surplus
   $$PS=\frac{1}{2}(13.33-5)(8.33)\approx \frac{1}{2}(8.33)(8.33)\approx \mathrm{34.72.}$$

4. Total Surplus
   $$TS=CS+PS\approx 69.44+34.72=\mathrm{104.16.}$$

Graphical Illustration

The blue area (consumer surplus) is larger than the red area (producer surplus), indicating that consumers gain more benefit than producers in this market.

Comparison Table

Advantages and Applications

• Policy Analysis: Governments use surplus calculations to evaluate the impact of taxes, subsidies, and price controls on welfare.
• Business Pricing: Firms determine optimal pricing by balancing demand elasticity and cost structure to maximize profit while considering consumer surplus.
• Market Design: Auction platforms (e.g., eSewa, Ncell) model supply and demand to set dynamic prices that maximize total surplus.
• Economic Forecasting: Economists predict how changes in income or preferences shift demand curves, affecting equilibrium and welfare.

In the real world

1. Daraz Order Queue

   • Product: Daraz’s online marketplace.
   • Idea Used: Demand and supply curves determine the price at which the number of orders matches the number of available sellers.
   • How It Works: When a flash sale starts, the sudden surge in demand shifts the demand curve rightward. Daraz’s algorithm adjusts prices in real time to restore equilibrium, ensuring sellers receive fair compensation while buyers get competitive prices.

2. Ncell Mobile Plans

   • Product: Ncell’s tiered data plans.
   • Idea Used: Consumer surplus analysis helps Ncell set plan prices that maximize revenue while keeping customers satisfied.
   • How It Works: By estimating the maximum price each customer segment is willing to pay (demand intercept), Ncell sets a price slightly below that threshold, capturing surplus without losing customers.

3. NEPSE Trading

   • Product: Nepal Stock Exchange.
   • Idea Used: Market equilibrium in a financial market where supply of shares equals demand at the equilibrium price.
   • How It Works: The exchange’s electronic trading system matches buy and sell orders, continuously updating the price until supply equals demand, thereby maximizing total surplus for investors and issuers.

Exam tip

• Always sketch the demand and supply curves before performing algebraic calculations; the sketch helps verify that the equilibrium lies in the positive quadrant.
• Remember the area formulas:
  • $CS=\frac{1}{2}(a-p^{\ast })Q^{\ast }$
  • $PS=\frac{1}{2}(p^{\ast }-c)Q^{\ast }$
  • $TS=CS+PS$
• Check units: Prices are in rupees, quantities in units; area units will be rupees × units (i.e., rupees).
• When given piecewise or nonlinear functions, break the integral into segments or use the general integral formula for consumer and producer surplus.
• Practice with past exam questions: They often involve linear functions; ensure you can solve for equilibrium, compute surpluses, and interpret the results in a real‑world context.