Question

Given the function \(y=a-\frac{b}{c+x} \quad(a, b, c>0 ; x \geq 0),\) determine the general shape of its graph by examining ( \(a\) ) its first and second derivatives, (b) its vertical intercept, and (c) the limit of \(y\) as \(x\) tends to infinity. If this function is to be used as a consumption function, how should the parameters be restricted in order to make it economically sensible?

Short answer

The graph of the function is increasing and concave down, with a vertical intercept at \( y = a - \frac{b}{c} \) and horizontal asymptote \( y = a \). Parameters \( a, b, \) and \( c \) must remain positive, with \( b < ac \) for economic sensibility.

Step-by-step solution

Calculate the First Derivative

The given function is \( y = a - \frac{b}{c+x} \). To find the first derivative, we use the quotient rule for the term \( \frac{b}{c+x} \). The derivative \( \frac{d}{dx} \left( \frac{b}{c+x} \right) \) is \( -\frac{b}{(c+x)^2} \). Hence, the first derivative of the function is:\[\frac{dy}{dx} = \frac{b}{(c+x)^2}.\] This indicates that the function is increasing as \( x \) increases because \( \frac{b}{(c+x)^2} > 0 \) for all \( x \geq 0 \).

Calculate the Second Derivative

Next, to find the concavity of the function, compute the derivative of the first derivative. The second derivative of \( -\frac{b}{(c+x)^2} \) is given by:\[\frac{d^2y}{dx^2} = -\frac{2b}{(c+x)^3}.\] The second derivative is negative, indicating that the function is concave down.

Determine the Vertical Intercept

The vertical intercept is found by evaluating the function at \( x=0 \). This gives:\[y(0) = a - \frac{b}{c} = a - \frac{b}{c}.\] This is the point where the function intersects the y-axis.

Determine the Limit as \( x \to \infty \)

Examine the behavior of \( y \) as \( x \to \infty \). As \( x\) increases without bound, \( \frac{b}{c+x} \) approaches zero. Therefore, the limit of \( y \) as \( x \to \infty \) is \( a \):\[ \lim_{x \to \infty} y = a.\]

Determine Economically Sensible Restrictions

For the function to be a sensible consumption function, as \( x \) (typically representing income) increases, consumption \( y \) should also increase, which is ensured by \( a - \frac{b}{c+x} \) with \( a, b, c > 0 \). Additionally, \( a \) should represent a level of consumption that is eventually approached as income becomes very large, while \( b \) must be smaller than \( ac \) to keep the initial consumption \( a - \frac{b}{c} \) positive.

Key concepts

Understanding the First Derivative

The first derivative of a function gives us valuable information about the rate of change or the slope of the function at any given point. In simpler terms, it tells us how the function behaves as we move along the x-axis and how steeply it is increasing or decreasing. For the function \( y = a - \frac{b}{c+x} \), we find the first derivative by differentiating the fraction \( \frac{b}{c+x} \). Applying the quotient rule, we find that the first derivative is \( \frac{dy}{dx} = \frac{b}{(c+x)^2} \).

This means:

• For all values \( x \geq 0 \), the first derivative \( \frac{b}{(c+x)^2} > 0 \).
• A positive first derivative indicates that the function is increasing as \( x \) increases.

Therefore, as the independent variable \( x \) grows, the function \( y \) also increases, which is consistent with behaviors typical of consumption functions in economics.

Exploring the Second Derivative

The second derivative of a function provides information about its concavity, which is essentially how the function curves. By finding the second derivative, we can determine whether a function is concave up (curving upwards) or concave down (curving downwards).

For our function \( y = a - \frac{b}{c+x} \), after finding the first derivative, we differentiate it again to get:
\[\frac{d^2y}{dx^2} = -\frac{2b}{(c+x)^3}.\]

This second derivative \(-\frac{2b}{(c+x)^3} \) is always negative for all \( x \geq 0 \), as all parameters are positive.

• This negative value indicates the function is concave down.
• A concave down graph implies the slope of the tangent line decreases as \( x \) increases, pointing to slower growth in function value.

Concave functions are crucial for identifying maximum points and understanding the function's overall growth behavior.

Concave Functions Explained

A function is said to be concave if its graph lies below its tangent lines at every point within a certain interval. Specifically, for concave down functions like ours, the second derivative is negative, which means the function is curving downward away from the tangent.

Important aspects of concave functions:

• The second derivative serves as a test for concavity.
• If \( \frac{d^2y}{dx^2} < 0 \), the function is concave down.

The concavity of this specific function, as indicated by \(-\frac{2b}{(c+x)^3} \), is fundamental because it shows that the rate of increase of \( y \) reduces as \( x \) increases, crucial for economic interpretations like diminishing returns in consumption.

Analyzing Limits at Infinity

When studying functions, especially in calculus and economics, it's essential to understand their behavior as the independent variable tends to very large values or infinity. This is where the concept of limits at infinity comes into play.

For our function \( y = a - \frac{b}{c+x} \), as \( x \to \infty \), the term \( \frac{b}{c+x} \) approaches zero, simplifying the function to \( \lim_{x \to \infty} y = a \).

• This tells us that no matter how large \( x \) gets, \( y \) will approach but never surpass \( a \).
• The value \( a \) can be viewed as the limiting or asymptotic behavior of the function.

Understanding limits at infinity helps us predict and interpret the stable long-term trends of the function, a vital tool for economic models and predictions.