1. Consumption, Saving, and Investment
Mr. Vaughan
Income and Employment Theory (402)

2. Lecture Goals
Use microeconomic building blocks to explore household choices of consumption/saving patterns.
Assumptions:
Household takes real wage as given (i.e., labor market is perfectly competitive)
Household takes interest rate as given (i.e., bond market is perfectly competitive)
Households choose a time path of consumption (C1, C2, C3, etc.) to maximize utility subject to a budget constraint.
Higher consumption in any year increases household utility.
Households prefer smooth consumption, even if income is irregular (i.e., they prefer C1, C2, C3, to relatively close together)

3. Consumption and Saving
Recall,
Household budget constraint:
C + (1/P)·∆B+ ∆K = π/P + (w/P)·L + i ·(B/P +K) (6.12)
π /P = 0
C + (1/P)·∆B+ ∆K = (w/P)·L + i·(B/P + K) (7.1)
(i.e., consumption + real saving = real income)

4. Consumption and Saving
Consumption over Two Years:
Year1
C1 + (B1/P + K1) − (B0/ P + K0) = (w/P)1·L + i0·(B0/P + K0) (7.2)
(i.e., real consumption in year1 + real saving in year1 = real income in year1)
Year2
C2 + (B2/P + K2) − (B1/P + K1) = (w/P)2·L + i1 · (B1/P + K1) (7.3)
(i.e., real consumption in year2 + real saving in year 2 = real income in year 2)
Now, combine budget constraints to describe household’s choice between consuming this year, C1, and next year, C2.
Notice both 7.2 and 7.3 include (B1/P + K1). So solve 7.2 for (B1/P + K1)
So solving 7.2 for (B1/P + K1), yields:
B1/P+K1 = B0/P+K0 + [i0·(B0/P + K0) + (w/P)1·L] - C1 (7.4)
(i.e., real assets end year1 = real assets end year0 + real income year1 − consumption year1)
Similarly,
B2/P+K2 = B1/P+K1 + [i1·(B1/P + K1) + (w/P)2·L] – C2 (7.5)
(i.e., real assets end year2 = real assets end year1 + real income year2 − consumption year2)

5. Consumption and Saving
Consumption over Two Years:
Factoring out “(B0/P + K0)” on the right-hand side of 7.4 leaves:
B1/P + K1 = (1+i0) · (B0/P + K0) + (w/P)1 · L − C1 (7.6)
Doing the same with “(B1/P + K1)” in 7.5 leaves:
B2/P + K2 = (1+i1) · (B1/P + K1) + (w/P)2 · L − C2 (7.7)
Now, substituting 7.6 into 7.1 yields:
B2/P+K2 = (1+i1)·[(1+i0)·(B0/P+K0)+(w/P)1·L - C1] + (w/P)2·L − C2 or
B2/P + K2 = (1+i1)·(1+i0)·(B0/P+K0)+(1+i1)·(w/P)1·L - (1+i1)·C1 + (w/P)2·L − C2 (7.8)
Dividing by (1+i1) and collecting consumption terms on left-hand side yields…

6. Consumption and Saving
Key Equation (two-year household budget constraint):
C1+[C2/(1+i1)] = (1+i0)·(B0/P+K0)+(w/P)1·L + [(w/P)2·L]/(1+i1) - (B2/P+K2)/(1+i1) (7.9)
In words:
Present value of consumption = Value of initial assets + Present value of of wage incomes − Present value of assets end year 2
Now, to simplify analysis algebraically, let:
V = ( 1 + i0)·(B0/P+K0) + (w/P)1·L + (w/P)2·L/(1+i1) (7.10)
Present value of sources of funds = Value of initial assets + Present value of wage incomes
So, (7.9) can be rewritten:
C1+[C2/(1+i1)] = V - (B2/P+K2)/(1+i1) (7.11)
Present value of consumption = Present value of initial assets − Present value of assets end year 2

7. Consumption and Saving
Choosing Consumption: Income Effects
To keep things simple, hold (B2/P+K2)/(1+ i1) constant.
Suppose V, present value of sources of funds, rises due to an increase in initial assets (B0/P + K0) or wage incomes (w/P)1·L and (w/P)2·L. Since (B2/P+K2)/(1+ i1) is held constant, total present value of consumption C1 + C2/(1 + i1) must rise by V.
Households prefer consuming at similar levels across years so C1 and C2 will rise by similar amounts.
This response of consumption to increases in initial assets/wage incomes are called income effects.

8. Consumption and Saving
Choosing Consumption: Inter-temporal Substitution Effects
C1 + C2/(1+i1) = V − (B2/P+K2)/(1+i1)
p.v. of consumption = p.v. of sources of funds − p.v. of assets end year 2
Again, to keep things simple hold (B2/P+K2)/(1+ i1) constant.
Higher i1 provides greater reward for deferring consumption. Therefore, household responds to increase in i1 by lowering C1 and raising C2.
Viewed another way, higher “i” induces households to save more today.
Response is called inter-temporal-substitution effect

9. Consumption and Saving
Choosing Consumption: Income Effects from Δi1
Again, to keep things simple hold (B2/P + K2) / (1 + i1) constant.
Recall, household budget constraint for year 2:
C2 + (B2/P + K2) − (B1/P + K1) = (w/P) 2·L + i1·(B1/P + K1) (7.3)
Income effect from Δi1 is [i1·(B1/P + K1)]
i1(B1/P)
i1K1
Income effect from change in interest rate, i1(B1/P)
For bondholder, income effect from increase in i1 is positive. For bond issuer, income effect from increase in i1 is negative. => For economy as a whole, lending/borrowing balances.
So income effect from term i1·(B1/P) is zero.
Income effect from change in interest rate (i1K1)
Average household’s holding of claims on capital, K1, exceeds zero.
i1K1 (income effect from increase in i1) is positive.

10. Consumption and Saving
Income effect from change in interest rate
In aggregate, income effect from increase in i1 consists of zero effect from i1·(B1/P) and positive effect from i1K1.
Result: Full income effect from increase in i1 is positive.

11. Consumption and Saving
Combining Income and Substitution Effects:
Effect of increase in current interest rate (i1) on year 1 consumption (C1)
Inter-temporal substitution effect motivates household to reduce consumption today (C1 ↓).
Positive income effect motivates household to raise consumption today (C1 ↑).
Overall impact on current consumption is ambiguous (C1 ?).

12. Consumption and Saving

13. Consumption and Saving
Consumption Over Many Years
Relax simplifying assumption households cannot change present value of assets held at end of year 2 (i.e., consider consumption over “n” years)
Consumption and income in future years.
Present value of lifetime consumption (“n” years) =
C1 + C2/(1 + i1) + C3/[(1 + i1)·(1 + i2) ] + · · ·
Present value of lifetime wage income (“n” years) =
(w/P)1·L + (w/ P)2·L/(1+ i1) + (w/P)2·L/[(1+i1)·(1+i2) ] + · · ·
Present value of assets in nth year) =
(Bn/P+Kn)/(1+i1)·(1+i2)·(1+i3)…

14. Consumption and Saving
Consumption Over Many Years
(Bn/P+Kn)/(1+i1)·(1+i2)·(1+i3)… is effectively zero and can be dropped.
Multiyear budget constraint (key equation, 7.12)
C1 + C2/(1+i1) + C3/[(1+i1)·(1+i2) ] + · · · =
(1+ i0)·(B0/P+K0) + (w/P)1·L + (w/P)2·L/(1+ i1) + (w/P)2·L/[(1+i1)·(1+i2) ] + · · ·
p.v. of lifetime consumption = value of initial assets + present value of lifetime wage income.
What is the point?
Multi-year budget constraint facilitates analysis of impact of temporary and permanent changes in income.

15. Consumption and Saving
Consumption Over Many Years: Temporary change in income
C1 + C2/(1+i1) + C3/[(1+i1)·(1+i2) ] + · · · =
(1+ i0)·(B0/P+K0) + (w/P)1·L + (w/P)2·L/(1+ i1) + (w/P)2·L/[(1+i1)·(1+i2) ] + · · ·
p.v. of lifetime consumption = value of initial assets + present value of lifetime wage income.
Household responds to rise in (w/P)1·L by raising consumption by similar amounts in each year: C1, C2, C3, and so on.
This response means consumption in any particular year, such as year 1, cannot increase very much.
Implication: If (w/P)1·L rises by one unit, C1 increases by much less than one unit.
Put it another way, Propensity to Consume in year 1 from extra unit of year 1 income is small when extra income is temporary.

16. Consumption and Saving
Consumption Over Many Years: Temporary change in income
If (w/P)1·L rises by one unit on right-hand side, C1 rises by much less than one unit on left-hand side.
Year 1 real saving, (B1/P + K1) − (B0/P + K0), must rise by nearly one unit on left-hand side.
Implication: Propensity to Save in year 1 from extra unit of year 1 income is nearly one when extra income is temporary.

17. Consumption and Saving
Consumption Over Many Years: Permanent Change in Income
(w/P)1·L, (w/P)2·L, (w/P)3·L, and so on each rise by one unit.
Households respond by increasing consumption by one unit in each year.
Implication: Propensity to consume from extra unit of year 1 income is high—close to one—when extra income is permanent. (Or, put another way, Propensity to Save in year 1 from extra unit of year 1 income is small when extra income is permanent.
Permanent-Income Hypothesis (Milton Friedman): Consumption depends on long-term average of incomes (permanent income) rather than current income.

18. Consumption, Saving, and Investment in Equilibrium
Determine aggregate quantities of consumption and saving.
Determine aggregate quantity of investment.

19. Consumption, Saving, and Investment in Equilibrium
Budget Constraint
C + (1/P)·∆B+ ∆K = (w/P)·L + i·(B/P) + iK
i = (R/P − δ )
C+ (1/P)·∆B+ ∆K =
(w/P)·L + i·(B/P) + (R/P) · K − δ K
B = 0 and ∆B = 0

20. Consumption, Saving, and Investment in Equilibrium
Budget Constraint
C+ ∆K = (w/P)·L + (R/P)·K − δ K
(w/P)·L + (R/P)·K = Y (Real GDP).
C + ∆K = Y − δ K
Consumption + net investment =
real GDP − depreciation =
real net domestic product

21. Consumption, Saving, and Investment in Equilibrium
Left-hand side of equation implies economy’s net investment, ∆K, is determined by households’ choices of consumption, C.
Given real net domestic product, one unit more of consumption, C, means one unit less of net investment, ∆K.
This choice of C determines ∆K