Demand for noodle produced by ABC Company in Nepal is affected by various determinants. Among them, price and sales expenditure are the key determinants. The following table shows the sales of noodle in Kathmandu for six different weeks at various levels of prices and sales expenditure. The data was collected for six weeks on Saturday of every week.
| First week | Price of noodle (per unit) | Sales expenditure (Rs. in Millions) | Sales of noodle ('000 units per day) |
|---|---|---|---|
| Second week | 290 | 30 | 350 |
| | 288 | 40 | 360 |
| Third week | 285 | 48 | 380 |
| Fourth week | 280 | | 400 |
| Fifth week | 274 | 60 | 415 |
| Sixth week | 260 | 85 | 435 |
a. Derive the linear demand function of noodle in Kathmandu and forecast the sales of noodle for the $7^{\text {th }}$ week when price is decreased Rs. 240. Also estimate the price elasticity of demand.
b. Derive the linear demand function of noodle in Kathmandu and forecast the sales of noodle for the $7^{\text {th }}$ week when sales expenditure is increased to Rs. 100 millions. Also estimate the sales elasticity of demand.
c. Which policy is more effective to promote sales of noodle to the ABC Company in Kathmandu? Justify it by using the value of price and sales elasticities.
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Ans: (a) $Q_d=1,172.6-2.8 P ; E_p=-1.343$ (b) $Y=273.87-2.08 X ; E_g=3.16$
Answer (2)
The demand function for noodles based on price is derived as Q_d = 1172.6 - 2.8P, predicting sales of about 500.6 '000 units at a price of Rs. 240. The elasticity of demand is estimated at -1.34, showing high responsiveness to price changes. Conversely, the sales function based on expenditure is Q_d = 273.87 + 2.08S, predicting sales of approximately 481.87 '000 units at Rs. 100 million expenditure, with a lower elasticity of 0.43, indicating a stronger effectiveness of price reduction to boost sales.
;
Derive the linear demand function with price: Q d = 1170.00 − 2.79 P .
Forecast sales for P = 240: Q d = 500.4 '000 units per day.
Estimate the price elasticity of demand: E p = − 1.34 .
The price policy is more effective. E p = − 1.34
Explanation
Problem Analysis We are given data on noodle sales, price, and sales expenditure over six weeks. Our goal is to derive linear demand functions based on price and sales expenditure, forecast sales for specific price and sales expenditure levels, estimate price and sales elasticities, and determine which policy (price reduction or sales expenditure increase) is more effective in promoting noodle sales.
Linear Demand Function with Price Let's derive the linear demand function with price. We assume the linear demand function is of the form Q d = a + b P , where Q d is the quantity demanded, P is the price, and a and b are coefficients to be determined. Using the provided data, we can calculate these coefficients using linear regression. After performing the calculations, we find that a = 1170.00 and b = − 2.79 . Therefore, the linear demand function with price is: Q d = 1170.00 − 2.79 P
Forecasting Sales with Price Now, let's forecast sales for the 7th week when the price is Rs. 240. Substituting P = 240 into the derived demand function, we get: Q d = 1170.00 − 2.79 ( 240 ) = 1170.00 − 669.6 = 500.4
So, the forecasted sales for the 7th week when the price is Rs. 240 is approximately 500.4 '000 units per day.
Price Elasticity of Demand Next, we estimate the price elasticity of demand. The formula for price elasticity is E p = d P d Q d ⋅ Q d P = b ⋅ Q d P . Using the price and quantity at the 7th week ( P = 240 and Q d = 500.4 ), we have: E p = − 2.79 ⋅ 500.4 240 = − 2.79 ⋅ 0.4796 = − 1.34
Thus, the price elasticity of demand is approximately -1.34.
Linear Demand Function with Sales Expenditure Now, let's derive the linear demand function with sales expenditure. We assume the linear demand function is of the form Q d = c + d S , where Q d is the quantity demanded, S is the sales expenditure, and c and d are coefficients to be determined. Using the provided data, we can calculate these coefficients using linear regression. After performing the calculations, we find that c = 303.30 and d = 1.65 . Therefore, the linear demand function with sales expenditure is: Q d = 303.30 + 1.65 S
Forecasting Sales with Sales Expenditure Now, let's forecast sales for the 7th week when sales expenditure is Rs. 100 million. Substituting S = 100 into the derived demand function, we get: Q d = 303.30 + 1.65 ( 100 ) = 303.30 + 165 = 468.30
So, the forecasted sales for the 7th week when sales expenditure is Rs. 100 million is approximately 468.3 '000 units per day.
Sales Elasticity of Demand Next, we estimate the sales elasticity of demand. The formula for sales elasticity is E s = d S d Q d ⋅ Q d S = d ⋅ Q d S . Using the sales expenditure and quantity at the 7th week ( S = 100 and Q d = 468.3 ), we have: E s = 1.65 ⋅ 468.3 100 = 1.65 ⋅ 0.2135 = 0.35
Thus, the sales elasticity of demand is approximately 0.35.
Comparing Policy Effectiveness Finally, let's compare the effectiveness of price and sales expenditure policies. We compare the absolute values of the price elasticity ( ∣ E p ∣ ) and sales elasticity ( ∣ E s ∣ ). We have ∣ E p ∣ = ∣ − 1.34∣ = 1.34 and ∣ E s ∣ = ∣0.35∣ = 0.35 . Since |E_s|"> ∣ E p ∣ > ∣ E s ∣ , the price policy is more effective.
Conclusion The linear demand function with price is Q d = 1170.00 − 2.79 P , and the forecasted sales for a price of Rs. 240 is 500.4 '000 units per day. The price elasticity of demand is -1.34. The linear demand function with sales expenditure is Q d = 303.30 + 1.65 S , and the forecasted sales for a sales expenditure of Rs. 100 million is 468.3 '000 units per day. The sales elasticity of demand is 0.35. The price policy is more effective because the absolute value of price elasticity is greater than the absolute value of sales elasticity.
Examples
Understanding demand elasticity is crucial for businesses. For instance, a coffee shop might consider lowering the price of their lattes to attract more customers. If the price elasticity of demand for lattes is high (greater than 1 in absolute value), a small price decrease will lead to a significant increase in the quantity demanded, boosting overall revenue. Conversely, if the elasticity is low (less than 1 in absolute value), the coffee shop might focus on increasing sales expenditure, such as advertising or loyalty programs, to drive sales more effectively. These strategies help businesses optimize their pricing and marketing efforts.
