Demand Elasticity & Revenue: Solving P = 1000 + 3Q - 4Q^2

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Hey guys! Today, we're diving into a fun economics problem involving demand elasticity and revenue. We've got this inverse demand function: P = 1000 + 3Q - 4Q^2. Our mission is to figure out a few key things: the price elasticity of demand at a specific quantity, and the equations for total revenue (TR) and marginal revenue (MR). Let's break it down step by step!

a. Determining Price Elasticity of Demand at Q = 10 Units

Okay, so first up, let's tackle the price elasticity of demand at Q = 10 units. What exactly is price elasticity of demand? Simply put, it measures how much the quantity demanded of a good changes when its price changes. A product with high elasticity means that a small price change leads to a big change in quantity demanded, while low elasticity means the opposite.

To calculate this, we'll need a formula. The price elasticity of demand (Ed) is calculated as:

Ed = (dQ/dP) * (P/Q)

Where:

  • dQ/dP is the derivative of quantity demanded with respect to price (the change in quantity demanded for a change in price).
  • P is the price.
  • Q is the quantity.

Step 1: Find the Price (P) at Q = 10

First things first, we need to find the price when Q = 10. We can plug this value into our inverse demand function:

P = 1000 + 3(10) - 4(10)^2 P = 1000 + 30 - 400 P = 630

So, when the quantity is 10 units, the price is 630. Got it!

Step 2: Find dQ/dP

Now for a little calculus! We need to find the derivative of quantity (Q) with respect to price (P). But hold on, our equation is in terms of P (price) as a function of Q (quantity). No sweat! We need to find dQ/dP, which is the inverse of dP/dQ. So, let's first find dP/dQ:

dP/dQ = d/dQ (1000 + 3Q - 4Q^2) dP/dQ = 3 - 8Q

Now, plug in Q = 10:

dP/dQ = 3 - 8(10) dP/dQ = 3 - 80 dP/dQ = -77

Remember, we want dQ/dP, which is the inverse of this. So:

dQ/dP = 1 / (dP/dQ) dQ/dP = 1 / (-77) dQ/dP = -1/77

Awesome! We've got dQ/dP. This tells us how quantity demanded changes for each unit change in price.

Step 3: Calculate Elasticity (Ed)

Alright, we've got all the pieces! Now, we can plug everything into our elasticity formula:

Ed = (dQ/dP) * (P/Q) Ed = (-1/77) * (630/10) Ed = (-1/77) * 63 Ed = -63/77 Ed β‰ˆ -0.818

Interpretation

The price elasticity of demand at Q = 10 units is approximately -0.818. The negative sign indicates that as price increases, quantity demanded decreases (which is the law of demand). The absolute value being less than 1 (| -0.818 | < 1) tells us that demand is inelastic at this point. This means that a 1% change in price will lead to a less than 1% change in quantity demanded. In other words, consumers aren't super sensitive to price changes at this quantity.

b. Determining Equations for Total Revenue (TR) and Marginal Revenue (MR)

Next up, let's figure out the equations for total revenue (TR) and marginal revenue (MR). These are crucial concepts for understanding how a firm's revenue changes with its output.

Step 1: Total Revenue (TR)

Total revenue is the total amount of money a firm receives from selling its products. It's calculated simply as price (P) multiplied by quantity (Q):

TR = P * Q

We already have our inverse demand function for P: P = 1000 + 3Q - 4Q^2. Let's plug that into the TR equation:

TR = (1000 + 3Q - 4Q^2) * Q TR = 1000Q + 3Q^2 - 4Q^3

Boom! That's our equation for total revenue. It shows how total revenue changes as the quantity sold changes.

Step 2: Marginal Revenue (MR)

Marginal revenue is the additional revenue a firm earns from selling one more unit of its product. Mathematically, it's the derivative of total revenue (TR) with respect to quantity (Q):

MR = d(TR)/dQ

We already found our TR equation: TR = 1000Q + 3Q^2 - 4Q^3. Let's take the derivative:

MR = d/dQ (1000Q + 3Q^2 - 4Q^3) MR = 1000 + 6Q - 12Q^2

And there you have it! This is the equation for marginal revenue. It tells us how much extra revenue the firm will get for each additional unit sold.

Key Takeaways

Let's recap what we've learned in this awesome economics adventure:

  • Price Elasticity of Demand: At Q = 10, the price elasticity of demand is approximately -0.818, which indicates that demand is inelastic at this quantity.
  • Total Revenue (TR): The equation for total revenue is TR = 1000Q + 3Q^2 - 4Q^3.
  • Marginal Revenue (MR): The equation for marginal revenue is MR = 1000 + 6Q - 12Q^2.

Understanding these concepts – price elasticity, total revenue, and marginal revenue – is super crucial for businesses to make informed decisions about pricing and output levels. By knowing how demand responds to price changes and how revenue is affected by production, companies can optimize their strategies and maximize their profits. Isn't economics fascinating?

I hope this breakdown was helpful and clear! If you have any more questions or want to explore other economics topics, just let me know. Keep learning and keep rocking it, guys! Remember, economics isn't just about numbers and graphs; it's about understanding the world around us and how we make choices in it.