The data file malawi_small contains survey data from Malawi during 2007-2008 on total household expenditures in the prior month (in Malawian Kwacha) as well as expenditures on categories of goods such as food, clothes, and fuel. Consider the following models.

i. Budget share: PTELEPHONE \(=\beta_{1}+\beta_{2} \ln (\) TOTEXP \()+e\)

ii. Expenditure: \(\ln (\) PTELEPHONE \(\times\) TOTEXP \()=\alpha_{1}+\alpha_{2} \ln (\) TOTEXP \()+e\)

iii. Budget share: \(P C L O T H E S=\beta_{1}+\beta_{2} \ln (\) TOTEXP \()+e\)

iv. Expenditure: \(\ln (P C L O T H E S \times\) TOTEXP \()=\alpha_{1}+\alpha_{2} \ln (\) TOTEXP \()+e\)

v. Budget share: \(P F U E L=\beta_{1}+\beta_{2} \ln (\) TOTEXP \()+e\)

vi. Expenditure: \(\ln (P F U E L \times T O T E X P)=\alpha_{1}+\alpha_{2} \ln (\) TOTEXP \()+e\)

a. Estimate each of the models (i) to (vi). Interpret the estimated coefficients of \(\ln (T O T E X P)\). Is each item a necessity, or a luxury?

b. For each commodity equation (ii), (iv), and (vi), calculate the expenditure elasticity with respect to total expenditure at the 25th and 75th percentiles of TOTEXP.

c. For the budget share equations, (i), (iii), and (v), find the elasticities that are given by \(\varepsilon=\frac{\beta_{1}+\beta_{2}[\ln (\text { TOTEXP })+1]}{\beta_{1}+\beta_{2} \ln (\text { TOTEXP })}\). Are the changes in elasticities between the two percentiles, noticeable? [A standard log-log expenditure model can be obtained using the data, by creating a dependent variable that is the logarithm of the budget share times total expenditure. That is, for example, \(\ln (\) TELEPHONE \()=\ln (\) PTELEPHONE \(\times\) TOTEXP \()\).]