13+14 Ramsey

Question 1

Which constraints is this optimization problem subject to?

Answer 1

• transition equation
• terminal condition
• initial condition

Question 2

What are the components of this optimization problem? What are their roles?

Answer 2

e<sup>-ρt</sup> = discount factor in continuous time
cₜ = consumption = control variable, which we maximize
kₜ = net assets, capital = state variable (as all other variables that are not control)
t = time

Question 3

What is the transition equation?

Answer 3

differential equation in k that shows how the choice of control variable translates into a pattern of movement for the state variable(s)

Question 4

What is the terminal condition?

Answer 4

chosen discrete value of the state variable at the end of the planning horizon

Question 5

What is the initial condition?

Answer 5

initially, the state variable must be non-negative

Question 6

Following our standard Lagrange approach, we would maximize the Lagrangian w.r.t c_t and k_t. Why can we not do that?

Answer 6

We don’t know what the partial derivative of k̇ₜ with respect to kₜ is

Question 7

What is the cookbook recipe for applying the Hamiltonian?

Answer 7

1. identify control + state variables
2. define the Hamiltonian: (function) + λₜ (transition equation)
3. take the partial derivatives w.r.t. control and state variables and set the one for the control variable equal to zero, the state variable equal to -dot λ_t
4. enforce the transversality condition

Question 8

What do the different parts of the Hamiltonian mean?

Answer 8

(1) the first part is the direct contribution to the objective of the control variable

(2) the second part shows how the choice of the control variable affects the state variable’s evolution. The value of the change in the state variable is captured by λg() = the rate of change weighted by its shadow value (Lagrangian multiplier)

Question 9

What is the transversality condition?

Answer 9

It describes what happens at the end of the planning horizon (T): if the state variable left at T is positive, then the shadow value (= λ_T) must be zero - otherwise it is not optimal not to use the remaining state variable any more. Vice versa, if there is not state variable left, it implies that the shadow value is positive.

Question 19

What are phase diagrams?

Answer 19

• Graphical device to solve a system of equations
• Allows us to visualize the dynamics of the system
• Objective: translate the differential equations into a system of arrows that describe the qualitative behavior of the economy over time

Question 20

In a phase diagram, what is the locus?

Answer 20

• It is the set of points for which ẏ₁(t) = 0
• It is called the y₁ = 0 schedule

Question 21

What are the steps to construct a Phase Diagram?

Answer 21

1. Plot the locus of the points for which ẏ₁(t) = 0
2. Analyze the dynamics of y₁ in the two regions generated by ẏ₁ = 0, i.e. where y₁ is positive & where y₁ is negative
3. Repeat 1.+2. for y₂
4. Join the two graphsand identify the steady state

Question 22

Which steady states are there in a system (phase diagram)?

Answer 22

• unstable steady state
• stable steady state
• saddle-path stable steady state

Question 23

What is an unstable steady state (phase diagram)?

Answer 23

A steady state, where if you are only a little bit off, the dynamics of the system will take you away from the steady state

Question 24

What is an stable steady state (phase diagram)?

Answer 24

No matter where in the system you start, the system will take you back to the steady state (at the origin)

Question 25

What is an saddle-path stable steady state (phase diagram)?

Answer 25

The steady state is neither stable nor unstable: depending on where the initial condition is, the dynamics will take you back to the steady state or away from it

Question 26

What is this max problem and what do its components mean?

Answer 26

Housholds in the Ramsey growth model.

c_t = per capita consumption, C_t = aggregate consumption,
B_t = assets, L_t = N_t (Labor = Population) , L_t = e^nt,
u (…) = utility function, ρ = time preference rate,
r_t = real return on assets, w_t = wage.

Question 27

What are 3 Features of the objective function?

Answer 27

1. Infinite horizon: altruism (Barro)
   (see below, where : U₁ = utility of the adult, U2 = utility of the child)
2. Discount: selfishness (Barro)
3. Derivatives and Inada conditions

Question 28

How can you formulate this constraint in per capita terms?

Answer 28

Question 29

What is the Hamiltonian of this?

Answer 29

Question 30

What is the intuition for this result in the Ramsey model?

Answer 30

Preference for consumption smoothing, as measured by
parameter θ, makes the HH want to shift consumption from the future to the present.

Question 31

What is R in the Ramsey firm’s maximization problem?

Answer 31

R = rental rate of a unit of capital

Question 32

Answer 32

How well did you check out the phase diagram?

Question 33

What is the intuition for the state and control variables?

Answer 33

• control = to maximize, what agent needs to make a decision on
• state = it is predetermined, from the previous period

Question 34

If A_t is the level of technology which increases at the exogenously given rate x (see below), how can you express A_t in terms of A₀?

Answer 34

Question 35

Do you need a Hamiltonian for this function? Why (not)?

Answer 35

As the problem is neither intertemporal nor constrained, no Hamiltonian Function is needed

Question 36

What is θ in the Ramsey model?

Answer 36

intertemporal elasticity of consumption = the higher θ, the more willing is the household to forgo current consumption for future consumption

(logical next step: the more rapid is the accumulation of capital towards the steady state)

Question 37

Where is the locus of ^c_t?

Answer 37

the red horizontal line

Question 38

Where is the locus of ^k_t?

Answer 38

the green half-circle function

Question 39

Which steady states are there? Are they stable?

Answer 39

• 1) -> unstable
• 2) -> saddle-path steady
• 3) -> stable

Question 40

What characterizes the steady state 1) ?

Answer 40

• it is an unstable steady state: only when the initial condition is c_t = 0, k_t = 0, this steady state exists
• Otherwise, the dynamics of the system will move away from the steady state

Question 41

What characterizes the steady state 2) ?

Answer 41

• it is a saddle-path steady state:
• If the starting point is on the stable arm (that leads to the steady state), the dynamics of the system will take you to the steady state
• Otherwise, move away

Question 42

What characterizes the steady state 3) ?

Answer 42

• it is an stable steady, where the dynamics of the system will move towards the steady state
• BUT:
  
  • it violates the transversality condition
  • it is inefficient, because you save so much that you are better off increasing consumption both tomorrow and today - so you would never end up here!

Question 43

What are the steady state growth rates of the variables in effective units:

γ_^k*, γ_^c*

?

Answer 43

γ_^k* = γ_^c* = 0

by definition of the steady state

Question 44

What are the steady state growth rates of the per capita variables:

γ_kt*, γ_ct*

?

Answer 44

γ_k* = γ_c* = x

the capital labor ratio and consumption per capita
grow with the constant rate x (= the rate of technological progress)

Question 45

What are the steady state growth rates of:

γ_K*, γ_C*

?

Answer 45

γ_K* = γ_C* = x + n

the exogenously given technological progress and
population growth are the drivers of the steady state growth of the capital stock and of consumption

Question 46

What are the corresponding steady state levels? Do they depend on the parameter θ?

Answer 46

in the steady state of the Ramsey model:

• consumption increases with increases in capital.
• an increase in the relative risk aversion of households, θ, (that strenghtens the motive for consumption smoothing) decreases both:
  
  • steady state capital and
  • steady state consumption.
• θ ↑ → ^k*↓ → ^c*↓