Understanding Total Revenue Functions in Microeconomics

What is the total revenue function for a good with the demand function P=200-0.1Q?

• A. TR=200Q-0.2Q^2
• B. TR=200Q+0.1Q^2
• C. TR=200Q-0.1Q^2
• D. TR=0.1Q^2-200Q

Answer: (C)

To find the total revenue function, we start with the demand function given as P = 200 - 0.1Q. Total revenue (TR) is calculated as the product of price (P) and quantity (Q), which can be expressed mathematically as TR = P × Q. By substituting the expression for P from the demand function into the TR equation, we have: TR = (200 - 0.1Q) × Q. Expanding this gives: TR = 200Q - 0.1Q^2. This shows that the total revenue function is a quadratic equation in the standard form TR = aQ + bQ^2, where a = 200 and b = -0.1. The significance of this formulation is that it captures how total revenue changes with varying levels of output (Q). The positive term (200Q) indicates that revenue increases with quantity sold, while the negative term (-0.1Q^2) accounts for diminishing returns at higher quantities due to the nature of the demand curve, where price reduces as quantity increases. Thus, the total revenue function accurately reflects the relationship between price, quantity, and total revenue, making it clear why the listed function is correct.

When diving into microeconomics, one of the fundamental concepts you’ll encounter is the total revenue function. If you’re gearing up for the UCF ECO2023 Principles of Microeconomics Final Exam, understanding this can give you a solid edge. Let’s unpack this whole total revenue (TR) concept using the demand function ( P = 200 - 0.1Q ) as our guide.

First things first, you might be wondering why this matters. Understanding how revenue behaves as you sell different quantities of products is key to making smart business decisions. If you think about it, it’s like driving a car; knowing how fast you can go (or revenue you can generate) at different speeds (or quantities) can help prevent crashes in your economic strategy!

Here’s the scoop: total revenue is calculated using the formula ( TR = P \times Q ) — that means price (P) multiplied by quantity (Q). So, how do we plug in this demand function from earlier? Simple!

We substitute the expression for price into our revenue equation:

[

TR = (200 - 0.1Q) \times Q.

]

When we expand this, it gives us:

[

TR = 200Q - 0.1Q^2.

]

This represents our total revenue function. It's essential to recognize that this formula is actually a quadratic equation, where the revenue depends on the quantity sold. You might be thinking, “What does that mess of numbers even mean?” Well, let’s break it down a bit more.

The term ( 200Q ) indicates that as you sell more (increase Q), your revenue is going up linearly. It's intuitive — the more of something you sell, the more money you make! However, don't forget that we also have a -0.1Q² term in there. This reflects diminishing returns. In simpler terms, as you sell even more, you can’t expect to keep that same high revenue growth because, for each additional unit sold, you’re likely lowering the price you can charge.

Imagine you’re at a farmer’s market. If you’re the only seller of apples, you might charge $2 per apple. But if suddenly, five other sellers pop up, that price may drop to $1.50. As more sellers enter the market, each additional apple sold brings in less revenue. That’s the heart of the diminishing returns concept captured in our revenue function!

What’s key to realize is that this total revenue formulation allows businesses to forecast how changes in price and quantity can affect their bottom line. By analyzing the quadratic equation, businesses can even determine the optimal output level where revenue maximizes—now that’s a strategist’s dream!

As you review your microeconomics concepts leading up to your exam, reflect on how price and quantity interact through this total revenue function. This relationship isn’t just a math problem; it’s at the core of how markets function and how businesses can thrive.

There you have it! Understanding total revenue through the lens of the demand function ( P = 200 - 0.1Q ) has both theoretical and practical applications. As you prepare for your final exam, remember that these concepts are interconnected and vital for grasping the broader economic picture. Good luck studying; you've got this!