College

As a data scientist, you collect 60,000 data points for an online real estate listing company on 5,000 newly listed single-family homes (SFH) within the Central Valley. You develop the following inverse demand models to help find undervalued homes:

Model 1:
[tex]\[
y = 300 - 0.1x_1 + 200x_2 + 100x_3 + 0.05x_4
\][/tex]

Model 2:
[tex]\[
y = 300 - 0.1x_1 + 200x_2 + 100x_3 + 0.05x_4 + 0.04x_5 + 75x_6 - 0.1x_7 + 150x_8 - 20x_9 - 30x_{10} - 50x_{11}
\][/tex]

**Data Table:** Inverse Demand ( [tex]\(P^D\)[/tex] ) Variables for Single Family Homes

**Tasks:**

(a) Using a ceteris paribus condition, determine the outcomes in the table above. For each variable [tex]\(x_n\)[/tex], determine the coefficient measure, whether [tex]\(y\)[/tex] will increase or decrease when [tex]\(x_n\)[/tex] increases, whether the variable is endogenous or exogenous, and whether a change in [tex]\(x_n\)[/tex] will cause a shift or movement along the model.

(b) Suppose the company is specifically targeting SFH with [tex]\(x_2 = 4\)[/tex] bedrooms, [tex]\(x_3 = 2\)[/tex] bathrooms, and with [tex]\(x_4 = 3,000 \, \text{ft}^2\)[/tex] lot size. The current number of listings for SFH is [tex]\(x_1 = 5,000\)[/tex]. What mortgage price would Model 1 estimate?

Answer :

Sure, let's go through the solution step by step for the given problem.

(a) Ceteris Paribus Analysis

To analyze the impact of each variable on the model, we'll consider the coefficient of each variable in the inverse demand model and determine how the changes in these variables affect the value of [tex]\( y \)[/tex]. We'll also classify the variables as endogenous or exogenous and explain whether they cause a shift or movement along the demand curve.

1. Model 1:
[tex]\[ y = 300 - 0.1x_1 + 200x_2 + 100x_3 + 0.05x_4 \][/tex]

- [tex]\( x_1 \)[/tex] (Number of listings): Coefficient = [tex]\(-0.1\)[/tex]
- Impact: As [tex]\( x_1 \)[/tex] increases, [tex]\( y \)[/tex] decreases.
- Type: Endogenous, part of the model itself.
- Change: Movement along the model.

- [tex]\( x_2 \)[/tex] (Bedrooms): Coefficient = [tex]\(200\)[/tex]
- Impact: As [tex]\( x_2 \)[/tex] increases, [tex]\( y \)[/tex] increases.
- Type: Exogenous, influences demand but not directly part of [tex]\( y \)[/tex].
- Change: Shift in the model.

- [tex]\( x_3 \)[/tex] (Bathrooms): Coefficient = [tex]\(100\)[/tex]
- Impact: As [tex]\( x_3 \)[/tex] increases, [tex]\( y \)[/tex] increases.
- Type: Exogenous.
- Change: Shift in the model.

- [tex]\( x_4 \)[/tex] (Lot size in square feet): Coefficient = [tex]\(0.05\)[/tex]
- Impact: As [tex]\( x_4 \)[/tex] increases, [tex]\( y \)[/tex] increases.
- Type: Exogenous.
- Change: Shift in the model.

(b) Calculate Estimated Mortgage Price Using Model 1

To estimate the mortgage price for a single-family home with the given features:

- [tex]\( x_1 = 5,000 \)[/tex]
- [tex]\( x_2 = 4 \)[/tex] bedrooms
- [tex]\( x_3 = 2 \)[/tex] bathrooms
- [tex]\( x_4 = 3,000 \)[/tex] sq ft lot size

Substitute these values into Model 1:

[tex]\[ y = 300 - 0.1(5,000) + 200(4) + 100(2) + 0.05(3,000) \][/tex]

Let's calculate each part step by step:

- Calculate [tex]\( -0.1 \times 5,000 = -500 \)[/tex].
- Calculate [tex]\( 200 \times 4 = 800 \)[/tex].
- Calculate [tex]\( 100 \times 2 = 200 \)[/tex].
- Calculate [tex]\( 0.05 \times 3,000 = 150 \)[/tex].

Now, plug these calculations into the equation for [tex]\( y \)[/tex]:

[tex]\[ y = 300 - 500 + 800 + 200 + 150 \][/tex]

Combine all the numbers:

[tex]\[ y = 950 \][/tex]

So, the estimated mortgage price using Model 1 is $950.

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