Macroeconomics, Volume I by David G. Tuerck

Author:David G. Tuerck
Language: eng
Format: epub
Publisher: Business Expert Press

c1

$50,495b

sav1

$9,505c

s1

15.84%d

sav1(1 + r)

9,980e

c2

$51,980f

Δc/c

2.94% g

a = $60,000 + $42,000/(1.05); b = 0.50495 × $100,000; c = $60,000 – $50,495;

d = $9,505/$60,000; e = $9,505 × (1 + 0.05); f = $42,000 + $9,980;

g = ($51,980 – $50,495)/$50,495

Table 4.2 Two-period model (r = 6%)

PVy

$99,623a

c1

$50,305b

sav1

$9,695c

s1

16.2%d

sav1(1 + r)

10,277e

c2

$52,277f

Δc/c

3.92%g

a = $60,000 + $42,000/(1.06); b = 0.50495 × $99,263; c = $60,000 – $50,305; d = $9,695/$60,000; e = $9,695 × (1 + 0.06); f = $42,000 + $10,277; g = ($52,277 – $50,305)/$50,305

Suppose that IES equals 1.5. Eve will want the percentage change in her consumption to equal 1.5 percentage points for every percentage point rise in r (again, holding ρ constant). The substitution effect of the rise in r will more than offset the income effect.

Let’s make our analysis more realistic now by assuming that Eve takes her first job on her 22nd birthday and that the job offers a starting salary of $50,000. Eve expects to retire in 40 years, when she is 62, and expects to get a 5% raise each year she is on the job (which means she’ll make about $335,000 dollars the last year she works!). We assume that, for convenience, the interest rate is initially also 5%. Finally, we assume that Eve has a rate of time preference of 2% and an IES of 1.5.

Discounted over Eve’s 40-year working life, the present value of her income is $2,000,000. Given our assumptions, Eve’s v is 2.77%, and her first-year consumption is $55,400 (see footnote a to Table 4.3). She plans that, during the first year of her career, her saving will be a negative $5,400 and her saving rate will therefore be –10.8%. Her planned year-2 consumption is $57,893, which, given the values assigned to r, ρ, and IES must be 4.5% greater than her year-1 consumption.

Now suppose that, just after Eve had planned out her current and future consumption, the interest rate rose unexpectedly to 6%. Eve’s income stream remains unchanged, but her v falls to 2.53%.

The present value of her income stream falls to $1,672,452. Thus, she revises her plans so that she will consume $42,313 in the first year and $44,852 in the second. She will increase her first-year saving from –$5,400 to $7,687 and her saving rate from –10.8 to 15.4%.

We can infer that a given rise in the return to saving produces a larger rise in Eve’s saving rate as her IES exceeds 1. But whatever the value of her IES, v will always be small when the planning period extends well out to the future. To see this, let’s consider a few more examples. In Table 4.4 we assume that ρ = 0.02, n = 40, r = 0.05, and lay1 = $50,000. The larger Eve’s IES, the smaller her v and the larger her s.

Table 4.3 Lifetime model: Effects of a rise in r

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