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Chapter 2: Problem 5

In the linear consumption function $$\widehat{c o n s}=\widehat{\beta}_{0}+\widehat{\beta}_{1} \text { inc, }$$ the (estimated) marginal propensity to consume (MPC) out of income is simply the slope, \(\widehat{\beta}_{1}\), while the average propensity to consume (APC) is cons/inc \(=\widehat{\beta}_{0} /\)inc\(+\widehat{\beta}_{1}\). Using observations for 100 families on annual income and consumption (both measured in dollars), the following equation is obtained: \begin{array}{c} \widehat{c o n s}=-124.84+0.853 \text { inc} \\ n=100, R^{2}=0.692. \end{array} i. Interpret the intercept in this equation, and comment on its sign and magnitude. ii. What is the predicted consumption when family income is \(\$ 30,000 ?\) iii. With inc on the \(x\) -axis, draw a graph of the estimated MPC and APC.

Short Answer

Expert verified
The intercept suggests unrealistic negative consumption when income is zero. Predicted consumption at $30,000 income is $25,465.16. Graph shows constant MPC and changing APC with income.

Step by step solution

01

Interpret the Intercept

The intercept \(\widehat{\beta}_{0} = -124.84\) in the estimated consumption function represents the consumption level when the income (inc) is zero. The negative sign indicates that if family income were zero, the estimated consumption would be negative, which is not realistic in practice. It suggests that there might be other factors (not included in the model) that contribute to consumption when income is extremely low or zero.
02

Calculate Predicted Consumption for $30,000 Income

To predict the consumption when family income is \(\\(30,000\), substitute \(\text{inc} = 30,000\) into the consumption function:\[\widehat{\text{cons}} = -124.84 + 0.853 \times 30,000\]Carry out the multiplication and addition:\[\widehat{\text{cons}} = -124.84 + 25,590 = 25,465.16\]Thus, the predicted consumption for a family with an income of \\)30,000 is \$25,465.16.
03

Draw the Graph of Estimated MPC and APC

To visualize the estimated MPC and APC, plot income (inc) on the \(x\)-axis and consumption on the \(y\)-axis:1. The slope of the linear equation, \(\widehat{\beta}_{1} = 0.853\), represents the Marginal Propensity to Consume (MPC). It indicates how much consumption changes with each additional dollar of income.2. To graph APC (Average Propensity to Consume) at a specific income level, calculate:\[\text{APC} = \frac{-124.84}{\text{inc}} + 0.853\]At various income levels, compute and plot these values to showcase how APC changes with income. Include a line for MPC which remains constant at 0.853 across income levels.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Linear Regression
Linear regression is a fundamental concept in econometrics and statistics, used to model the relationship between two variables. In the context of the linear consumption function, we are examining the relationship between family income and consumption. The equation \( \widehat{c o n s} = \widehat{\beta}_{0} + \widehat{\beta}_{1} \text{ inc} \) serves as our estimated linear regression model. Here, \(\widehat{\beta}_{0}\) is the intercept, and \(\widehat{\beta}_{1}\) is the slope of the line.
Linear regression helps to determine how changes in the independent variable (income) affect the dependent variable (consumption). The simplicity of linear regression makes it a popular choice for understanding relationships in economic data.
  • The slope \(\widehat{\beta}_{1}\) indicates the change in consumption for each additional dollar of income.
  • R-squared \(R^{2}\) tells us how well the data fit the model.
Marginal Propensity to Consume
The Marginal Propensity to Consume (MPC) is a crucial economic concept that measures the change in consumer spending resulting from a change in income. In our linear model, MPC is represented by the slope of the equation, \(\widehat{\beta}_{1} = 0.853\).
This value tells us that for every additional dollar received in income, consumption increases by approximately 85.3 cents. MPC is vital for understanding consumer behavior and the impact of income changes on economic consumption levels.
  • MPC helps in modeling demand in the economy.
  • High MPC indicates robust consumer spending in response to income changes.
Considering MPC is constant in our simple model, it highlights the direct link between income and consumption in this scenario.
Average Propensity to Consume
Average Propensity to Consume (APC) is another important concept, reflecting the proportion of total income that is spent on consumption. In our given equation, \( \text{APC} = \frac{\widehat{\beta}_{0}}{\text{inc}} + \widehat{\beta}_{1} \),we see how APC can be calculated.
APC changes with different income levels, unlike the constant MPC. It provides insights into how much of their income individuals spend, and how this spending varies with changes in income.
  • APC provides a broader view of consumption patterns over the entire spectrum of income.
  • A declining APC with increasing income suggests that a smaller fraction of income is being used for consumption at higher income levels.
Intercept Interpretation
The interpretation of the intercept \(\widehat{\beta}_{0} = -124.84\) in a linear regression model requires careful consideration. In our model, the intercept represents the theoretical level of consumption when income is zero.
The negative intercept is not typically practical or possible, as it implies negative consumption. This suggests that there may be omitted variables or baseline consumption that the model does not account for when income is zero.
  • A negative intercept might indicate an unrealistic scenario in real-world applications.
  • It underscores the limitations of simple linear models in capturing all influencing factors on consumption.
Graphical Analysis
Graphical analysis is a powerful tool in econometrics to visualize relationships and model dynamics. When plotting the linear equation \( \widehat{c o n s} = -124.84 + 0.853 \text{ inc} \) on a graph, we set income (inc) on the x-axis and consumption on the y-axis.
The constant slope representing MPC is plotted as a straight line. To visualize APC, it's helpful to compute and plot its values across different income levels, showing how it varies while MPC remains constant.
  • Graphically, MPC is shown as a fixed slope, aiding in assessing linear relationships.
  • APC plots provide a dynamic view of changing consumption proportions related to income.
Using graphs allows for clearer, more intuitive understanding of the consumption-income relationship.

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Most popular questions from this chapter

In the potential outcomes framework with heterogeneous (nonconstant) treatment effect, write the error as$$u_{i}=\left(1-x_{i}\right) u_{i}(0)+x_{i} u_{i}(1).$$Let \(\sigma_{0}^{2}=\operatorname{Var}\left[u_{i}(0)\right]\) and \(\sigma_{1}^{2}=\operatorname{Var}\left[u_{i}(1)\right] .\) Assume random assignment. i. Find \(\operatorname{Var}\left(u_{i} | x_{i}\right)\).ii. When is \(\operatorname{Var}\left(u_{i} | x_{i}\right)\) constant?

Using data from 1988 for houses sold in Andover, Massachusetts, from Kiel and Mcclain (1995) , the following equation relates housing price (price) to the distance from a recently built garbage incinerator (dist): $$\begin{aligned} \widehat{\log (\text {price})} &=9.40+0.312 \log (\text {dist}) \\ n &=135, R^{2}=0.162. \end{aligned}$$ i. Interpret the coefficient on log (dist). Is the sign of this estimate what you expect it to be? ii. Do you think simple regression provides an unbiased estimator of the ceteris paribus elasticity of price with respect to dist? (Think about the city's decision on where to put the incinerator. iii. What other factors about a house affect its price? Might these be correlated with distance from the incinerator?

Consider the savings function $$\operatorname{sav}=\beta_{0}+\beta_{1} \operatorname{inc}+u, u=\sqrt{i n c} \cdot e$$ where \(e\) is a random variable with \(\mathrm{E}(e)=0\) and \(\operatorname{Var}(e)=\sigma_{e}^{2} .\) Assume that \(e\) is independent of inc. i. Show that \(\mathrm{E}(u | \text { inc })=0,\) so that the key zero conditional mean assumption (Assumption SLR.4) is satisfied. [Hint: If \(e \text { is independent of inc, then } \mathrm{E}(e | i n c)=\mathrm{E}(e) .]\) ii. Show that \(\operatorname{Var}(u | \text { inc })=\sigma_{c}^{2}\) inc, so that the homoskedasticity Assumption SLR.5 is violated. In particular, the variance of sav increases with inc. [Hint: Var(elinc) = Var(e) if \(e\) and inc are independent.] iii. Provide a discussion that supports the assumption that the variance of savings increases with family income.

The data set BWGHT contains data on births to women in the United States. Two variables of interest are the dependent variable, infant birth weight in ounces (bwght), and an explanatory variable, average number of cigarettes the mother smoked per day during pregnancy (cigs). The following simple regression was estimated using data on \(n=1,388\) births: $$\widehat{b w g h t}=119.77-0.514 \mathrm{cigs}$$ i. What is the predicted birth weight when \(\operatorname{cigs}=0\) ? What about when \(\operatorname{cigs}=20\) (one pack per day)? Comment on the difference. ii. Does this simple regression necessarily capture a causal relationship between the child's birth weight and the mother's smoking habits? Explain. iii. To predict a birth weight of 125 ounces, what would cigs have to be? Comment. iv. The proportion of women in the sample who do not smoke while pregnant is about. \(85 .\) Does this help reconcile your finding from part (iii)?

Let \(x\) be a binary explanatory variable and suppose \(P(x=1)=\rho\) for \(0<\rho<1\). i. If you draw a random sample of size \(n\), find the probability-call it \(\gamma_{n}-\) that Assumption \(\mathrm{SLR} .3\) fails. [Hint: Find the probability of observing all zeros or all ones for the \(x_{i} .\) ] Argue that \(\gamma_{n} \rightarrow 0\) as \(n \rightarrow \infty\). ii. If \(\rho=0.5,\) compute the probablity in part (i) for \(n=10\) and \(n=100 .\) Discuss. iii. Do the calculations from part (ii) with \(\rho=0.9 .\) How do your answers compare with part (ii)?
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