Answer Key - Problem Set 3

2.
If f(x) is a linear function of x then it can be mathematical represented in the following form:
f(x)=a+b*x, where a (intercept) and b (slope) are two arbitrary constants. Thus, we can
represent consumption and saving function with the same analytical expression. For a linear
function it is sufficient to know two arbitrary point on the line to find the values of both a and b.
For example, took two points form table 6-3 (page 109) in your text:
Disposable
Income
24000
25000
Consumption
Saving
First, let find a and b for the consumption function. We know that the slope of the consumption
function b is equal to the MPC, thus we can find b by dividing a change in consumption (800)
with a change in income (1000):
thus our consumption function is:
C = 5 000 + 0.8*DI (1)
You may use point B to check that it is actually true:
A
B
24200
25000
b=MPC=ΔC/ΔDI=800/1000=0.8
To find a simply use either point A or B in the following way: since C=a+b*DI, consumption at
point A is equal to:
C(B) = 5 000 + 0.8*25 000 = 5 000 + 20 000 = 25 000
The same logic applied to the savings function: S=a+b*DI, where b is the slope of the saving
function which is equal to the marginal propensity to save (0.2 in this table). To find a for this
function use same steps as with consumption function:
-200
24 200 = a+0.8*24 000 or a = 24 200 - 0.8*24 000 = 24 200 - 19 200 = 5 000
-200 = a+0.2*24 000 or a = -200 - 0.2*24 000 =- 200 - 4 800 = - 5 000
as a result, the saving function is
S = - 5 000 + 0.2*DI (2)
Alternatively, you can use your consumption function to derive the savings function:
S = DI-C, S = DI - 5000 - 0.8*DI = - 5000 + 0.2*DI
0
i.
ii.
Using data from the problem 1 fill the following table:
Consumption
function
Saving function
intercept a slope b intercept a slope b
A-B
B-C
300
450
0.95
0.85
-300
-450
0.05
0.15
C-D
D-E
E-F
750
1000
1300
0.70
0.60
0.50
-750
-1000
-1300
0.30
0.40
0.50
What is the economic meaning of the intercepts a of savings and consumption functions?
ANSWER: The value of intercept shows the consumption (savings) when disposable income is
equal to zero. That is in case of consumption function a represents the amount household
needs to consume in order to survive.
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