The non-linear demand model can be expressed as:

$Q = c e^{-bp}$

Where:

• $Q$ = demand
• $c$ = constant (given as 15,000 when price is 0)
• $p$ = price
• $b$ = demand sensitivity constant
• $e$ = Euler’s number (~2.718)

Step 1: Determine the Value of $b$

When $p = 0$, we know that $Q = c$, so:

$Q = 15,000 = 15,000 e^{-b(0)}$

Since $e^0 = 1$, this confirms the equation holds, but does not define $b$. If more data points were available, we could estimate $b$. For now, we assume $b$ is known or estimated.

Step 2: Calculate Demand for $p = 75$

If $b$ is known, we substitute $p = 75$:

$Q_{75} = 15,000 e^{-b(75)}$

Step 3: Calculate Demand for $p = 90$

Similarly, for $p = 90$:

$Q_{90} = 15,000 e^{-b(90)}$

Step 4: Interpretation

• If $b$ is small, the demand decreases slowly as price increases.
• If $b$ is large, the demand drops sharply with price increases.
• Comparing $Q_{75}$ and $Q_{90}$, a higher price results in lower demand due to the exponential nature of the model.
• The company should assess elasticity, meaning how sensitive demand is to price changes.

Would you like me to assume a value for $b$ and compute exact demand values?

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Related Questions:

1. How does the exponential decay model differ from a linear demand model?
2. What real-world factors could influence the constant $b$ in the demand model?
3. How can businesses optimize pricing based on non-linear demand models?
4. What is the significance of price elasticity of demand, and how does it relate to this model?
5. How does machine learning help in demand forecasting beyond mathematical models?

Tip:

Understanding price elasticity is crucial—if demand is highly sensitive to price changes, small price increases might lead to significant drops in sales.