Marginal Product of Labor

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Marginal Product of Additional Workers (Discrete Workers)

Suppose we can not hire fractions of workers, but have to hire 1, 2, 3, etc.. What is the marginal product of each additional worker?

% fixed capital level

K = 1;

% current labor level

L = [1,2,3,4,5,6,7,8,9,10];

% Cobb Douglas Production Parameters

alpha = 0.5;

beta = 1-alpha;

% Output at x0

fx0 = (K^alpha)*(L.^beta);

% a vector of h

h = 1;

% output at fx0plush

x0plush = L+h;

fx0plush = (K^alpha)*((x0plush).^beta);

% derivatie

outputIncrease = (fx0plush - fx0)./h;

% Show Results in table

T = table(L', x0plush', fx0plush', outputIncrease');

T.Properties.VariableNames = {'L', 'x0plush', 'fx0plush', 'outputIncrease'};

disp(T);

L x0plush fx0plush outputIncrease __ _______ ________ ______________ 1 2 1.4142 0.41421 2 3 1.7321 0.31784 3 4 2 0.26795 4 5 2.2361 0.23607 5 6 2.4495 0.21342 6 7 2.6458 0.19626 7 8 2.8284 0.18268 8 9 3 0.17157 9 10 3.1623 0.16228 10 11 3.3166 0.15435

% Graph

close all;

figure();

hold on;

plot(L, outputIncrease);

scatter(L, outputIncrease,'filled');

grid on;

ylabel('Marginal Output Increase from each Additional Worker (h=1)')

xlabel('L, previous/existing number of workers')

title('Discrete Labor Unit, Marginal Product of Each Worker')

Using Derivative to approximate Increase in Output from More Workers

We know the MPL formula, so we can evaluate MPL at the vetor of L

% fixed capital level
K = 1;
% current labor level
L = [1,2,3,4,5,6,7,8,9,10];
% Cobb Douglas Production Parameters
alpha = 0.5;
% Output at x0
fprimeX0 = (1-alpha)*(K^alpha)*(L.^(-alpha));
T = table(L', outputIncrease', fprimeX0');
T.Properties.VariableNames = {'L', 'outputIncrease','fprimeX0'};
disp(T);
    L     outputIncrease    fprimeX0
    __    ______________    ________

     1       0.41421            0.5 
     2       0.31784        0.35355 
     3       0.26795        0.28868 
     4       0.23607           0.25 
     5       0.21342        0.22361 
     6       0.19626        0.20412 
     7       0.18268        0.18898 
     8       0.17157        0.17678 
     9       0.16228        0.16667 
    10       0.15435        0.15811 

Marginal Product of Additional Workers Different Capital (Discrete Workers)

Suppose we can not hire fractions of workers, but have to hire 1, 2, 3, etc.. What is the marginal product of each additional worker?

% fixed capital level

K1 = 1;

[fprimeX0K1, L] = MPKdiscrete(K1);

K2 = 2;

[fprimeX0K2, L] = MPKdiscrete(K2);

K3 = 3;

[fprimeX0K3, L] = MPKdiscrete(K3);

% Graph

close all;

figure();

hold on;

plot(L, fprimeX0K1);

scatter(L, fprimeX0K1,'filled');

plot(L, fprimeX0K2);

scatter(L, fprimeX0K2,'filled');

plot(L, fprimeX0K3);

scatter(L, fprimeX0K3,'filled');

grid on;

ylabel('Marginal Output Increase from each Additional Worker (h=1)')

xlabel('L, previous/existing number of workers')

title('Discrete Labor Unit, Marginal Product of Each Worker')

legend(['k=',num2str(K1)], ['k=',num2str(K1)],...

['k=',num2str(K2)],['k=',num2str(K2)],...

['k=',num2str(K3)],['k=',num2str(K3)]);

function [fprimeX0, L] = MPKdiscrete(K)
% current labor level
L = [1,2,3,4,5,6,7,8,9,10,11,12,13,14,15];
% Cobb Douglas Production Parameters
alpha = 0.5;
beta = 1-alpha;
% Output at x0
fx0 = (K^alpha)*(L.^beta);
% a vector of h
h = 1;
% output at fx0plush
x0plush = L+h;
fx0plush = (K^alpha)*((x0plush).^beta);
% derivatie 
fprimeX0 = (fx0plush - fx0)./h;
end