Advanced Macro
Assignment 1
Hans Martinez
Sep 23, 2020
Competitive equilibrium for this economy
Definition: A sequential markets equilibrium is a a sequence of prices \(\{w_t, r_t\}^{\infty}_{t=0}\), allocations for the representative agent \(\{c_t, k_{t+1}^s\}^{\infty}_{t=0}\) and allocations for the representative firm \(\{l_t^d, k_{t}^d\}^{\infty}_{t=0}\) such that:
- Given \(k_0\) and \(\{w_t, r_t\}^{\infty}_{t=0}\), allocations for the representative agent \(\{c_t, k_{t+1}^s\}^{\infty}_{t=0}\) solves the agent’s maximization problem:
\[\begin{aligned} \max_{\{c_t, k_{t+1}\}_{t=0}^\infty} & \sum_{t=0}^{\infty}\beta^t U(c_t)\\ s.t. \; & c_t+k_{t+1}^s = r_tk_{t}^s+w_t\\ & c_t, k_{t+1}^s\geq0 \; \forall \, t=0,1,\dots\\ & k_0 \; \text{given} \end{aligned}\]
- For each \(t \ge 0\), given \((w_t, r_t)\) the firm allocation \((l_t^d,k_t^d)\) solves the firms’ maximization problem:
\[ \max_{k_t,l_t\, \ge \, 0} \; F(k_t^d,l_t^d)-r_tk_t^d-w_tl_t^d \]
- Markets clear, for all \(t \ge 0\)
\[\begin{aligned} l_t^d &= 1 \\ k_t^s &= k_t^d \\ c_t+k_{t+1}^s&=F(k_t^d,l_t^d) \\ \end{aligned}\]
Show that the equilibrium allocation of consumption, capital, and labor coincides with those of the planner’s.
- CE FOC
\[\begin{aligned} -u'(c_{t})+\beta u'(c_{t+1})r_t&=0 \\ -u'(c_{t})+\beta u'(c_{t+1})f'(k_{t+1})&=0 \; \text{ from firm's static optimization problem}\\ -u'(f(k_t)-k_{t+1})+\beta u'(f(k_{t+1})-k_{t+2})f'(k_{t+1})&=0 \;\text{ from market clearing}\\ \end{aligned}\]
- SPP FOC
\[\begin{aligned} -u'(f(k_t)-k_{t+1})+\beta u'(f(k_{t+1})-k_{t+2})f'(k_{t+1})&=0 \end{aligned}\]
Therefore, as long as the transversality condition holds, both solutions are equivalent.
\[ \lim_{t \to \infty} \beta^tu'(f(k_t)-k_{t+1})f'(k_t)k_t=0 \]
Planner’s dynamic problem and the Bellman equation
\[\begin{aligned} v(k') &= \max_{0\ge k' \ge f(k)} \{U(f(k)-k')+\beta v(k')\}\\ \end{aligned}\]
Solve planner’s problem.
Assume \(u(c)=log(c)\), \(f(k,l)=z k^\alpha l^{1-\alpha}\).
Using the guess and verify method, let \(v(k)=A+Bln(k)\).
FOC: \(k'=\frac{B\beta zk^\alpha}{1+B\beta}\)
Plugging into orginial problem and solving for coefficients, we get:
\[\begin{aligned} A &= \frac{1}{1-\beta}\left[ \frac{\alpha\beta}{1-\alpha\beta}ln(\alpha\beta)+ln(1-\alpha\beta) \right] \\ B &= \frac{\alpha}{1-\alpha\beta} \end{aligned}\]
Then, our value function and our policy function would be:
\[\begin{aligned} v^*&=A+Bln(k) \; \text{with A and B as defined above} \\ g^*(k)&=\alpha\beta zk^\alpha \end{aligned}\]
Obtain {c,k,r,w,y} at the steady state
\[\begin{aligned} k^*&=(\alpha\beta z)^{\frac{1}{1-\alpha}} \\ r^*&=\beta^{-1} \\ w^*&=z(1-\alpha)(\alpha\beta z)^{\frac{\alpha}{1-\alpha}} \\ y^*&=z(\alpha\beta z)^{\frac{\alpha}{1-\alpha}} \\ c^*&=z(\alpha\beta z)^{\frac{\alpha}{1-\alpha}}- (\alpha\beta z)^{\frac{1}{1-\alpha}} \end{aligned}\]
Obtain the path of {c,k,r,w,y} in the following cases
Let \(\alpha=\frac{1}{3}, z=1\). Use planner’s solution, starting from steady state:
A. Capital decreases to 80% of its steady state.
# Julia code
# setting working directory
cd("/Volumes/SSD Hans/Github/MacroFall2020/AdvMacro")
#Defining vars
a = 1/3
b = 0.6
k_star = (a*b)^(1/(1-a))
r = 1/b
# Initializing array for loop
m, n = 100,2
K = fill(0.0, (m, n))
#defining policy fn
g = x -> a*b*(x^a)
# loop for capital
k = 0.8*k_star
for i in 1:m
K[i, 1] = k
K[i, 2] = g(K[i,1])
global k = K[i, 2]
end
# defining fn's for the rest of the variables
w = x -> (1-a)x^a
y = x -> x^a
c = (p,q) -> y(p)-q
r = x -> a*(x^(a-1))
int = r.(K[1:100,2])
wage = w.(K[1:100,1])
output = y.(K[1:100,1])
cons = c.(K[1:100,1],K[1:100,2])
# importing packages for plotting
import Pkg; Pkg.add("Plots")
using Plots
gr()
# plotting capital
plot(1:10,fill(k_star,10), label="capital SS")
plot!(1:10,K[1:10,1], label="capital path", linestyle = :dot, legend = :bottomright)
xlabel!("t")
title!("Capital")
savefig("graphs/capital")
# plotting output
plot(1:10,fill(y(k_star),10), label="output SS")
plot!(1:10,output[1:10], label="output path", linestyle = :dot, legend = :bottomright)
xlabel!("t")
title!("Output")
savefig("graphs/output")
# plotting consumption
plot(1:10,fill(c(k_star,k_star),10), label="consumption SS")
plot!(1:10,cons[1:10], label="consumption path", linestyle = :dot, legend = :bottomright)
xlabel!("t")
title!("Consumption")
savefig("graphs/consumption")
# plotting wage
plot(1:10,fill(w(k_star),10), label="wage SS")
plot!(1:10,wage[1:10], label="wage path", linestyle = :dot, legend = :bottomright)
xlabel!("t")
title!("Wage")
savefig("graphs/wage")
# plotting capital rental rate
plot(1:10,fill(r(k_star),10), label="capital rental rate SS")
plot!(1:10,int[1:10], label="rate path", linestyle = :dot, legend = :topright)
xlabel!("t")
title!("Capital rental rate")
savefig("graphs/capitalr")
Wage and capital
Capital rate and output
Consumption
B. Productivity increases permanently by 5%
# Julia code
# Increase in productivity permanently by 5%
k_starp = (a*b*1.5)^(1/(1-a))
#defining policy fn with productivity increases
g1 = x -> a*b*1.05*(x^a)
# Initializing array for loop
K1 = fill(0.0, (m, n))
# loop for capital
k1 = k_starp
for i in 1:m
K1[i, 1] = k1
K1[i, 2] = g(K[i,1])
global k1 = K[i, 2]
end
# defining fn's for the rest of the variables
w1 = x -> (1-a)*(x^a) #does not change
y1 = x -> 1.5*(x^a)
c1 = (p,q) -> y(p)-q #does not change
r1 = x -> a*(1.5*x^(a-1))
int_p = r1.(K1[1:100,2])
wage_p = w1.(K1[1:100,1])
output_p = y1.(K1[1:100,1])
cons_p = c1.(K1[1:100,1],K1[1:100,2])
# plotting capital
plot(1:10,fill(k_star,10), label="capital SS")
plot!(1:10,K1[1:10,1], label="capital path", linestyle = :dot, legend = :bottomright)
xlabel!("t")
title!("Capital (productivity increase)")
savefig("graphs/capitalp")
# plotting output
plot(1:10,fill(y1(k_star),10), label="output SS")
plot!(1:10,output_p[1:10], label="output path", linestyle = :dot, legend = :bottomright)
xlabel!("t")
title!("Output (productivity increase)")
savefig("graphs/outputp")
# plotting consumption
plot(1:10,fill(c1(k_star,k_star),10), label="consumption SS")
plot!(1:10,cons_p[1:10], label="consumption path", linestyle = :dot, legend = :bottomright)
xlabel!("t")
title!("Consumption (productivity increase)")
savefig("graphs/consumptionp")
# plotting wage
plot(1:10,fill(w1(k_star),10), label="wage SS")
plot!(1:10,wage_p[1:10], label="wage path", linestyle = :dot, legend = :bottomright)
xlabel!("t")
title!("Wage (productivity increase)")
savefig("graphs/wagep")
# plotting capital rental rate
plot(1:10,fill(r1(k_star),10), label="capital rental rate SS")
plot!(1:10,int_p[1:10], label="rate path", linestyle = :dot, legend = :topright)
xlabel!("t")
title!("Capital rental rate (productivity increase)")
savefig("graphs/capitalrp")
Wage and capital (productiviy increase)
Capital rate and output (productivity increase)
Consumption (productivity increase)
Social Planner’s Problem
\[\begin{aligned} w^T(\bar{k_0}) = \max_{\{k_{t+1}\}_{t=0}^\infty} & \sum_{t=0}^{\infty}\beta^t U(f(k_t)-k_{t+1})\\ s.a.&\\ & 0 \le k_{t+1} \le f(k_t)\\ & k_0 = \bar{k_0} > 0 \; \text{given} \end{aligned}\]
Where \(f(k_t) = F(k_t,1)\).