K_borrow_firm.m

%% Profit Maximization over Capital and Labor
% *back to* <https://fanwangecon.github.io *Fan*>*'s* <https://fanwangecon.github.io/Math4Econ/ 
% *Intro Math for Econ*>*,*  <https://fanwangecon.github.io/M4Econ/ *Matlab Examples*>*, 
% or* <https://fanwangecon.github.io/MEconTools/ *MEconTools*> *Repositories*
%% Model Components and Maximization Problem
% Assume that the firm has fixed free labor, but can choose capital input. At 
% the start of a period, a firm rents capital inputs and combines capital with 
% labor to produce. At the end of the period, the firm sells its output and pays 
% interest rates based on how much capital it rented (no wage costs). Profit is 
% denoted by π, period interest rate is $r$, the price of output is $p$, the firm 
% makes $y$ units of output, and the production function is Cobb-Douglas: $A\cdot 
% K^{\alpha}\cdot L^{0.5}$
%% 
% * *Profit:* $\Pi =  p\cdot A\cdot K^{\alpha}\cdot L^{0.5}-r\cdot K$
%% 
% *Profit Maximization:*
%% 
% * $\max_{K} \left( p\cdot A\cdot K^{\alpha}\cdot L^{0.5}-r\cdot K \right)$
%% 
% The $r$ here is just the interest on loans, in another word, if the borrowing 
% rate is 2 percent, $r=0.02$. Alternatively, one could replace the $r$ in the 
% equation above by $1+r$. The implication of just having $r$ is that the $K$ 
% that was borrowed could be resold and so the firm does not need to generate 
% revenue to pay for the principle borrowed, only the interest rate. If however, 
% during the process of production, ther capital depreciates, then the firm would 
% have to pay back more than $r$. In the extreme case where the capital fully 
% gets used up and can not be resold for any value, then the cost term in the 
% equation above becomes: $(1+r)\cdot K$.
%% Finding Optimal Choices--Brute Force Grid
% We can visualize the solution here like we do for the household savings problem

clear all;
alpha=0.25;
A=1;
r=1.05;
p=1;
L=2;
choice_grid_count=100;
capital_grid=linspace(0, 1,choice_grid_count);
profit_at_capitalgrid=p*A*(capital_grid.^alpha)*(L^0.5)-(r*capital_grid);
[max_profit, max_profit_index]=max(profit_at_capitalgrid);
max_profit
max_profit_index
optimal_capital_choice = capital_grid(max_profit_index);
figure();
hold on;
plot(capital_grid, profit_at_capitalgrid);
scatter(optimal_capital_choice, max_profit, 'filled');
xlabel('Capital Choices along Grid');
ylabel('Profit at different capital choices');
title({'Optimal capital choice (red dot) on feasible choice grid';...
 ['optimal capital = ', num2str(optimal_capital_choice)]});
grid on;
%% Analytical Solution
% You can use the symbolic toolbox to take derivative and find root:

syms r p alpha L K A
fpi = p*A*K^(alpha)*L^0.5 - r*K
dPIdK = diff(fpi, K)
KOpti = solve(dPIdK == 0, K, 'REAL',true)
%% Demand Curve For Capital
% With the optimal capital choice as a function of interest rate, we can plot 
% out the demand for capital.

p=1.15; %From the question.
L=2; %From the question.
A=3; %You can pick a random number.
alpha=0.45; %You can pick a random number.
grid_points = 21;
% Vector of interest rates
r = linspace(1.0,1.2,grid_points);
% Demand Curve
K= (r/(p*A*alpha*(L^0.5))).^(1/(alpha-1));
% Plot
figure();
plot(K,r);
xlabel('Capital (Borrowed from Banks)');
ylabel('Interest Rate');
title({'Inverse Demand For Capital'});
grid on;
%% Demand and Supply Intersection
% Combining the Firm's problem here, and the household's problem from the other 
% file, we have the equilibrium result. 
% 
% Note that you should adjust your interest rate range so that you can see the 
% intersection. For the problem here, there exists an interest rate that clears 
% the market for capital.

z=10;% from household problem
beta=0.80; % from household problem
p=1.15; %From the question.
L=2; %From the question.
A=3; %You can pick a random number.
alpha=0.45; %You can pick a random number.
grid_points = 21;
% Vector of interest rates
r = linspace(1.0,1.2,grid_points);
% Demand Curve
Demand = (r/(p*A*alpha*(L^0.5))).^(1/(alpha-1));
Supply = (z*beta*(1+r)-(z/2))./((1+r)*(1+beta));
% Plot
figure();
hold on;
plot(Demand,r);
plot(Supply,r);
xlabel('Capital Demand and Supply');
ylabel('Interest Rate');
title({'Inverse Demand and Supply For Capital'});
legend({'Demand', 'Supply'});
grid on;