Week 2: 15/02/21
ECN302 – Advanced Macroeconomics - Uncertainty
Video 1
Since there is now uncertainty, the individual does not know in period t=0 what their future
endowment income will be.
The desired outcome of the individual is perfect consumption smoothing.
We will look at how consumption allocation changes with uncertainty.
Again, we will be using a Dynamic General Equilibrium (DGE) model.
This is an endowment economy with no government and no foreign sector.
The mathematical tools that we will use are the first order difference equations (FODEs) & dynamic
optimisation. These are the same tools that we used in topic 1.
We will focus on endowment income uncertainty alone.
-Environment
This is the exact same as it is in topic 1 (deterministic). The only difference in this model is that
future income is unknown. the representative agent can make expectations on future income. The
assumption is that all individuals form the same expectations.
1) Future income expectations are based on (and are therefore a function of) current income.
Therefore: Xt+1=f(Xt). This is called adaptive expectation.
2) Φt is the information set – the assumption is that this contains all the info on the given
variable up until time t. Expectations are conditional on Φ t, meaning that E[Xt+1 l Φt]. This can
also be written as Et[Xt+1]. This is called rational expectation as it uses all info in order to
produce a forecast.
In period t+1, we compare both expectations to get the forecast error:
Xt+1 - Et[Xt+1] = et+1
The rational expectation hypothesis implies that the expectation today of the forecast error is equal
to zero, meaning that people do not make forecast errors.
Forecast errors are uncorrelated, meaning that making a forecast error today does not automatically
mean a forecast error will be made in the next period.
The assumption is that the sequence of income is expected to be bounded (no one expects to
become infinitely rich in the future), therefore:
, Week 2: 15/02/21
-Intertemporal budget constraint
We’ll use forward substitution between 2 subsequent periods.
The budget constraints in period t=0 is:
c0 + a0 = y0 + (1 + r) a-1 (E0x0 = x0)
The expectation of the budget constraint in t=1 based on the b.c. of t=0 is:
E0c1 + E0a1 = E0y1 + (1 + r) a0
The economy knows how much financial income will be available in the next period since a 0 is
known.
Rearrange and solve for a0 using the t=1 b.c. to get:
Then substitute this into the t=0 b.c. to get:
Rearrange this to get:
This is the 1 period ahead (1.p.a.) solution. The LHS of this equation is the expected present value of
lifetime consumption up to period 1. The RHS is the expected present value of lifetime income up to
period 1, plus the gross return on the initial asset holding position, minus the expectation of the
asset value at the end of period 1 discounted.
Video 2
-Two-period ahead solution
The two-period ahead solution is calculated by taking the b.c. in t=2 and the expectation in t=0:
Solve b.c. in t = 2 for E0a1 to get:
Substitute this into the t=0 expectation b.c. to get:
This is the two-period ahead (2.p.a.) solution.
ECN302 – Advanced Macroeconomics - Uncertainty
Video 1
Since there is now uncertainty, the individual does not know in period t=0 what their future
endowment income will be.
The desired outcome of the individual is perfect consumption smoothing.
We will look at how consumption allocation changes with uncertainty.
Again, we will be using a Dynamic General Equilibrium (DGE) model.
This is an endowment economy with no government and no foreign sector.
The mathematical tools that we will use are the first order difference equations (FODEs) & dynamic
optimisation. These are the same tools that we used in topic 1.
We will focus on endowment income uncertainty alone.
-Environment
This is the exact same as it is in topic 1 (deterministic). The only difference in this model is that
future income is unknown. the representative agent can make expectations on future income. The
assumption is that all individuals form the same expectations.
1) Future income expectations are based on (and are therefore a function of) current income.
Therefore: Xt+1=f(Xt). This is called adaptive expectation.
2) Φt is the information set – the assumption is that this contains all the info on the given
variable up until time t. Expectations are conditional on Φ t, meaning that E[Xt+1 l Φt]. This can
also be written as Et[Xt+1]. This is called rational expectation as it uses all info in order to
produce a forecast.
In period t+1, we compare both expectations to get the forecast error:
Xt+1 - Et[Xt+1] = et+1
The rational expectation hypothesis implies that the expectation today of the forecast error is equal
to zero, meaning that people do not make forecast errors.
Forecast errors are uncorrelated, meaning that making a forecast error today does not automatically
mean a forecast error will be made in the next period.
The assumption is that the sequence of income is expected to be bounded (no one expects to
become infinitely rich in the future), therefore:
, Week 2: 15/02/21
-Intertemporal budget constraint
We’ll use forward substitution between 2 subsequent periods.
The budget constraints in period t=0 is:
c0 + a0 = y0 + (1 + r) a-1 (E0x0 = x0)
The expectation of the budget constraint in t=1 based on the b.c. of t=0 is:
E0c1 + E0a1 = E0y1 + (1 + r) a0
The economy knows how much financial income will be available in the next period since a 0 is
known.
Rearrange and solve for a0 using the t=1 b.c. to get:
Then substitute this into the t=0 b.c. to get:
Rearrange this to get:
This is the 1 period ahead (1.p.a.) solution. The LHS of this equation is the expected present value of
lifetime consumption up to period 1. The RHS is the expected present value of lifetime income up to
period 1, plus the gross return on the initial asset holding position, minus the expectation of the
asset value at the end of period 1 discounted.
Video 2
-Two-period ahead solution
The two-period ahead solution is calculated by taking the b.c. in t=2 and the expectation in t=0:
Solve b.c. in t = 2 for E0a1 to get:
Substitute this into the t=0 expectation b.c. to get:
This is the two-period ahead (2.p.a.) solution.
