When a mutual fund company charges a fee of 0.47 % on its...

When a mutual fund company charges a fee of $0.47 \%$ on its index funds, its assets in the fund are $\$ 41$ billion. And when it charges a fee of $0.18 \%,$ its assets in the fund are S 300 billion. (Source: The Boston Globe.) (a) Let $x \%$ denote the fee that the company charges the index fund and $A(x)$ its assets in the fund. Express $A(x)$ as a linear function of $x$. (b) In September 2004 , Fidelity Mutual lowered its fees on various index funds from an average of $0.3 \%$ to $0.10 \%$ Let $R(x)$ denote the revenue of the company from fees when its index-fund fee is $x \% .$ Compare the revenue of the company before and after lowering the fees. [Hint: Revenue is $x \%$ of assets.] (c) Find the fee that maximizes the revenue of the company and determine the maximum revenue.

When a mutual fund company charges a fee of $0.47 \%$ on its index funds, its assets in the fund are $\$ 41$ billion. And when it charges a fee of $0.18 \%,$ its assets in the fund are S 300 billion. (Source: The Boston Globe.)
(a) Let $x \%$ denote the fee that the company charges the index fund and $A(x)$ its assets in the fund. Express $A(x)$ as a linear function of $x$.
(b) In September 2004 , Fidelity Mutual lowered its fees on various index funds from an average of $0.3 \%$ to $0.10 \%$ Let $R(x)$ denote the revenue of the company from fees when its index-fund fee is $x \% .$ Compare the revenue of the company before and after lowering the fees. [Hint: Revenue is $x \%$ of assets.]
(c) Find the fee that maximizes the revenue of the company and determine the maximum revenue.

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Key Concepts

Linear Functions

A linear function expresses a relationship between two variables with a constant rate of change. In contexts like this problem, the assets in the mutual fund are modeled as a linear function of the fee percentage, which means that knowing the asset values at two fee rates allows one to determine the slope (representing the rate at which assets change with fee alterations) and the y-intercept of the line.

Revenue as a Product Function

Revenue in such contexts is calculated as a percentage fee applied to the asset value. When modeling revenue, you multiply the fee rate (expressed as a decimal) by the asset amount. This concept is crucial for understanding how changes in fee percentages affect overall revenue and for setting up the function that needs to be optimized.

Quadratic Optimization

When the revenue function is expressed in terms of the fee percentage, it often becomes a quadratic function. Quadratic optimization involves finding the vertex of the parabola defined by the quadratic function, allowing you to determine the fee that maximizes revenue and the corresponding maximum revenue value. Techniques such as completing the square or using calculus (finding and setting the derivative to zero) are typically used for this purpose.

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