Exam Prep Flashcards
Name ways to measure inequality
- Gini
- Percentile based measurement: 90 - 50
-Variance based measurement: log - var
What is the PIH? Give some intuition to the theory.
Permanent Income Hypothesis: consumers consume approximately the same level of consumption over their lifetime.
Intuition: the agent spreads initial wealth + human capital evenly across all periods and consumes its annuity value (thus, he probably borrows at the beginning of the lifetime, then saves and dissaves after retirement)
How do we introduce uncertainty in the model as discussed in class?
Through income (y)
Name the budget constraint for the model as discussed in class. Explain each component.
a_t = (1+r) a_ (t-1) + y_t - c_t
a: savings
y: income
c: consumption
r: return
Write and explain the typical model with which we could measure and investigate the PIH (with income uncertainty)
Maximize:
max {c_t, a_t} E_0 Σ (from t=0 to T) β^t u(c_t)
Subject to:
a_t = (1 + r) a_{t-1} + y_t - c_t, for all t in {0,1, …, T}
a_T ≥ 0
a_{-1} given
Explain the difference between the deterministic and the stochastic approach.
Deterministic: no income uncertainty
Stochastic: includes income uncertainty, thus we have to work with expectation operators
What is the Euler for the deterministic model? What does it imply?
u’(c_t) = ß (1+r) u’(c_(t+1))
- This characterises the optimal decision. Typically ß (impatience) <1, (1+r) is the price of moving consumption across periods (r is small positive number)
- c_0 determines the entire consumption path until period T (higher c_0 implies higher uniform consumption profile)
- c_0 must be picked s.t. budget constraint is satisfied and a_T = 0
What is the consolidated budget constraint in the deterministic model? Explain the equation.
Σ c_t/(1+r)^t = w_0 + Σ y_t/(1+r)^t
This is the present value of total consumption equals the sum of PVs of total income and initial wealth.
What is the consumption in period t in a deterministic model given that (1+r)ß = 1?
c_t = R_0 [w_0 + Σ y_s/(1+r)^s]
where R_0 = Σ 1/(1+r)^s
Does PIH hold in the real world? If yes, how? If no, where does it fail?
- consumption tracks labor income over the lifecycle (consumption is hump-shaped)
- consumption drops at retirement (consumption retirement puzzle)
What fixes have been proposed to move from the PIH?
- introduce uncertainty & borrowing constraints
Explain the stochastic life-cycle model using z_t to introduce uncertainty. State the assumptions and the model specifications.
Problem: consumption in next period depends on income in last period
Let z_t = [ k_1, …, k_t] be the history of all relevant variables up to period t and pi(z_t ) be the unconditional probability of z_t realizing.
Then,
E [u(c_t)] = Σ pi (z_t) u(c_t (z_t))
a_t (z_t) = a_(t-1) (z_t-1) (1+r) + y_t(1+r) - c_t(z_t)
We then sum over all histories that are a continutation of z_t
What is the Euler equation of the stochastic model? Give some intuition.
u’(c_t(z_t)) = ß (1+r) Σ pi(z_(t+1)/ pi (z_t) u’(c_(t+1)(z_(t+1))
Simplified: u’(c_t) = ß(1+r) E( u’(c_(t+1))
Where the RHS term in the conditional expectation of the marginal utility
What is the stochastic model with quadratic utility? Give the utility function, the Euler and the implication about consumption growth.
Utility function: u(c) = - (c - č)^2
Euler: u’(c_t) = ß (1+r) E_t[u’(c_(t+1)]
Implication about consumption: c_t = E_t(c_(t+1)
Thus, consumption growth: c_t+1 - c_ t = c_t+1 - E_t(c_t+1) –> is the forecast error
Explain the model as introduced by Campbell & Mankiw (1989). State their hypothesis, the model and their findings. Give an interpretation of their findings.
Hypothesis: (1)hand-to_mouth consumers (lambda) & (2) PIH consumers ( 1- lambda)
Model: based on consumption growth
change in consumption = lambda change of c(1) + (1- lambda) change of c(2) = lambda change of y + (1 - lambda) e_t
Test: regress consumption growth on income growth and see if lambda = 0
Findings: lambda somewhere between 0.35 - 0.7
Interpretation: there is a substantial fraction of hand-to-mouth consumers (or a substantial failure of the PIH)
Give an intuition why consumption is less cyclical than GDP?
People save to self-insure against negative income shocks.
State the lifecycle model with borrowing limits (but not income uncertainty. Give an intuition on how we expand our model.
Also write up the Lagrangian and the resulting Euler.
Give some intuition on the parameters and the model.
Maximize:
max {c_t, a_t} Σ (from t=0 to T) β^t u(c_t)
Subject to:
a_t = (1 + r) a_{t-1} + y_t - c_t, for all t in {0,1, …, T}
a_T ≥ 0
a_t>= - b_t –> this gives an upper limit to debt
a_{-1} given
Lagrangian: L =Σ (from t=0 to T) ß^t {c(c_t) + lambda_t [(1 + r) a_{t-1} + y_t - c_t - a_t] + mu_t [ a_t + b_t]}
Euler: u’(c_t) - mu_t >= ß (1+r) u’(c_t+1)
Intuition:
- marginal utility might be lower in the next period than now
- this can happen with borrowing limits
- complementary slackness: if mu = 0 –> model w/o borrowing constraints, mu >0: borrowing constraint is binding, you eat everything you can (and still not consume enough today)
Explain the model as introduced by Zeldes (1989). Specifically, explain the utility function, give our testable version of the Euler equation (as derived from stochastic Euler), explain this more complicated model specification by Zeldes:
ln( c_t+1/c_t) = X_tß + gamma ln (1+r) + v_t + ln (1 + lambda)
Model: with income uncertainty
Utility function: with preference shifter Θ: U(c,Θ) = c^(1-a) / (1-a) * E(Θ)
Testable Euler: 1 = (ßE_t ( (1+r) u*(c_t+1)))/u’(c_t) *(1 + lambda’_t)
where lambda’ denotes stuff that we cannot measure (so the entire term around mu)
His version:
X: controls
r: interest
v: error term
lambda: unobservables
Explain the three tests as used by Zeldes. Also elaborate on how he does about testing these.
Process: creation of two samples using PSID
Sample 1: low wealth to income ratio, borrowing constraints hold
Sample 2: high ratio, borrowing constraints do not hold
Test 1: Does the log of disposable income enter the equation significantly?
–> current income does not influence consumption growth if PIH holds
–> high current income influences consumption growth negatively if borrowing constraints hold (y up, c_t up, c_t+1/c_t down)
Results: confirm borrowing constraints story
Test 2: Sum of residuals from regression should be positive for constrained agents
1) Estimate coefficients for group 2
2) Using the coefficients, compute the residuals in sample 1
3) Average residuals over population
Results: weakly significant positive residuals
Test 3: when income is larger (all else equal), borrowing constraints should bind less
- regress residuals for group 1 and test whether sign is negative, tells us the sign of the correlation between lambda and y
Results: confirms the borrowing constraints hypothesis
State the full consumption-saving model with uncertainty & borrowing constraints using the notation with histories (not in Bellman notation).
V_0 (z_0) = max {c_s(Z_s), a_s(z_s} Σ (from s=0 to T) pi(z_s) β^s u(c_s(z_s)
Subject to:
a_s(z_s) = (1 + r) a_{s-1}(z_t-1) + y_s(z_s) - c_s(z_s), for all s in {0,1, …, T}
a_T(z_T) ≥ 0
a_s(z_s) >= - b
a_{-1} given
State the Bellman equation for the full consumption-saving model with uncertainty & borrowing constraints using z as the state variable. Give some intuition about the equation.
V_t (z_t) = max {c_t(z_t), a_t(z_t} {u(c_t(z_t) + ßE_t(V_(t+1) (z_(t+1)}
Subject to:
a_t(z_t) = (1 + r) a_{t-1}(z_(t-1) + y_t(z_t) - c_t(z_t)
a_t(z_t) >= - b
Intuition: V_t+1 incorporates everything about the future effects of your choice you need to care about
Give the Bellman equation for the for the full consumption-saving model with uncertainty & borrowing constraints using wealth and income as the state variables. Give some intuition.
V_t (a_(t-1), y_t) = max {c_t(a_(t-1), y_t) >= b {u((1+r)a_(t-1) + y_t - a_t) + ßE_t(V_(t+1) (a_(t-1), y_t))}
Subject to:
a_t(a_(t-1), y_t)) = (1 + r) a_{t-1} + y_t - c_t
a_t(a_(t-1), y_t) >= - b
The current policy is a function of the current state (a_(t-1), y_t))
Now, give the Bellman equation for the for the full consumption-saving model with uncertainty & borrowing constraints using cash-on-hand as the state variables. Give some intuition.
Intuition: we do not care about where our resources come from, we just care how much we have
Let coh: x t = (1+r) a(t-1) + y_t
V_t (x_t) = max {c_t,a_t {u(c_t) + ßE_t(V_(t+1) (x_t))}
Subject to:
a_t = x_t - c_t
x_(t+1) = (1+r) a_t + y_(t+1)
a_t >= - b
How do you solve the consumption-saving model using the Bellman equation?
Intuition: you solve the value functions backwards
1) Solve for period T: a_t(x_T) = 0, c_T(x_T) = x_t and V_T(x_T) = u (x_T)
2) Solve for T -1: V_T-1(x_T-1) = max {a_T-1 {u(x_T-1 - a_T-1) + ßE_T-1(V_T (a_T-1(1+r) - y_T))}
