Week 8
Group 5
Matthew Kourlinins, Nicola Simpson, Nicholas Russell, Jack Nguyen
Part 1
Q1 - What’s Modigliani’s life-cycle theory of consumption?
People make intelligent decisions about their level of consumption spending at each age subject to the resources available over their lives, these two factors are positively correlated. A key motive for saving is to create a provision for retirement and a buffer for emergencies. A person starts off with very little wealth and accumulates it over their working life, it then slowly dissipates throughout their retirement. Economic growth creates wealth and total wealth is influenced on the length of retirements. Modigliani explains why some contradictions in the data exist. Young people often exclude contributions to pension plans by their employers when the household surveys are conducted and retired people include the entire payment from their plans as income, when a substantial proportion actually represents the drawing down of assets. This understates saving by young people and understates dissaving by old people. He also dismissed the theory that young people will consume more than their income due to expectations of income increasing as sound in theory but impractical, they were ‘unlikely’ to be capable to ‘borrow enough to secure living standards that were much beyond there means’. However Modigliani does cite some factors that somewhat undermine the life-cycle theory such as the uncertainty about one’s date of death limiting the degree to which assets are run down by the elderly. Additional concerns have also been raised by other economists such as the validity of the Modigliani insight that consumption profiles can be detached from income profiles. Over time Modigliani’s life-cycle model has evolved to deal with the current economic issues faced and new trains of thought, however the ‘life-cycle model will still be a baseline to which people aspire’ (Angus Deaton). 
At which periods are they saving & dissaving?
In periods of high population growth there are expected to be high rates of saving as young people save and accumulate wealth where old people spend it during retirement and vice-versa. In periods of economic growth there are also savings to be expected. In the economic cycle, they are savings in expansion periods and dissaving in downturns. The saving/dissaving is zero at the peak of a boom or valley of a recession. This is explained in greater detail in the last section of question 1.
Q2 - What effects do population growth & economic growth have on saving rates?
With higher rates of population growth there is ‘more young people than old’. Young people tend to accumulate wealth (save) in preparation for retirement when they’re old and consume it (spend), with more young people than old savings rates are expected to be higher. Therefore population growth has a positive effect on the rate of saving. Saving is also positively correlated with the rate of economic growth (rate of increase in total income). That is the faster incomes are rising (economy is growing) the greater the scale of the young saving (accumulating wealth) is than old dissaving (spending wealth) and hence a higher amount of net saving is present.
Q3 - In Modigliani’s model do consumers consume from current income?
Modigliani’s model contends that when a person is earning an income (working) they will be accumulating wealth (saving). In other words, people have a positive savings rate when earning their current income (except in some cases outliers in the lower tail of income distribution). He contends that these ‘savings’ accumulated are used to finance current consumption by arguing that current income has a lag effect present in relation to consumption.
Q4 - What’s the impact in the model of an increase in transitory income?
Transitory income is not permanent income, it is short-lived i.e. an inheritance or lottery winnings. Therefore the increase in the savings rate caused by the increase in transitory income is also temporary and vice-versa.
Q5 - What are the theory’s implications for savings rates over the economic cycle?
The economic cycle, commonly referred to as the business cycle, has 4 distinct stages of economic activity: Recession (low point), expansion (increasing), boom (peak) and downturn (decreasing). Modigliani argued that in the long-run (when cyclical factors aren’t present) savings ratios will not change if economic growth does not change. The theory argues that in at the peak of the boom and bottom of the recession the gradient of the curve is zero and we therefore have a maximum or minimum point where the growth rate is equal to zero. With an economic growth rate of zero Modigliani contends that no wealth will be created and hence “savings = consumption” and the savings rate will be equal to zero. A downturn and expansion will have opposite effects, the effects of an expansion are described here. In an expansion economic growth is positive and incomes as a whole are rising, with this increase in total income Modigliani states that the savings rate will be positive. The steeper the gradient, that is the higher the rate of economic growth, the higher this savings rate will be.
Part 2
library(dplyr); library(ggplot2)##
## Attaching package: 'dplyr'
##
## The following objects are masked from 'package:stats':
##
## filter, lag
##
## The following objects are masked from 'package:base':
##
## intersect, setdiff, setequal, union##Data Preperation
#Question 1
#import data
data <- read.csv("FCE GDP.csv")
#Question 2
#adding extra columns with logs and lags of Consumption and GDP
data <- data %>% mutate(lnC = log(Consumption),
lnGDP = log(GDP),
lnC_lagged = lag(lnC),
lnGDP_lagged = lag(lnGDP)) %>%
mutate(lnC_diff = c(NA, diff(lnC)), # difference between lnC and lags
lnGDP_diff = c(NA, diff(lnGDP))) #difference between lnGDP and lags
##Modelling
#Question 1
#plot of Consumption and GDP
ggplot() +
geom_line(aes(x = Date, y = GDP, group = 1, color = "GDP"), data = data) +
geom_line(aes(x = Date, y = Consumption, group = 1, color = "Consumption"), data = data) +
ylab("Expenditure") +
ggtitle("Consumption and GDP") +
scale_color_discrete(name = "")
#this doesn't seem to be stationary as it is trending upwards
#plot of lnC_diff and lnGDP_diff
ggplot() +
geom_line(aes(x = Date, y = lnC_diff, group = 1, colour = "lnC_Diff"), data = data) +
geom_line(aes(x = Date, y = lnGDP_diff, group = 1, colour = "lnGDP_Diff"), data = data) +
ylab("Difference") +
ggtitle("lnC_diff and lnGDP_diff") +
scale_color_discrete(name = "")## Warning: Removed 1 rows containing missing values (geom_path).## Warning: Removed 1 rows containing missing values (geom_path).
#this plot does seem to be stationary, having values around a certain value of 0.025
#Question 2
#linear model of logged values (mod1)
mod1 <- lm(lnC ~lnC_lagged, data = data)
summary(mod1)##
## Call:
## lm(formula = lnC ~ lnC_lagged, data = data)
##
## Residuals:
## Min 1Q Median 3Q Max
## -0.0298837 -0.0115955 -0.0000182 0.0116133 0.0262341
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 0.129859 0.048578 2.673 0.01 *
## lnC_lagged 0.992566 0.003806 260.816 <2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.0151 on 52 degrees of freedom
## (1 observation deleted due to missingness)
## Multiple R-squared: 0.9992, Adjusted R-squared: 0.9992
## F-statistic: 6.803e+04 on 1 and 52 DF, p-value: < 2.2e-16coef(mod1)[2] #value of b1## lnC_lagged
## 0.9925655#Question 3
#attempt of forecasting linear model (mod2)
mod2 <- lm(lnC ~lnGDP_lagged, data = data)
#adding residuals to table from mod1
data <- mutate(data, Residuals = NA)
data$Residuals[-1] <- resid(mod2)
#plotting residuals vs time with average trajectory
ggplot() +
geom_line(aes(x = Date, y = Residuals, group = 1), data = data) +
geom_smooth(aes(x = Date, y = Residuals, group = 1), data = data)+
ggtitle("Residuals vs Time")## geom_smooth: method="auto" and size of largest group is <1000, so using loess. Use 'method = x' to change the smoothing method.## Warning: Removed 1 rows containing missing values (stat_smooth).## Warning: Removed 1 rows containing missing values (geom_path).
#it does appear to be stationary, always returning to around 0
#If the residual is high in a given period, it does indicate that the future periods would have lower residuals
#Question 4
#lagging the residuals into a new column
data <- mutate(data,
R_lag = lag(Residuals))
#Question 5
#linear model
mod3 <- lm(lnC_diff ~ lnC_lagged + lnGDP, data = data)
summary(mod3)##
## Call:
## lm(formula = lnC_diff ~ lnC_lagged + lnGDP, data = data)
##
## Residuals:
## Min 1Q Median 3Q Max
## -0.0214939 -0.0087999 -0.0008341 0.0061723 0.0226861
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) -0.23982 0.06818 -3.517 0.000926 ***
## lnC_lagged -0.34195 0.05220 -6.551 2.79e-08 ***
## lnGDP 0.34557 0.05384 6.418 4.52e-08 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.01134 on 51 degrees of freedom
## (1 observation deleted due to missingness)
## Multiple R-squared: 0.4846, Adjusted R-squared: 0.4644
## F-statistic: 23.98 on 2 and 51 DF, p-value: 4.562e-08coef(mod3) #coefficients## (Intercept) lnC_lagged lnGDP
## -0.2398198 -0.3419508 0.3455688#Question 6
#linear model in q5 with lag in residuals
mod4 <- lm(lnC_diff ~ lnC_lagged + lnGDP + R_lag, data = data)
summary(mod4)##
## Call:
## lm(formula = lnC_diff ~ lnC_lagged + lnGDP + R_lag, data = data)
##
## Residuals:
## Min 1Q Median 3Q Max
## -0.0249962 -0.0079000 -0.0003246 0.0051651 0.0195132
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) -0.25304 0.07806 -3.242 0.00214 **
## lnC_lagged -0.36242 0.06061 -5.980 2.52e-07 ***
## lnGDP 0.36603 0.06263 5.844 4.07e-07 ***
## R_lag 0.08405 0.09028 0.931 0.35647
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.01139 on 49 degrees of freedom
## (2 observations deleted due to missingness)
## Multiple R-squared: 0.4933, Adjusted R-squared: 0.4623
## F-statistic: 15.9 on 3 and 49 DF, p-value: 2.371e-07coef(mod4) #coefficients## (Intercept) lnC_lagged lnGDP R_lag
## -0.25304379 -0.36242466 0.36603133 0.08404618#Question 7
#The marginal propensity to consume is dc/dy, so assuming we are using the equation from mod4 in question 6, the slope (B1) equals
coef(mod4)[3]## lnGDP
## 0.3660313