Elasticity and Marginal Analysis in Economics

Should a tea company raise its price? How many loaves should a bakery bake each day? Economists answer such questions with two linked tools: elasticity, which measures how strongly buyers respond to change, and marginal analysis, which looks at the effect of one more unit. Together they form the mathematical core of microeconomics and managerial economics, and this article develops both with calculations you can follow line by line.

Price Elasticity of Demand

Price elasticity of demand compares the percentage change in quantity demanded with the percentage change in price:

$$ E_d = \frac{\%\Delta Q}{\%\Delta P} $$

Because demand curves slope downward, \( E_d \) is negative; many textbooks quote its absolute value. The interpretation:

• \( |E_d| > 1 \): elastic – quantity responds more than proportionally.
• \( |E_d| < 1 \): inelastic – quantity responds less than proportionally.
• \( |E_d| = 1 \): unit elastic.

The Midpoint (Arc) Formula

Using the starting value as the base gives different answers for a price rise and a price fall. The midpoint formula avoids this by dividing by the average:

$$ E_d = \frac{(Q_2 - Q_1) / \left(\frac{Q_1 + Q_2}{2}\right)}{(P_2 - P_1) / \left(\frac{P_1 + P_2}{2}\right)} $$

Worked Example 1: A Tea Brand Raises Its Price

A tea company in Lahore raises the price of a pack from PKR 500 to PKR 550. Weekly sales in a market fall from 1,000 packs to 900 packs.

Step 1: Percentage change in quantity (midpoint).

$$ \%\Delta Q = \frac{900 - 1000}{(1000 + 900)/2} = \frac{-100}{950} \approx -10.53\% $$

Step 2: Percentage change in price (midpoint).

$$ \%\Delta P = \frac{550 - 500}{(500 + 550)/2} = \frac{50}{525} \approx 9.52\% $$

Step 3: Elasticity.

$$ E_d = \frac{-10.53}{9.52} \approx -1.11 $$

Demand is (slightly) elastic.

Step 4: Effect on revenue.

$$ \text{Before: } 500 \times 1{,}000 = \text{PKR } 500{,}000, \qquad \text{After: } 550 \times 900 = \text{PKR } 495{,}000 $$

Revenue fell by PKR 5,000. This is the key rule: when demand is elastic, raising the price lowers total revenue; when demand is inelastic, raising the price raises revenue. The result matches our elasticity, which was just above 1 in absolute value, so revenue fell only slightly.

Other Elasticities

• Income elasticity \( E_Y = \%\Delta Q / \%\Delta Y \). Positive for normal goods; above 1 for luxuries such as foreign holidays; negative for inferior goods.
• Cross-price elasticity \( E_{XY} = \%\Delta Q_X / \%\Delta P_Y \). Positive for substitutes (two brands of cooking oil), negative for complements (cars and petrol).
• Price elasticity of supply measures how producers respond to price.

What makes demand elastic? Close substitutes, a large share of the buyer's budget, a non-essential product and a long time to adjust all raise elasticity. That is why demand for one particular brand of tea is far more elastic than demand for tea as a whole: shoppers can easily switch brands, but few households give up tea altogether.

Point Elasticity With Calculus

For a smooth demand function \( Q(P) \), elasticity at a single point is

$$ E_d = \frac{dQ}{dP} \cdot \frac{P}{Q}. $$

For a linear demand curve the slope is constant, but \( P/Q \) changes along the line, so elasticity varies: demand is elastic at high prices and inelastic at low prices.

Marginal Analysis

"Marginal" means "the effect of one more unit". With calculus:

• Marginal revenue \( MR = \dfrac{dTR}{dQ} \)
• Marginal cost \( MC = \dfrac{dTC}{dQ} \)
• Marginal profit \( = MR - MC \)

If producing one more unit adds more to revenue than to cost (\( MR > MC \)), the firm should expand. If \( MR < MC \), it should cut back. Profit is maximised where

$$ MR = MC, $$

with the second-order condition that \( MC \) cuts \( MR \) from below.

Linking MR and Elasticity

Since \( TR = P \cdot Q \), differentiating gives a famous result:

$$ MR = P\left(1 + \frac{1}{E_d}\right). $$

When demand is elastic (\( E_d < -1 \)), \( MR > 0 \), so selling more raises revenue. When inelastic, \( MR < 0 \). A profit-maximising firm with positive marginal cost therefore always operates on the elastic part of its demand curve.

Worked Example 2: A Bakery's Profit-Maximising Output

A bakery in Islamabad faces daily demand for its speciality cake given by \( Q = 200 - 2P \), where \( P \) is price in PKR and \( Q \) is number of cakes. Its daily cost is \( TC = 500 + 20Q + 0.5Q^2 \) rupees. (Prices are kept small for clarity; think of PKR as hundreds of rupees if you prefer realism.)

Step 1: Write price in terms of quantity.

$$ P = 100 - \frac{Q}{2} $$

Step 2: Total and marginal revenue.

$$ TR = PQ = 100Q - \frac{Q^2}{2}, \qquad MR = 100 - Q $$

Step 3: Marginal cost.

$$ MC = \frac{d}{dQ}\left(500 + 20Q + 0.5Q^2\right) = 20 + Q $$

Step 4: Set MR = MC.

$$ 100 - Q = 20 + Q \;\Rightarrow\; 2Q = 80 \;\Rightarrow\; Q = 40 $$

Step 5: Price.

$$ P = 100 - \frac{40}{2} = 80 $$

Step 6: Profit.

$$ TR = 80 \times 40 = 3{,}200, \qquad TC = 500 + 20(40) + 0.5(40)^2 = 500 + 800 + 800 = 2{,}100 $$

$$ \text{Profit} = 3{,}200 - 2{,}100 = 1{,}100 $$

Step 7: Second-order check. The profit function is \( \pi = TR - TC = 80Q - Q^2 - 500 \), so \( \pi''(Q) = -2 < 0 \): a maximum.

Step 8: Elasticity at the optimum. From \( Q = 200 - 2P \), \( dQ/dP = -2 \), so

$$ E_d = -2 \times \frac{80}{40} = -4. $$

Demand is elastic, as theory predicts. Check the MR–elasticity formula: \( MR = 80(1 - 1/4) = 60 \), which equals \( 100 - 40 = 60 \) and also \( MC = 20 + 40 = 60 \). Three independent routes give the same answer.

Sanity check: at \( Q = 39 \), profit is \( 80(39) - 39^2 - 500 = 3{,}120 - 1{,}521 - 500 = 1{,}099 \), and at \( Q = 41 \) it is \( 3{,}280 - 1{,}681 - 500 = 1{,}099 \). Both neighbours give less than 1,100, confirming the peak.

Where Businesses Use These Ideas

1. Pricing: firms estimate elasticity from sales data to decide whether discounts will raise revenue.
2. Taxation: governments tax goods with inelastic demand (tobacco, fuel) because revenue is stable; the burden falls mostly on buyers.
3. Price discrimination: students and senior citizens with more elastic demand are offered lower prices.
4. Production decisions: managers expand output only while marginal revenue exceeds marginal cost.

Did You Know?

• Alfred Marshall formalised price elasticity of demand in his Principles of Economics in 1890.
• Antoine Augustin Cournot used calculus to find a monopolist's profit-maximising output as early as 1838.
• Petrol demand tends to be inelastic in the short run but noticeably more elastic in the long run, as people change cars and commuting habits.

Key Takeaways

• Price elasticity is the ratio of percentage changes; use the midpoint formula for arcs and \( (dQ/dP)(P/Q) \) at a point.
• Elastic demand means a price rise lowers revenue; inelastic demand means it raises revenue.
• Profit is maximised where \( MR = MC \), with MC cutting MR from below.
• \( MR = P(1 + 1/E_d) \) links marginal analysis to elasticity.
• A profit-maximising firm always operates where demand is elastic.

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