Two Famous Economic Models Solved by 3D-FOA

3D-FOA が解決しようと二つの有名な経済モデル


In addition, we also use two popular economic examples to find the optimal solutions based on the 3D-FOA as follows:

Let us first postulate a two-product firm under circumstances of monopoly.

1. Problem of a Multiproduct Firm

Suppose that the demands facing the monopolist firm are as follows:

  Q_1= 40 - 2 P_1  + P_2                    (17)

  Q_2= 15  +  P_1   - P_2                    (18)

The firm total revenue function can be written as

R=P_1 Q_1+P_2 Q_2                         (19)

And the total cost function is

C=Q_1^2+Q_1 Q_2+ Q_2^2              (20)

Then the profit function will be:

Π = R-C= 55Q_1+ 70 Q_2 - 3Q_1 Q_2  - 2 Q_1^2 - 3 Q_2^2       (21)

which is the objective function with two choice variables (Q_1, Q_2). Thus the optimal solution of output levels and profit can be found by Calculus as following:

(Q_1 *, Q_2 *, Π*) = (8, 7 2/3, 488 1/3))

And our computer simulation result is shown in Figure 14:

We find the solution of Q_1 * = x1 = 7.91, Q_2 * =  x2  =7.75, and profit  Π* = 488.32 by 3D-FOA. It takes 0.129 second.

 

 

  Fig. 14 Find the maximal value of monopolistic profit

 

2. Input Decisions of a Firm

Next, consider a competitive firm with the following profit function: 

R-C=PQ-(wL+rK)                   (22)
 

where P=price, Q=output, K=capital, L=labor; w and r denote input price for L and K, respectively, Π= profit.

Since the firm operates in a competitive market, the exogenous variables are P, w, and r, There are three endogenous (decision) variables, K, L, and Q in this example. Output Q is in turn a function of K and L via the Cobb-Douglas production function

Q = f (K,L) = AL^α K^β          (23)

For simplicity, we shall consider the symmetric case where α=β <1/2, Therefore, the cost function and profit function are defined as:

C = w L+r K                               (24)

 Π = P Q- C                                (25)

Traditional Calculus gives us an expression (closed solution) for the optimal inputs and output as a function of the exogenous variables P, w, r respectively, i.e.

L* = (( P α w^ ( α-1) r ^ (-α))) ^ ( 1/(1-2α))           (26)

K* = (( P α r^ (α-1) w ^ (-α))) ^ (1/(1-2α))             (27)

Q* = (( α^2 P ^2 / w r)) ^ (α/(1-2α))                       (28)

Assume that the competitive price is $100 (P=100), wage rate $10 (w=10), Interest rate 10% (r=0.1), α=β=0.4. Then the optimal solutions are L*= 1024, K*=1024, Q*= 256, C*= 2048, and  Π*=512.

Fig. 15 Find the maximal value of competitive firm’s profit

Similarly, from 3D-FOA, It is easier to find the optimal solution of  L*=1026,  K*=1026 and profit Π*= 512 only 0.486 second, which is shown in Figure 15.

These two firm’s profits and their contours are also shown in Figure 16:

Fig. 16 Two firm’s optimal profits and their contours

From the simulation results of these two economic examples, we could easily find the optimal solutions by 3D-FOA compare to traditional calculus. Therefore, our method can be further applied in other economic applications in the future.

References:

1. Wei-Yuan Lin (2013), “3D-Novel fruit fly optimization algorithm and its applications in economics,” Working paper, Department of Economics, Soochow University, Taiwan.
2. Chiang AC, Wainwright K (2005) Fundamental methods of mathematical economics, 4th edn. McGraw Hill.
3. Nien Benjamin (2011) Application of data mining and fruit fly optimization algorithm to construct financial crisis early warning model – A case study of listed companies in Taiwan, Master Thesis, Department of Economics, Soochow University, Taiwan (in chinese).
4. Wei-Yuan Lin (2012),“A Hybrid Approach of 3D Fruit Fly Optimization Algorithm and General Regression Neural Network for Financial Distress  Forecasting,” Working Paper, Jan. 2012, Soochow University, Taiwan.

Jing Si Aphorism:

While working, learn;

While learning, awaken to many truths of life.

 

  Soochow University EMA