Question

Happyhols sells packaged holidays and has observed the following pattern of sales over the last 8 years in terms of the relationship with average income in the market:

                              2010    2011    2012    2013    2014    2015    2016    2017

Income ($000)        28        27      29       31       30        33        34         34

Sales (units)          380      430      410      420      450      440      480      500

1. Is it appropriate to use a lagged relationship in the above situation? Explain your reasoning.
2. Estimate an appropriate relationship.    
3. Estimate sales in 2018 stating any relevant assumptions.         
4. Estimate the income elasticity of demand for the period as a whole.
5. How much of the variation in sales is explained by the relationship with income?

Answer 1

Let's draw two diagrams: One without lag, and the other with lag (2010 Income and 2011 Sales, as we would expect people to use this year's income to plan next year's travel, and not the other way around).

Sales on primary (left) axis and Income on secondary (right) axis

Without lag

With lag

a. It is clear that the lagged series follow each other more closely (the reason could be that people plan their next year's travel based on current year's income)

b. If we plot the lagged relationship in a table as follows, and calculate the period to period changes in both income and sales, and see the ratio we find that sales go up or down by about 11.67 units in the current period for each 1000 increase or decrease in income in the previous period.

Year	2010	2011	2012	2013	2014	2015	2016	2017	Average
Income (T-1)		28	27	29	31	30	33	34	30.28571
Sales (T)	380	430	410	420	450	440	480	500	447.1429
Absolute change in income from T-1 to T			-1	2	2	-1	3	1	1
Absolute change in sales from T to T+1			-20	10	30	-10	40	20	11.66667
Ratio of change (sales / income)			20	5	15	10	13	20

c. Sales in 2018 will depend on the income in 2017 (remember lagged). Since the income in 2017 (34) is the same as the income in 2016 (34), we can expect Sales in 2018 to be equal to the Sales in 2017, i.e., 500

d. Income elasticity, i.e., % change in sales / % change in income = 0.78. Pls see table below for calculations

Year	2010	2011	2012	2013	2014	2015	2016	2017	Average
Income (T-1)		28	27	29	31	30	33	34	30.28571
Sales (T)	380	430	410	420	450	440	480	500	447.1429
% change in income from T-1 to T			-3.6%	7.4%	6.9%	-3.2%	10.0%	3.0%	3.4%
% change in Sales from T to T+1			-4.7%	2.4%	7.1%	-2.2%	9.1%	4.2%	2.7%
Ratio of change (sales / income)			1.30	0.33	1.04	0.69	0.91	1.38	0.78

e. Pls see below.

Year	2010	2011	2012	2013	2014	2015	2016	2017	Average
Income (T-1)		28	27	29	31	30	33	34	30.28571
Sales (T)	380	430	410	420	450	440	480	500	447.1429
Absolute change in income from T-1 to T			-1	2	2	-1	3	1	1
Absolute change in sales from T to T+1			-20	10	30	-10	40	20	11.66667
Ratio of change (sales / income)			20	5	15	10	13	20
Estimated sales (using 11.67 as the factor, according to the answer in b.)			418.33	433.34	443.34	438.33	475.01	491.67
Delta from actual sales			-8.33	-13.34	6.66	1.67	4.99	8.33