Question

A financial services company is interested in examining the relationship between the age of an individual and their wealth in order to make more informed recommendations. They use client data to estimate the following models. Note, Age is measured in years and Wealth is the total dollar amount the individual has saved. Assume each of the explanatory variables are significant at the 5% level.

Model 1:  Wealthˆ= 5,450 + 4,589Age, Se = 11,550, R² = 0.59, Adjusted-R² = 0.50

Model 2:  Wealthˆ= 4,265.20 + 10,400.1241Age − 73.7597Age^2, Se = 7,421, R² = 0.79, Adjusted-R² = 0.77

Using the estimates in Model 2, complete the formula for the marginal effect. Use four decimals when entering your answers.

Marginal Effect = __________ - _____________ Age

Using the estimates in Model 2, calculate the marginal effect (similar to “As x increases by 1 unit”) for someone who is 50 years old. Round your answer to 2 decimals

Using the estimates in Model 2, calculate the marginal effect (similar to “As x increases by 1 unit) for someone who is 75 years old. Round your answer to 2 decimals

Using the estimates in Model 2, find the age at which wealth is maximized. Round your answer to 2 decimals.

Using the estimates in Model 2, find the maximum wealth. Round your answer to 2 decimals.

Answer 1

The regression model 2 is given as,

Part A:

The marginal effect of age on wealth is given by the partial derivative of wealth with respect to age, which is

    [ From model 2]

                                   

So, Marginal Effect = 10400.1241 - 147.5194 Age

Part B:

Marginal effect for someone who is 50 years old is,

So Marginal effect for someone who is 50 years old is 3024.15.

Part C:

Marginal effect for someone who is 75 years old is,

So Marginal effect for someone who is 50 years old is -663.83.

Part D:

For maximizing a function, f(x), we know that at points at which is either a maximum or a minimum.

So here,

If wealth is maximum at age = x, then

or,

or,

Hence the age at which wealth will be maximum is 70.50.

Part E:

We know wealth will be maximum at age = 70.50.

So the maximum wealth will be,

                        

Hence the maximum wealth will be 370869.80.

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