# 1. Measure the own price elasticity of X
modelX <- log(oz_X) ~ log(pX)
fitX <- lm(modelX)
summary(fitX)##
## Call:
## lm(formula = modelX)
##
## Residuals:
## Min 1Q Median 3Q Max
## -0.6564 -0.2395 -0.0788 0.1190 1.2732
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) -16.5304 0.3840 -43.05 <2e-16 ***
## log(pX) -6.9446 0.1074 -64.63 <2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.3671 on 726 degrees of freedom
## Multiple R-squared: 0.8519, Adjusted R-squared: 0.8517
## F-statistic: 4177 on 1 and 726 DF, p-value: < 2.2e-16#Self price elasticity for X is -6.9.
# 2. Measure the own price elasticity of Y
modelY <- log(oz_Y) ~ log(pY)
fitY <- lm(modelY)
summary(fitY)##
## Call:
## lm(formula = modelY)
##
## Residuals:
## Min 1Q Median 3Q Max
## -2.9059 -0.6063 0.0550 0.6057 3.4900
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) -15.5981 0.8990 -17.35 <2e-16 ***
## log(pY) -6.6942 0.2519 -26.57 <2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.9671 on 726 degrees of freedom
## Multiple R-squared: 0.4931, Adjusted R-squared: 0.4924
## F-statistic: 706.1 on 1 and 726 DF, p-value: < 2.2e-16#Self price elasticity for Y is -6.69
# 3. Qualitatively compare the elasticities. What do you infer?
## The self price elasticity for X is -6.9. That is, for every 1% increase in the price of X (Coke), the demand of X drops by 6.9%.
## The self price elasticity for Y is -6.69. That is, for every 1% increase in the price of Y (Pepsi), the demand of Y drops by -6.69%
## Hence, if the retailer wants to offer a discount, he should offer it on X(Coke) because demand will increase more in comparison to Y (Pepsi).# 4. Measure the cross-price elasticity of X w.r.t Y
modelXY <- log(oz_X) ~ log(pY)
fitXY <- lm(modelXY)
summary(fitXY)##
## Call:
## lm(formula = modelXY)
##
## Residuals:
## Min 1Q Median 3Q Max
## -1.3777 -0.5510 -0.4059 -0.1456 2.5414
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 9.4395 0.8857 10.658 <2e-16 ***
## log(pY) 0.3268 0.2482 1.317 0.188
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.9528 on 726 degrees of freedom
## Multiple R-squared: 0.002383, Adjusted R-squared: 0.001008
## F-statistic: 1.734 on 1 and 726 DF, p-value: 0.1883#Cross price elasticity of X wrt Y is 0.32. However, the p-value is not significant (it is 0.188). Hence, we cannot reject our null hypotheses.
# 5. Measure the cross-price elasticity of Y w.r.t X
modelYX <- log(oz_Y) ~ log(pX)
fitYX <- lm(modelYX)
summary(fitYX)##
## Call:
## lm(formula = modelYX)
##
## Residuals:
## Min 1Q Median 3Q Max
## -3.3541 -0.9954 -0.0906 0.9914 3.6652
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 11.1712 1.4169 7.884 1.16e-14 ***
## log(pX) 0.8120 0.3965 2.048 0.0409 *
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 1.354 on 726 degrees of freedom
## Multiple R-squared: 0.005745, Adjusted R-squared: 0.004376
## F-statistic: 4.195 on 1 and 726 DF, p-value: 0.0409#Cross price elasticity of Y wrt X is 0.8.
# 6. Are they the same? Or different? What do you infer?
# The cross price elasicities are different. Also, as already stated, we can't comment on cross-price elasticity of X w.r.t Y. The cross price elasticity of Y wrt X is 0.8. That is, with 1% increase in the price of X, the demand off Y increase by 0.8%.# 7. Measure the price elasticity of X when a deal is (not) offered?
modelDealX <- log(oz_X) ~ log(pX)*deal_X
fitDealX <- lm(modelDealX)
summary(fitDealX)##
## Call:
## lm(formula = modelDealX)
##
## Residuals:
## Min 1Q Median 3Q Max
## -0.47804 -0.11484 -0.01953 0.06748 1.28798
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) -2.4229 0.5048 -4.800 1.93e-06 ***
## log(pX) -2.9133 0.1435 -20.302 < 2e-16 ***
## deal_XYes -3.9397 1.3448 -2.930 0.0035 **
## log(pX):deal_XYes -1.4255 0.3588 -3.973 7.80e-05 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.2288 on 724 degrees of freedom
## Multiple R-squared: 0.9426, Adjusted R-squared: 0.9424
## F-statistic: 3966 on 3 and 724 DF, p-value: < 2.2e-16#The difference in the price elasticity when the deal is offered vs not offered is -1.4. The price elasticity when deal is not offered is -2.9. The price elasticity when the deal is offered is -4.3.
# 8. Is the price elasticity of X when a deal is offered different than the price elasticity when a deal is not offered?
## The difference in the price elasticity when the deal is offered vs not offered is -1.4. Since p-value is significant (<0.05) we can comment that the elasticities are statistically significant. It is a good idea to offer a deal on X.
# 9. Measure the price elasticity of Y when a deal is (not) offered?
modelDealY <- log(oz_Y) ~ log(pY)*deal_Y
fitDealY <- lm(modelDealY)
summary(fitDealY)##
## Call:
## lm(formula = modelDealY)
##
## Residuals:
## Min 1Q Median 3Q Max
## -2.83238 -0.61046 0.02982 0.60163 3.06752
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) -3.5422 1.7345 -2.042 0.0415 *
## log(pY) -3.2352 0.4947 -6.539 1.17e-10 ***
## deal_YYes -7.0003 6.7669 -1.034 0.3013
## log(pY):deal_YYes -2.2084 1.7845 -1.238 0.2163
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.927 on 724 degrees of freedom
## Multiple R-squared: 0.5355, Adjusted R-squared: 0.5335
## F-statistic: 278.2 on 3 and 724 DF, p-value: < 2.2e-16# The difference in the price elasticity when the deal is offered vs not offered is -2.2. The price elasticity when deal is not offered is -3.2. The price elasticity when the deal is offered is -5.4.
# 10 Is the price elasticity of Y when a deal is offered different than the price elasticity when a deal is not offered?
# The difference in the price elasticity when the deal is offered vs not offered is -2.2. However, the p-value is not less than 0.05. It is 0.2163. Hence, we cannot reject the null hypotheses and can't comment if the price elasticity of Y when a deal is offered different than the price elasticity when a deal is not offered.```