7.19. The demand for chicken in the United States, 1960-1982. To study the per capita consumption of chicken in the United States, you are given the data in Table 7.9, where   Y=   per capita consumption of chickens, lb
\[
$\begin{aligned}
X_{2} &=\text { real disposable income per capita, } \$ \\
X_{3} &=\text { real retail price of chicken per } $\mathrm{lb}, \phi$ \\
X_{4} &=\text { real retail price of pork per } $\mathrm{lb}, \phi$ \\
X_{5} &=\text { real retail price of beef per } $\mathrm{lb}, \phi$
\end{aligned}$
\]
  $X_{6}=$   composite real price of chicken substitutes per   $\mathrm{lb}, \phi$  , which is a weighted average of the real retail prices per lb of pork and beef, the weights being the relative consumptions of beef and pork in total beef and pork consumption>Note: The real prices were obtained by dividing the nominal prices by the Consumer Price Index for food>Now consider the following demand functions:
\[
$\begin{array}{l}
\ln Y_{t}=\alpha_{1}+\alpha_{2} $\ln X_{2 t}$+\alpha_{3} $\ln X_{3 t}$+u_{t} \\
\ln Y_{t}=\gamma_{1}+$\gamma_{2}$ $\ln X_{2 t}$+$\gamma_{3}$ $\ln X_{3 t}$+\gamma_{4} \ln X_{4 t}+u_{t} \\
\ln Y_{t}=\lambda_{1}+\lambda_{2} $\ln X_{2 t}$+\lambda_{3} $\ln X_{3 t}$+\lambda_{4} \ln X_{5 t}+u_{t} \\
\ln Y_{t}=\theta_{1}+\theta_{2} $\ln X_{2 t}$+\theta_{3} $\ln X_{3 t}$+\theta_{4} \ln X_{4 t}+\theta_{5} \ln X_{5 t}+u_{t} \\
\ln Y_{t}=\beta_{1}+$\beta_{2}$ $\ln X_{2 t}$+$\beta_{3}$ $\ln X_{3 t}$+\beta_{4} \ln X_{6 t}+u_{t}
\end{array}$
\]
From microeconomic theory it is known that the demand for a commodity generally depends on the real income of the consumer, the real price of the commodity, and the real prices of competing or complementary commodities. In view of these considerations, answer the following questions.
a. Which demand function among the ones given here would you choose, and why?
b. How would you interpret the coefficients of   $\ln X_{2 t}$   and   $\ln X_{3 t}$   in these models?
c. What is the difference between specifications (2) and (4)?
d. What problems do you foresee if you adopt specification (4)? (Hint: Prices of both pork and beef are included along with the price of chicken.)
e. Since specification (5) includes the composite price of beef and pork, would you prefer the demand function (5) to the function (4)? Why?
f. Are pork and/or beef competing or substitute products to chicken? How do you know?
g. Assume function (5) is the "correct" demand function. Estimate the parameters of this model, obtain their standard errors, and   $$R^{2}$, \bar{R}^{2}$  , and modified   $R^{2}$  . Interpret your results.
  h  . Now suppose you run the "incorrect" model (2). Assess the consequences of this mis-specification by considering the values of   $\gamma_{2}$   and   $\gamma_{3}$   in relation to   $\beta_{2}$   and   $\beta_{3}$  , respectively. (Hint: Pay attention to the discussion in Section 7.7.)

7.19. The demand for chicken in the United States, 1960-1982. To study the per capita consumption of chicken in the United States, you are given the data in Table 7.9, where $Y=$ per capita consumption of chickens, lb $X_{2}=$ real disposable income per capita, $\$$ $X_{3}=$ real retail price of chicken per lb, $\notin$ $X_{4}=$ real retail price of pork per lb, $\notin$ $X_{5}=$ real retail price of beef per lb, $\notin$ $X_{6}=$ composite real price of chicken substitutes per $\mathrm{lb}, \boldsymbol{\ell}$, which is a weighted average of the real retail prices per $\mathrm{lb}$ of pork and beef, the weights being the relative consumptions of beef and pork in total beef and pork consumption
Note: The real prices were obtained by dividing the nominal prices by the Consumer Price Index for food
Now consider the following demand functions: \[ \begin{array}{l} \ln Y_{t}=\alpha_{1}+\alpha_{2} \ln X_{2 t}+\alpha_{3} \ln X_{3 t}+u_{t} \\ \ln Y_{t}=\gamma_{1}+\gamma_{2} \ln X_{2 t}+\gamma_{3} \ln X_{3 t}+\gamma_{4} \ln X_{4 t}+u_{t} \\ \ln Y_{t}=\lambda_{1}+\lambda_{2} \ln X_{2 t}+\lambda_{3} \ln X_{3 t}+\lambda_{4} \ln X_{5 t}+u_{t} \\ \ln Y_{t}=\theta_{1}+\theta_{2} \ln X_{2 t}+\theta_{3} \ln X_{3 t}+\theta_{4} \ln X_{4 t}+\theta_{5} \ln X_{5 t}+u_{t} \\ \ln Y_{t}=\beta_{1}+\beta_{2} \ln X_{2 t}+\beta_{3} \ln X_{3 t}+\beta_{4} \ln X_{6 t}+u_{t} \end{array} \] From microeconomic theory it is known that the demand for a commodity generally depends on the real income of the consumer, the real price of the commodity, and the real prices of competing or complementary commodities. In view of these considerations, answer the following questions. a. Which demand function among the ones given here would you choose, and why? b. How would you interpret the coefficients of $\ln X_{2 t}$ and $\ln X_{3 t}$ in these models? c. What is the difference between specifications (2) and (4)? d. What problems do you foresee if you adopt specification (4)? (Hint: Prices of both pork and beef are included along with the price of chicken.) e. Since specification (5) includes the composite price of beef and pork, would you prefer the demand function (5) to the function (4)? Why? f. Are pork and/or beef competing or substitute products to chicken? How do you know? g. Assume function (5) is the "correct" demand function. Estimate the parameters of this model, obtain their standard errors, and $R^{2}, \bar{R}^{2}$, and modified $R^{2}$. Interpret your results. h. Now suppose you run the "incorrect" model (2). Assess the consequences of this mis-specification by considering the values of $\gamma_{2}$ and $\gamma_{3}$ in relation to $\beta_{2}$ and $\beta_{3}$, respectively. (Hint: Pay attention to the discussion in Section 7.7.)