This lecture continues our analysis in this lecture Cass-Koopmans Planning Model about the model that Tjalling Koopmans [Koopmans, 1965] and David Cass [Cass, 1965] used to study optimal capital accumulation.
This lecture illustrates what is, in fact, a more general connection between a planned economy and an economy organized as a competitive equilibrium or a market economy.
The earlier lecture Cass-Koopmans Planning Model studied a planning problem and used ideas including
The present lecture uses additional ideas including
Let’s start with some standard imports:
import matplotlib.pyplot as plt
plt.rcParams["figure.figsize"] = (11, 5) #set default figure size
from numba import njit, float64
from numba.experimental import jitclass
import numpy as np
The physical setting is identical with that in Cass-Koopmans Planning Model.
Time is discrete and takes values $ t = 0, 1 , \ldots, T $.
Output of a single good can either be consumed or invested in physical capital.
The capital good is durable but partially depreciates each period at a constant rate.
We let $ C_t $ be a nondurable consumption good at time t.
Let $ K_t $ be the stock of physical capital at time t.
Let $ \vec{C} $ = $ \{C_0,\dots, C_T\} $ and $ \vec{K} $ = $ \{K_0,\dots,K_{T+1}\} $.
A representative household is endowed with one unit of labor at each $ t $ and likes the consumption good at each $ t $.
The representative household inelastically supplies a single unit of labor $ N_t $ at each $ t $, so that $ N_t =1 \text{ for all } t \in \{0, 1, \ldots, T\} $.
The representative household has preferences over consumption bundles ordered by the utility functional:
$$ U(\vec{C}) = \sum_{t=0}^{T} \beta^t \frac{C_t^{1-\gamma}}{1-\gamma} $$where $ \beta \in (0,1) $ is a discount factor and $ \gamma >0 $ governs the curvature of the one-period utility function.
We assume that $ K_0 > 0 $.
There is an economy-wide production function
$$ F(K_t,N_t) = A K_t^{\alpha}N_t^{1-\alpha} $$with $ 0 < \alpha<1 $, $ A > 0 $.
A feasible allocation $ \vec{C}, \vec{K} $ satisfies
$$ C_t + K_{t+1} \leq F(K_t,N_t) + (1-\delta) K_t \quad \text{for all } t \in \{0, 1, \ldots, T\} $$where $ \delta \in (0,1) $ is a depreciation rate of capital.
In this lecture Cass-Koopmans Planning Model, we studied a problem in which a planner chooses an allocation $ \{\vec{C},\vec{K}\} $ to maximize (40.2) subject to (40.5).
The allocation that solves the planning problem reappears in a competitive equilibrium, as we shall see below.
We now study a decentralized version of the economy.
It shares the same technology and preference structure as the planned economy studied in this lecture Cass-Koopmans Planning Model.
But now there is no planner.
There are (unit masses of) price-taking consumers and firms.
Market prices are set to reconcile distinct decisions that are made separately by a representative consumer and a representative firm.
There is a representative consumer who has the same preferences over consumption plans as did a consumer in the planned economy.
Instead of being told what to consume and save by a planner, a consumer (also known as a household) chooses for itself subject to a budget constraint.
Note
Again, we can think of there being unit measures of identical representative consumers and
identical representative firms.
The representative household and the representative firm are both price takers.
The household owns both factors of production, namely, labor and physical capital.
Each period, the firm rents both factors from the household.
There is a single grand competitive market in which a household trades date $ 0 $ goods for goods at all other dates $ t=1, 2, \ldots, T $.
There are sequences of prices $ \{w_t,\eta_t\}_{t=0}^T= \{\vec{w}, \vec{\eta} \} $ where
In addition there is a vector $ \{q_t^0\} $ of intertemporal prices where
We call $ \{q^0_t\}_{t=0}^T $ a vector of Hicks-Arrow prices, named after the 1972 economics Nobel prize winners.
Because is a relative price. the unit of account in terms of which the prices $ q^0_t $ are stated is; we are free to re-normalize them by multiplying all of them by a positive scalar, say $ \lambda > 0 $.
Units of $ q_t^0 $ could be set so that they are
$$ \frac{\text{number of time 0 goods}}{\text{number of time t goods}} $$In this case, we would be taking the time $ 0 $ consumption good to be the numeraire.
At time $ t $ a representative firm hires labor $ \tilde n_t $ and capital $ \tilde k_t $.
The firm’s profits at time $ t $ are
$$ F(\tilde k_t, \tilde n_t)-w_t \tilde n_t -\eta_t \tilde k_t $$where $ w_t $ is a wage rate at $ t $ and $ \eta_t $ is the rental rate on capital at $ t $.
As in the planned economy model
$$ F(\tilde k_t, \tilde n_t) = A \tilde k_t^\alpha \tilde n_t^{1-\alpha} $$Zero-profits conditions for capital and labor are
$$ F_k(\tilde k_t, \tilde n_t) =\eta_t $$and
$$ F_n(\tilde k_t, \tilde n_t) =w_t \tag{41.1} $$
These conditions emerge from a no-arbitrage requirement.
To describe this no-arbitrage profits reasoning, we begin by applying a theorem of Euler about linearly homogenous functions.
The theorem applies to the Cobb-Douglas production function because we it displays constant returns to scale:
$$ \alpha F(\tilde k_t, \tilde n_t) = F(\alpha \tilde k_t, \alpha \tilde n_t) $$for $ \alpha \in (0,1) $.
Taking partial derivatives $ \frac{\partial }{\partial \alpha} $ on both sides of the above equation gives
$$ F(\tilde k_t,\tilde n_t) = \frac{\partial F}{\partial \tilde k_t} \tilde k_t + \frac{\partial F}{\partial \tilde n_t} \tilde n_t $$Rewrite the firm’s profits as
$$ \frac{\partial F}{\partial \tilde k_t} \tilde k_t + \frac{\partial F}{\partial \tilde n_t} \tilde n_t-w_t \tilde n_t -\eta_t k_t $$or
$$ \left(\frac{\partial F}{\partial \tilde k_t}-\eta_t\right) \tilde k_t + \left(\frac{\partial F}{\partial \tilde n_t}-w_t\right) \tilde n_t $$Because $ F $ is homogeneous of degree $ 1 $, it follows that $ \frac{\partial F}{\partial \tilde k_t} $ and $ \frac{\partial F}{\partial \tilde n_t} $ are homogeneous of degree $ 0 $ and therefore fixed with respect to $ \tilde k_t $ and $ \tilde n_t $.
If $ \frac{\partial F}{\partial \tilde k_t}> \eta_t $, then the firm makes positive profits on each additional unit of $ \tilde k_t $, so it would want to make $ \tilde k_t $ arbitrarily large.
But setting $ \tilde k_t = + \infty $ is not physically feasible, so equilibrium prices must take values that present the firm with no such arbitrage opportunity.
A similar argument applies if $ \frac{\partial F}{\partial \tilde n_t}> w_t $.
If $ \frac{\partial \tilde k_t}{\partial \tilde k_t}< \eta_t $, the firm would want to set $ \tilde k_t $ to zero, which is not feasible.
It is convenient to define $ \vec{w} =\{w_0, \dots,w_T\} $ and $ \vec{\eta}= \{\eta_0, \dots, \eta_T\} $.
A representative household lives at $ t=0,1,\dots, T $.
At $ t $, the household rents $ 1 $ unit of labor and $ k_t $ units of capital to a firm and receives income
$$ w_t 1+ \eta_t k_t $$At $ t $ the household allocates its income to the following purchases between the following two categories:
Here $ \left(k_{t+1} -(1-\delta)k_t\right) $ is the household’s net investment in physical capital and $ \delta \in (0,1) $ is again a depreciation rate of capital.
In period $ t $, the consumer is free to purchase more goods to be consumed and invested in physical capital than its income from supplying capital and labor to the firm, provided that in some other periods its income exceeds its purchases.
A consumer’s net excess demand for time $ t $ consumption goods is the gap
$$ e_t \equiv \left(c_t + (k_{t+1} -(1-\delta)k_t)\right)-(w_t 1 + \eta_t k_t) $$Let $ \vec{c} = \{c_0,\dots,c_T\} $ and let $ \vec{k} = \{k_1,\dots,k_{T+1}\} $.
$ k_0 $ is given to the household.
The household faces a single budget constraint that requires that the present value of the household’s net excess demands must be zero:
$$ \sum_{t=0}^T q^0_t e_t \leq 0 $$or
$$ \sum_{t=0}^T q^0_t \left(c_t + (k_{t+1} -(1-\delta)k_t)\right) \leq \sum_{t=0}^T q^0_t(w_t 1 + \eta_t k_t) \ $$The household faces price system $ \{q^0_t, w_t, \eta_t\} $ as a price-taker and chooses an allocation to solve the constrained optimization problem:
$$ \begin{aligned}& \max_{\vec{c}, \vec{k} } \sum_{t=0}^T \beta^t u(c_t) \\ \text{subject to} \ \ & \sum_{t=0}^T q_t^0\left(c_t +\left(k_{t+1}-(1-\delta) k_t \right) - (w_t -\eta_t k_t) \right)\leq 0 \notag \end{aligned} $$Components of a price system have the following units:
The vision here is that an equilibrium price system and allocation are determined once and for all.
In effect, we imagine that all trades occur just before time $ 0 $.
We compute a competitive equilibrium by using a guess and verify approach.
In this lecture Cass-Koopmans Planning Model, we computed an allocation $ \{\vec{C}, \vec{K}, \vec{N}\} $ that solves a planning problem.
We use that allocation to construct a guess for the equilibrium price system.
Note
This allocation will constitute the Big $ K $ to be in the present instance of the Big $ K $ , little $ k $ trick
that we’ll apply to a competitive equilibrium in the spirit of this lecture and this lecture.
In particular, we shall use the following procedure:
Thus, we guess that for $ t=0,\dots,T $:
$$ q_t^0 = \beta^t u'(C_t) \tag{41.2} $$
$$ w_t = f(K_t) -K_t f'(K_t) \tag{41.3} $$
$$ \eta_t = f'(K_t) \tag{41.4} $$
At these prices, let capital chosen by the household be
$$ k^*_t(\vec {q}, \vec{w}, \vec{\eta)} , \quad t \geq 0 \tag{41.5} $$
and let the allocation chosen by the firm be
$$ \tilde k^*_t(\vec{q}, \vec{w}, \vec{\eta}), \quad t \geq 0 $$and so on.
If our guess for the equilibrium price system is correct, then it must occur that
$$ k_t^* = \tilde k_t^* \tag{41.6} $$
$$ 1 = \tilde n_t^* \tag{41.7} $$
$$ c_t^* + k_{t+1}^* - (1-\delta) k_t^* = F(\tilde k_t^*, \tilde n_t^*) $$We shall verify that for $ t=0,\dots,T $ allocations chosen by the household and the firm both equal the allocation that solves the planning problem:
$$ k^*_t = \tilde k^*_t=K_t, \tilde n_t=1, c^*_t=C_t \tag{41.8} $$
Our approach is firsts to stare at first-order necessary conditions for optimization problems of the household and the firm.
At the price system we have guessed, we’ll then verify that both sets of first-order conditions are satisfied at the allocation that solves the planning problem.
To solve the household’s problem, we formulate the Lagrangian
$$ \mathcal{L}(\vec{c},\vec{k},\lambda) = \sum_{t=0}^T \beta^t u(c_t)+ \lambda \left(\sum_{t=0}^T q_t^0\left(\left((1-\delta) k_t -w_t\right) +\eta_t k_t -c_t - k_{t+1}\right)\right) $$and attack the min-max problem:
$$ \min_{\lambda} \max_{\vec{c},\vec{k}} \mathcal{L}(\vec{c},\vec{k},\lambda) $$First-order conditions are
$$ c_t: \quad \beta^t u'(c_t)-\lambda q_t^0=0 \quad t=0,1,\dots,T \tag{41.9} $$
$$ k_t: \quad -\lambda q_t^0 \left[(1-\delta)+\eta_t \right]+\lambda q^0_{t-1}=0 \quad t=1,2,\dots,T+1 \tag{41.10} $$
$$ \lambda: \quad \left(\sum_{t=0}^T q_t^0\left(c_t + \left(k_{t+1}-(1-\delta) k_t\right) -w_t -\eta_t k_t\right)\right) \leq 0 \tag{41.11} $$
$$ k_{T+1}: \quad -\lambda q_0^{T+1} \leq 0, \ \leq 0 \text{ if } k_{T+1}=0; \ =0 \text{ if } k_{T+1}>0 \tag{41.12} $$
Now we plug in our guesses of prices and do some algebra in the hope of recovering all first-order necessary conditions (40.9)-(40.12) for the planning problem from this lecture Cass-Koopmans Planning Model.
Combining (41.9) and (41.2), we get:
$$ u'(C_t) = \mu_t $$which is (40.9).
Combining (41.10), (41.2), and (41.4), we get:
$$ -\lambda \beta^t \mu_t\left[(1-\delta) +f'(K_t)\right] +\lambda \beta^{t-1}\mu_{t-1}=0 \tag{41.13} $$
Rewriting (41.13) by dividing by $ \lambda $ on both sides (which is nonzero since u’>0) we get:
$$ \beta^t \mu_t [(1-\delta+f'(K_t)] = \beta^{t-1} \mu_{t-1} $$or
$$ \beta \mu_t [(1-\delta+f'(K_t)] = \mu_{t-1} $$which is (40.10).
Combining (41.11), (41.2), (41.3) and (41.4) after multiplying both sides of (41.11) by $ \lambda $, we get
$$ \sum_{t=0}^T \beta^t \mu_{t} \left(C_t+ (K_{t+1} -(1-\delta)K_t)-f(K_t)+K_t f'(K_t)-f'(K_t)K_t\right) \leq 0 $$which simplifies to
$$ \sum_{t=0}^T \beta^t \mu_{t} \left(C_t +K_{t+1} -(1-\delta)K_t - F(K_t,1)\right) \leq 0 $$Since $ \beta^t \mu_t >0 $ for $ t =0, \ldots, T $, it follows that
$$ C_t+K_{t+1}-(1-\delta)K_t -F(K_t,1)=0 \quad \text{ for all }t \text{ in } \{0, 1, \ldots, T\} $$which is (40.11).
Combining (41.12) and (41.2), we get:
$$ -\beta^{T+1} \mu_{T+1} \leq 0 $$Dividing both sides by $ \beta^{T+1} $ gives
$$ -\mu_{T+1} \leq 0 $$which is (40.12) for the planning problem.
Thus, at our guess of the equilibrium price system, the allocation that solves the planning problem also solves the problem faced by a representative household living in a competitive equilibrium.
We now turn to the problem faced by a firm in a competitive equilibrium:
If we plug (41.8) into (41.1) for all t, we get
$$ \frac{\partial F(K_t, 1)}{\partial K_t} = f'(K_t) = \eta_t $$which is (41.4).
If we now plug (41.8) into (41.1) for all t, we get:
$$ \frac{\partial F(\tilde K_t, 1)}{\partial \tilde L_t} = f(K_t)-f'(K_t)K_t=w_t $$which is exactly (41.5).
Thus, at our guess for the equilibrium price system, the allocation that solves the planning problem also solves the problem faced by a firm within a competitive equilibrium.
By (41.6) and (41.7) this allocation is identical to the one that solves the consumer’s problem.
Note
Because budget sets are affected only by relative prices,
$ \{q^0_t\} $ is determined only up to multiplication by a positive constant.
Normalization: We are free to choose a $ \{q_t^0\} $ that makes $ \lambda=1 $ so that we are measuring $ q_t^0 $ in units of the marginal utility of time $ 0 $ goods.
We will plot $ q, w, \eta $ below to show these equilibrium prices induce the same aggregate movements that we saw earlier in the planning problem.
To proceed, we bring in Python code that Cass-Koopmans Planning Model used to solve the planning problem
First let’s define a jitclass that stores parameters and functions
the characterize an economy.
planning_data = [
('γ', float64), # Coefficient of relative risk aversion
('β', float64), # Discount factor
('δ', float64), # Depreciation rate on capital
('α', float64), # Return to capital per capita
('A', float64) # Technology
]
@jitclass(planning_data)
class PlanningProblem():
def __init__(self, γ=2, β=0.95, δ=0.02, α=0.33, A=1):
self.γ, self.β = γ, β
self.δ, self.α, self.A = δ, α, A
def u(self, c):
'''
Utility function
ASIDE: If you have a utility function that is hard to solve by hand
you can use automatic or symbolic differentiation
See https://github.com/HIPS/autograd
'''
γ = self.γ
return c ** (1 - γ) / (1 - γ) if γ!= 1 else np.log(c)
def u_prime(self, c):
'Derivative of utility'
γ = self.γ
return c ** (-γ)
def u_prime_inv(self, c):
'Inverse of derivative of utility'
γ = self.γ
return c ** (-1 / γ)
def f(self, k):
'Production function'
α, A = self.α, self.A
return A * k ** α
def f_prime(self, k):
'Derivative of production function'
α, A = self.α, self.A
return α * A * k ** (α - 1)
def f_prime_inv(self, k):
'Inverse of derivative of production function'
α, A = self.α, self.A
return (k / (A * α)) ** (1 / (α - 1))
def next_k_c(self, k, c):
''''
Given the current capital Kt and an arbitrary feasible
consumption choice Ct, computes Kt+1 by state transition law
and optimal Ct+1 by Euler equation.
'''
β, δ = self.β, self.δ
u_prime, u_prime_inv = self.u_prime, self.u_prime_inv
f, f_prime = self.f, self.f_prime
k_next = f(k) + (1 - δ) * k - c
c_next = u_prime_inv(u_prime(c) / (β * (f_prime(k_next) + (1 - δ))))
return k_next, c_next
@njit
def shooting(pp, c0, k0, T=10):
'''
Given the initial condition of capital k0 and an initial guess
of consumption c0, computes the whole paths of c and k
using the state transition law and Euler equation for T periods.
'''
if c0 > pp.f(k0):
print("initial consumption is not feasible")
return None
# initialize vectors of c and k
c_vec = np.empty(T+1)
k_vec = np.empty(T+2)
c_vec[0] = c0
k_vec[0] = k0
for t in range(T):
k_vec[t+1], c_vec[t+1] = pp.next_k_c(k_vec[t], c_vec[t])
k_vec[T+1] = pp.f(k_vec[T]) + (1 - pp.δ) * k_vec[T] - c_vec[T]
return c_vec, k_vec
@njit
def bisection(pp, c0, k0, T=10, tol=1e-4, max_iter=500, k_ter=0, verbose=True):
# initial boundaries for guess c0
c0_upper = pp.f(k0)
c0_lower = 0
i = 0
while True:
c_vec, k_vec = shooting(pp, c0, k0, T)
error = k_vec[-1] - k_ter
# check if the terminal condition is satisfied
if np.abs(error) < tol:
if verbose:
print('Converged successfully on iteration ', i+1)
return c_vec, k_vec
i += 1
if i == max_iter:
if verbose:
print('Convergence failed.')
return c_vec, k_vec
# if iteration continues, updates boundaries and guess of c0
if error > 0:
c0_lower = c0
else:
c0_upper = c0
c0 = (c0_lower + c0_upper) / 2
pp = PlanningProblem()
# Steady states
ρ = 1 / pp.β - 1
k_ss = pp.f_prime_inv(ρ+pp.δ)
c_ss = pp.f(k_ss) - pp.δ * k_ss
The above code from this lecture Cass-Koopmans Planning Model lets us compute an optimal allocation for the planning problem.
Now we’re ready to bring in Python code that we require to compute additional objects that appear in a competitive equilibrium.
@njit
def q(pp, c_path):
# Here we choose numeraire to be u'(c_0) -- this is q^(t_0)_t
T = len(c_path) - 1
q_path = np.ones(T+1)
q_path[0] = 1
for t in range(1, T+1):
q_path[t] = pp.β ** t * pp.u_prime(c_path[t])
return q_path
@njit
def w(pp, k_path):
w_path = pp.f(k_path) - k_path * pp.f_prime(k_path)
return w_path
@njit
def η(pp, k_path):
η_path = pp.f_prime(k_path)
return η_path
Now we calculate and plot for each $ T $
T_arr = [250, 150, 75, 50]
fix, axs = plt.subplots(2, 3, figsize=(13, 6))
titles = ['Arrow-Hicks Prices', 'Labor Rental Rate', 'Capital Rental Rate',
'Consumption', 'Capital', 'Lagrange Multiplier']
ylabels = ['$q_t^0$', '$w_t$', '$\eta_t$', '$c_t$', '$k_t$', '$\mu_t$']
for T in T_arr:
c_path, k_path = bisection(pp, 0.3, k_ss/3, T, verbose=False)
μ_path = pp.u_prime(c_path)
q_path = q(pp, c_path)
w_path = w(pp, k_path)[:-1]
η_path = η(pp, k_path)[:-1]
paths = [q_path, w_path, η_path, c_path, k_path, μ_path]
for i, ax in enumerate(axs.flatten()):
ax.plot(paths[i])
ax.set(title=titles[i], ylabel=ylabels[i], xlabel='t')
if titles[i] == 'Capital':
ax.axhline(k_ss, lw=1, ls='--', c='k')
if titles[i] == 'Consumption':
ax.axhline(c_ss, lw=1, ls='--', c='k')
plt.tight_layout()
plt.show()
Now we see how our results change if we keep $ T $ constant, but allow the curvature parameter, $ \gamma $ to vary, starting with $ K_0 $ below the steady state.
We plot the results for $ T=150 $
T = 150
γ_arr = [1.1, 4, 6, 8]
fix, axs = plt.subplots(2, 3, figsize=(13, 6))
for γ in γ_arr:
pp_γ = PlanningProblem(γ=γ)
c_path, k_path = bisection(pp_γ, 0.3, k_ss/3, T, verbose=False)
μ_path = pp_γ.u_prime(c_path)
q_path = q(pp_γ, c_path)
w_path = w(pp_γ, k_path)[:-1]
η_path = η(pp_γ, k_path)[:-1]
paths = [q_path, w_path, η_path, c_path, k_path, μ_path]
for i, ax in enumerate(axs.flatten()):
ax.plot(paths[i], label=f'$\gamma = {γ}$')
ax.set(title=titles[i], ylabel=ylabels[i], xlabel='t')
if titles[i] == 'Capital':
ax.axhline(k_ss, lw=1, ls='--', c='k')
if titles[i] == 'Consumption':
ax.axhline(c_ss, lw=1, ls='--', c='k')
axs[0, 0].legend()
plt.tight_layout()
plt.show()
Adjusting $ \gamma $ means adjusting how much individuals prefer to smooth consumption.
Higher $ \gamma $ means individuals prefer to smooth more resulting in slower convergence to a steady state allocation.
Lower $ \gamma $ means individuals prefer to smooth less, resulting in faster convergence to a steady state allocation.
We return to Hicks-Arrow prices and calculate how they are related to yields on loans of alternative maturities.
This will let us plot a yield curve that graphs yields on bonds of maturities $ j=1, 2, \ldots $ against $ j=1,2, \ldots $.
We use the following formulas.
A yield to maturity on a loan made at time $ t_0 $ that matures at time $ t > t_0 $
$$ r_{t_0,t}= -\frac{\log q^{t_0}_t}{t - t_0} $$A Hicks-Arrow price system for a base-year $ t_0\leq t $ satisfies
$$ q^{t_0}_t = \beta^{t-t_0} \frac{u'(c_t)}{u'(c_{t_0})}= \beta^{t-t_0} \frac{c_t^{-\gamma}}{c_{t_0}^{-\gamma}} $$We redefine our function for $ q $ to allow arbitrary base years, and define a new function for $ r $, then plot both.
We begin by continuing to assume that $ t_0=0 $ and plot things for different maturities $ t=T $, with $ K_0 $ below the steady state
@njit
def q_generic(pp, t0, c_path):
# simplify notations
β = pp.β
u_prime = pp.u_prime
T = len(c_path) - 1
q_path = np.zeros(T+1-t0)
q_path[0] = 1
for t in range(t0+1, T+1):
q_path[t-t0] = β ** (t-t0) * u_prime(c_path[t]) / u_prime(c_path[t0])
return q_path
@njit
def r(pp, t0, q_path):
'''Yield to maturity'''
r_path = - np.log(q_path[1:]) / np.arange(1, len(q_path))
return r_path
def plot_yield_curves(pp, t0, c0, k0, T_arr):
fig, axs = plt.subplots(1, 2, figsize=(10, 5))
for T in T_arr:
c_path, k_path = bisection(pp, c0, k0, T, verbose=False)
q_path = q_generic(pp, t0, c_path)
r_path = r(pp, t0, q_path)
axs[0].plot(range(t0, T+1), q_path)
axs[0].set(xlabel='t', ylabel='$q_t^0$', title='Hicks-Arrow Prices')
axs[1].plot(range(t0+1, T+1), r_path)
axs[1].set(xlabel='t', ylabel='$r_t^0$', title='Yields')
T_arr = [150, 75, 50]
plot_yield_curves(pp, 0, 0.3, k_ss/3, T_arr)
Now we plot when $ t_0=20 $
plot_yield_curves(pp, 20, 0.3, k_ss/3, T_arr)