You Are Working Trying To Estimate The Proper Price To Charg

You Are Working Trying To Estimate The Proper Price To Charge A Market

You are working trying to estimate the proper price to charge a market for a firm that sells beer in Lancaster, Pennsylvania. The estimated demand curve for this market is given by: Quantity demanded = 20 - 2P. The firm is currently pricing the product at $8, and they are considering lowering the price to $6 in an attempt to increase profits. This assignment involves calculating price elasticity of demand at the specified prices, as well as cross-price elasticity with a related product.

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Understanding the concept of price elasticity of demand is fundamental in making pricing decisions. It measures the responsiveness of quantity demanded to a change in price, guiding firms on how to set prices that maximize revenue or profit. In this context, the firm is analyzing how changing the price of beer in Lancaster, PA, influences consumer demand, and how similar products can affect each other's demand through cross-price elasticity.

1. Calculating Price Elasticity of Demand for Lancaster Market when Price Changes from $8 to $6

The demand curve provided is Qd = 20 - 2P. To determine the price elasticity of demand when the price drops from $8 to $6, we first calculate the quantity demanded at these prices:

  • At P = $8: Qd = 20 - 2(8) = 20 - 16 = 4
  • At P = $6: Qd = 20 - 2(6) = 20 - 12 = 8

The midpoint method for elasticity uses the average quantities and prices to obtain a more accurate measurement, especially when the change is significant. The formula is:

Elasticity = [(Q2 - Q1) / ((Q2 + Q1)/2)] ÷ [(P2 - P1) / ((P2 + P1)/2)]

Substituting the values:

Numerator (change in quantity): (8 - 4) / [(8 + 4)/2] = 4 / 6 = 0.6667

Denominator (change in price): (6 - 8) / [(6 + 8)/2] = -2 / 7 = -0.2857

Elasticity = 0.6667 / -0.2857 ≈ -2.33

The absolute value of elasticity is approximately 2.33, indicating that the demand is elastic at this price range. A 1% decrease in price results in approximately a 2.33% increase in quantity demanded, suggesting that lowering the price from $8 to $6 could significantly increase total revenue, provided other factors remain constant.

2. Elasticity of Demand for the Scranton Market Moving from $2 to $5

The same demand curve applies here: Qd = 20 - 2P. Calculating demand quantities at the two prices:

  • At P = $2: Qd = 20 - 2(2) = 20 - 4 = 16
  • At P = $5: Qd = 20 - 2(5) = 20 - 10 = 10

Applying the midpoint method again:

Change in quantity: (10 - 16) / [(10 + 16)/2] = (-6) / 13 = -0.4615

Change in price: (5 - 2) / [(5 + 2)/2] = 3 / 3.5 ≈ 0.8571

Elasticity = -0.4615 / 0.8571 ≈ -0.538

The absolute value around 0.538 indicates that demand in Scranton at these prices is relatively inelastic. A 1% decrease in price would increase quantity demanded by approximately 0.54%, suggesting that lowering prices might not substantially increase total revenue in this scenario.

3. Cross-Price Elasticity of Demand for a Related Product

The relationship between the quantity demanded of the good and the price of a related product is given by: Qd = 15 + Pb. The initial price of the related good is $5, increasing to $6.

Calculating the quantities demanded at each related product price:

  • At Pb = $5: Qd = 15 + 5 = 20
  • At Pb = $6: Qd = 15 + 6 = 21

Cross-price elasticity is computed by:

Elasticity = [(Q2 - Q1) / ((Q2 + Q1)/2)] ÷ [(Pb2 - Pb1) / ((Pb2 + Pb1)/2)]

Numerator: (21 - 20) / [(21 + 20)/2] = 1 / 20.5 ≈ 0.0488

Denominator: (6 - 5) / [(6 + 5)/2] = 1 / 5.5 ≈ 0.1818

Cross-price elasticity = 0.0488 / 0.1818 ≈ 0.268

The positive value (~0.27) indicates a substitute relationship—when the price of the related good increases, demand for the other good also increases. This suggests that the two products are substitutes rather than complements.

Conclusion

The elasticity calculations demonstrate the sensitivity of demand in different markets and highlight strategic pricing implications. The elastic demand in Lancaster suggests that lowering prices could boost total revenue, while the inelastic demand in Scranton implies limited gains from price cuts. The positive cross-price elasticity confirms substitutability, affecting demand dynamics when related products' prices change. Firms should leverage these insights for optimal pricing strategies tailored to regional demand sensitivities, maximizing profitability while accounting for market-specific elasticities.

References

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