KL_borrowhire_firm.m

%% Cobb Douglas Profit Maximization
% *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*
%% 
% In the example here, we will solve a firm optimization problem using a system 
% of linear equations (2 equations and 2 unknowns). The solution method is the 
% same for N inputs with Cobb-Douglas Production Function.
%% Firm and Capital and Labor
% Assume that the firm can choose capital and labor inputs. 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, and also pays wage. Total wage bill is 
% $L \cdot w$, interest payment is $K \cdot r$ (Here we assume that there is no 
% depreciation of capital, so firm can repay principle by returning capital and 
% just pay interest rate). Profit is denoted by $\pi$, 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: $A \cdot K^\alpha \cdot L^{\beta}$
% 
% The profit maximization problem is:
%% 
% * $\max_{K, L} \left( p\cdot A\cdot K^{\alpha}\cdot L^{\beta}-r\cdot K-w\cdot 
% L \right)$
%% 
% To find optimal choices, we will assume that $\alpha + \beta < 1$
%% Two First Order Conditions
%% 
% * $\frac{\partial\Pi}{\partial K}=\alpha\cdot p\cdot A\cdot K^{\alpha-1}\cdot 
% L^{\beta}-r$
% * $\frac{\partial\Pi}{\partial L}=\beta\cdot p\cdot A\cdot K^{\alpha}\cdot 
% L^{\beta-1}-w$
%% 
% _Components of profit first oder conditions_: $MPL$ and $MPK$ are always both 
% positive, but they are decreasing with higher $L$ and higher $K$ respectively. 
% On the other hand, the marginal cost of capital and labor are fixed.
%% 
% * $\text{MPK} = \alpha \cdot A\cdot K^{\alpha-1}\cdot L^{\beta}$
% * $\text{MPL} = \beta\cdot A\cdot K^{\alpha}\cdot L^{\beta-1}$
% * $\text{MC}_{K}=r$
% * $\text{MC}_{L}=w$
%% Log Linearizing Optimality Conditions
% To find optimal choices, set the first order conditions you obtained above 
% to be equal to zero.
%% 
% # $\alpha\cdot p\cdot A\cdot K^{\alpha-1}\cdot L^{\beta}=r$
% # $\beta\cdot p\cdot A\cdot K^{\alpha}\cdot L^{\beta -1}=w$
%% 
% A generic system of 2 linear equations and 2 unknowns:
%% 
% * $\left[ {\begin{array}{cc} a & b  \\ d & e  \\ \end{array} } \right] \cdot 
% \left[ {\begin{array}{c} x_1 \\ x_2  \end{array} } \right] = \left[ {\begin{array}{cc} 
% a \cdot x_1 + b \cdot x_2 \\ d \cdot x_1 + e \cdot x_2   \\ \end{array} } \right]  
% = \left[ {\begin{array}{c} o \\ p\\ \end{array} } \right]$
%% 
% Take log of the first order conditions (log linearize), and the two equations 
% above become (Now we can solve for optimal choices using _linsolve._ Note that 
% log linearizing works regardless of how many terms there are in the cobb-douglas 
% production function):
%% 
% * $\left[ {\begin{array}{cc} (\alpha-1) & \beta  \\ \alpha & (\beta-1) \\ 
% \end{array} } \right] \cdot \left[ {\begin{array}{c} \log(K) \\ \log(L)  \end{array} 
% } \right] = \left[ {\begin{array}{cc} (\alpha-1) \cdot \log(K) + \beta \cdot 
% \log(L) \\ \alpha \cdot \log(K) + (\beta-1) \cdot \log(L)   \\ \end{array} } 
% \right]  = \left[ {\begin{array}{c} \log\left(\frac{r}{\alpha p A}\right) \\ 
% \log\left(\frac{w}{\beta p A}\right)\\ \end{array} } \right]$
%% 
% We can by hand solve by elementary row operation (linsolve).
%% Solving Linear System to find Optimal Choices
% The solution to the problem, with parameter values filled in could be obtained 
% like this:

clear all
% Parameters
w = 1;
r = 1.05;
p = 5;
alpha = 0.3;
beta = 0.5;
A = 1.0;

%% Matrix Form of linear system
B = [log(r/(p*A*alpha)); log(w/(p*A*beta))];
A = [(alpha-1), beta;alpha, beta-1];
%% Solve linear equations, and then exponentiate
linSolu = exp(linsolve(A, B));
%% Solution was for log(K*) and log(L*), exponentiate to get K* and L*
KOpti = linSolu(1)
LOpti = linSolu(2)
%% Relative Choices
% For Cobb-Douglas production functions, how do optimal capital and labor choices 
% relate to each other?

syms w r p alpha beta A 
% Matrix Form of linear system, same as before
B = [log(r/(p*A*alpha)); log(w/(p*A*beta))];
A = [(alpha-1), beta;alpha, beta-1];
% Solve linear equations, and then exponentiate, same as before
% We can use the simplify command to simplify this solution, get rid of exp and log:
linSolu = simplify(exp(linsolve(A, B)))
KOpti = linSolu(1)
LOpti = linSolu(2)
KOpti/LOpti
%% 
% The expressions from Matlab look a little convoluted, but you will notice 
% a lot of similar terms inside the equation. If you try to simplify things a 
% little bit, you will end up with a simple fraction below, which says the ratio 
% of optimal capital to labor choices is not related to $A$ and $p$, but determined 
% by the elasticity parameters $\alpha$ and $\beta$ as well as prices $w$ and 
% $r$. If wage increases, you will increase the relative demand of capital vs 
% labor. Similarly, if $\alpha$ is higher (each unit of capital is more productive), 
% you will have higher relative demand for capital vs labor as well.
%% 
% * $\frac{K^*(r, w, A, \alpha, \beta, p)}{L^*(r, w, A, \alpha, \beta, p)} = 
% \frac{w}{r}\cdot \frac{\alpha}{\beta}$
%% Choices as a Function of $w$ and $r$
% How do we solve for demand for capital and labor as a function of prices? 
% We can use the code above, except replace numerical values of $r$ and $w$ with 
% symbols. And we can easily derive demand elasticities of prices (which are constant 
% for cobb-douglas production functions)

p = 5;
alpha = 0.3;
beta = 0.5;
A = 1.0;
syms w r
% Matrix Form of linear system, same as before
B = [log(r/(p*A*alpha)); log(w/(p*A*beta))];
A = [(alpha-1), beta;alpha, beta-1];
% Solve linear equations, and then exponentiate, same as before
% We can use the simplify command to simplify this solution, get rid of exp and log:
linSolu = simplify(exp(linsolve(A, B)));
% The solution we get here is in terms of fractions, let's write them out:
KOpti = linSolu(1)
LOpti = linSolu(2)
%% *Own and Cross Price Elasticity*
% The price of labor and capital both impact the demand for labor as well as 
% for capital. 
% 
% The elasticity of capital demand with respect to interest rate is the _own_ 
% price elasticity of demand, and the elasticity of demand for capital with respect 
% to wage is the _cross_ price elasticity of demand. Similarly the elasticity 
% of labor demand with respect to wage is the _own_ price elasticity of demand, 
% and the elasticity of labor demand with respect to interest rate is the _cross_ 
% price elasticity of demand. 
%% 
% * If the _own_ and _cross_ price elastcities are in the same direction, then 
% the two inputs are complements.
%% 
% That is the case here as shown below. This means that with Cobb-Douglas production 
% function labor and capital are complements. When the price of labor, wage increases, 
% the demand for both labor and capital will decrease.  If they were substitutes, 
% when labor price increases, capital demand would increase. 
% 
% _Note_ that the elasticities below are not a function of anything, just a 
% _constant_. This is a feature of Cobb-Douglas production function, which has 
% constant demand elasticities of demand of inputs with respect to prices (also 
% constant elasticity of output with respect to inputs). This means that capital 
% and labor are always completements. 
% 
% _Note_ also that earlier on this page, we showed that with changes in prices, 
% relative choice for capital and labor will shift, that does not contradict the 
% fact that they are complements. In another word, as wage increases, firms demand 
% both less capital and less labor, but the effect is greater on labor, leading 
% to higher share of optimal capital choice.

% Elasticity of KOpti with respect to prices?
elasKoptiW = simplify((diff(KOpti, w)*w)/KOpti)
% elasKoptiW = -5/2 for alpha = 0.3 and A = 1.0
elasKoptiR = simplify((diff(KOpti, r)*r)/KOpti)
% elasKoptiR = -5/2 for alpha = 0.3 and A = 1.0
elasLoptiW = simplify((diff(LOpti, w)*w)/LOpti)
% elasLoptiW = -7/2 for alpha = 0.3 and A = 1.0
elasLoptiR = simplify((diff(LOpti, r)*r)/LOpti)
% elasLoptiR = -3/2 for alpha = 0.3 and A = 1.0
%% Graphical Results for Optimal Choices
% We can visualize the optimal choices with these codes below using mesh plot 
% and contour plot

% Number of grid points (points along x and y axis)
grid_points = 100;
% Cobb Douglas Utility
alpha = 0.30;
beta = 0.5;

% Budget
p0 = 5; % p0 is price of output
p1 = 1.05; % p1 is r
p2 = 1; % p2 is wage
maxX1 = 50; % this is max domain of capital to plot
maxX2 = 80; % this is max domain of labor to plot
% This generates a vector between 0 and 10 with grid_points number of points
x1 = linspace(0,maxX1,grid_points);
% This generates another vector between 0 and 10 with grid_points number of points
x2 = linspace(0,maxX2,grid_points);
% This creates all possible combinations of the x1 and x2 vectors, fills up the grid
[x1mesh, x2mesh] = meshgrid(x1,x2);
% Evaluate the utility function at all x1 and x2 combination points
PI = p0*(x1mesh.^alpha).*(x2mesh.^beta) - p1.*x1mesh - p2.*x2mesh;
% Graph "hi35ll" using mesh
close all;
figure();
mesh(x1mesh,x2mesh,PI);
% Labeling
xlabel('Capital');
ylabel('Labor');
zlabel('Cobb Douglas Firm Profit');
title('Profit Function for Labor and Capital Choices')
%% To see the results more easily, contour plot
figure();
hold on;
% contour plot, 100 is how many contour lines
contour = contourf(x1mesh, x2mesh, PI, 100);
clabel(contour);
% Labeling
xlabel('Capital');
ylabel('Labor');
zlabel('Cobb Douglas Firm Profit');
title('Profit Function for Labor and Capital Choices')