Chapter 1 Supply and demand
Chapter 1 Supply and demand O. Afonso, P. B. Vasconcelos
Computational Economics: a concise introduction
O. Afonso, P. B. Vasconcelos
Computational Economics
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Overview 1
Introduction
2
Economic model (in autarky)
3
First computer program
4
Economic model with international-trade policy
5
Numerical solution
6
Computational implementation
7
Numerical results and simulation
8
Highlights
9
Main references
O. Afonso, P. B. Vasconcelos
Computational Economics
2 / 38
Introduction
The supply–demand model is used to understand the determination of the price and of the quantity of a good sold on the market. In a context with a high degree of competition, buyers (sellers) bid against each other and thereby raise (lower) the price. The equilibrium is a point at which all the bidding has been done and nobody has an incentive to offer higher (or accept lower) prices. The closed–economy scenario is extended to an open one to accommodate international trade policy. Market effects as well as social welfare are analysed. MATLAB/Octave is used to solve the model in the different cases.
O. Afonso, P. B. Vasconcelos
Computational Economics
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Economic model (in autarky)
The model is composed of endogenous and exogenous variables, parameters and functional forms. The solution provides the market equilibrium price and quantity. It is straightforward to evaluate the impacts of changes in exogenous factors that affect demand or supply and to measure those impacts on social welfare. The equations that characterise the model are: demand, Qd = Q d − aP; supply, Qs = Q s + bP.
O. Afonso, P. B. Vasconcelos
Computational Economics
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Economic model (in autarky)
Endogenous variables: quantity demanded, Qd ; quantity offered, Qs ; price of the good, P.
Exogenous variables: independent/autonomous quantity demanded, Q d ; independent/autonomous quantity offered, Q s .
Parameters: a > 0 is the sensitivity of the demand to price; b > 0 is the sensitivity of the supply to price.
O. Afonso, P. B. Vasconcelos
Computational Economics
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Economic model (in autarky)
Demand Qd is the total amount of a good that buyers would choose to purchase at a given price P as well as other (exogenous) variables Q d , such as income and wealth, prices of substitutes and complements, population, preferences (tastes) and expectations of future prices Qd = Q d − aP.
(1)
The Law of Demand states that when the price rises, the quantity demanded falls (everything else remains constant); Changes or shifts in demand occur when one of the determinants of demand other than price changes; Representing (P, Qd ), respectively, on the x–axis and y–axis, it can be stated that: demand position depends on Q d ; ∂Qd < 0 (negative slope). ∂P
O. Afonso, P. B. Vasconcelos
Computational Economics
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Economic model (in autarky)
Supply Qs is the total amount of a good that sellers would choose to produce and sell given the price P, as well as other (exogenous) variables Q s , such as prices of factors of production, prices of alternative products the firm could produce, technology, productive capacity and expectations of future prices Qs = Q s + bP.
(2)
The Law of Supply states that when the price rises, the quantity supplied also rises (everything else remains constant). Changes or shifts in supply occur when one of the determinants of supply other than price changes. Supply position depends on Q s ; ∂Qs ∂P
> 0 (positive slope).
O. Afonso, P. B. Vasconcelos
Computational Economics
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Economic model (in autarky)
Demand and supply curves
The equilibrium is reached when the supply and demand curves cross Qd = Qs , determining the equilibrium price Pe and the equilibrium quantity Qe Pe =
Q d −Q s a+b
(3)
Q = Q − aP = Q + bP . e s e e d If price is below Pe , then there is ‘excess demand’ or ‘shortage’, Qd > Qs , and the quantity that actually occurs will be Qs . If price is above Pe , then there is ‘excess supply’ or ‘surplus’, Qd < Qs , and the quantity that actually occurs will be Qd .
O. Afonso, P. B. Vasconcelos
Computational Economics
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Economic model (in autarky)
Demand and supply curves
Thus, four basic laws of supply and demand are stressed. If demand increases (demand curve shifts to the right) and supply remains unchanged, a shortage occurs, leading to a higher equilibrium price. If demand decreases (demand curve shifts to the left) and supply remains unchanged, a surplus occurs, leading to a lower equilibrium price. If demand remains unchanged and supply increases (supply curve shifts to the right), a surplus occurs, leading to a lower equilibrium price. If demand remains unchanged and supply decreases (supply curve shifts to the left), a shortage occurs, leading to a higher equilibrium price.
O. Afonso, P. B. Vasconcelos
Computational Economics
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Economic model (in autarky)
Social welfare Consumer surplus, Cs , is the consumers’ utility gain when what they are willing to pay for the good is higher than what they really have to pay; graphically, for a linear demand function, it is the area given by: Cs =
Qe (Pmax − Pe ) , 2
where Pmax ≡ Qad is the price in which the quantity demanded is zero. Producer surplus, Ps , is the gain that producers’ obtain when the price of the good that they sell in the market is higher than what they would be willing to sell; graphically, for a linear supply function, it is the area of: s )Pe Pe Q s + (Qe −Q , if Q s ≥ 0 2 Ps = Qe (Pe −Pmin ) , otherwise, 2 where Pmin ≡ − Qbs . The social welfare, Sw , is the sum of the consumer surplus and the producer surplus: Sw = Cs + Ps . O. Afonso, P. B. Vasconcelos
Computational Economics
10 / 38
First computer program
Consider Q d = 1000, Q s = 250, a = 10 and b = 5. %% Supply−demand model % Implemented by : P . B . Vasconcelos and O. Afonso disp ( ’−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−− ’ ) ; disp ( ’ Supply−demand model i n a u t a r k y ’); disp ( ’−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−− ’ ) ; %% parameters a = 1 0 ; % s e n s i t i v i t y o f t h e demand t o p r i c e b = 5; % s e n s i t i v i t y of the supply to p r i c e %% exogenous v a r i a b l e s Qd_bar = 1000; % independent / autonomous q u a n t i t y demanded Qs_bar = 250; % independent / autonomous q u a n t i t y o f f e r e d
O. Afonso, P. B. Vasconcelos
Computational Economics
11 / 38
First computer program
%% % % %
endogenous v a r i a b l e s Qd, q u a n t i t y demanded Qs , q u a n t i t y o f f e r e d P, price
%% model f p r i n t f ( ’Qd = %g − %g∗P \ n ’ , Qd_bar , a ) % demand f p r i n t f ( ’ Qs = %g + %g∗P \ n ’ , Qs_bar , b ) % s u p p l y %% compute t h e endogenous v a r i a b l e s % solving a n a l i t i c a l l y Pe = ( Qd_bar−Qs_bar ) / ( a+b ) ; Qe = Qd_bar−a∗Pe ; disp ( ’ computed endogenous v a r i a b l e s : ( e q u i l i b r i u m p o i n t ) ’ ) fprintf ( ’ q u a n t i t y , Q: %g \ n ’ , Qe) ; fprintf ( ’ price , P: %g \ n ’ , Pe ) ; %% show t h e curves ( i n t h i s case t h e l i n e s ) Q = 0:2∗Qe ; p l o t (Q, ( Qd_bar−Q) / a , ’ b ’ ,Q, ( Q−Qs_bar ) / b , ’ r−− ’ ) ; t i t l e ( ’ demand and s u p p l y curves ’ ) ; legend ( ’ demand ’ , ’ s u p p l y ’ ) ; xlabel ( ’ quantity , Q ’ ) ; ylabel ( ’ price , P ’ ) ; y l i m ( [ 0 , Qd_bar / a ] ) ; x l i m ( [ 0 , 2 ∗Qe ] )
O. Afonso, P. B. Vasconcelos
Computational Economics
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First computer program
First numerical results and simulations
--------------------------------------------------------Supply-demand model in autarky --------------------------------------------------------Qd = 1000 - 10*P Qs = 250 + 5*P computed endogenous variables: (equilibrium point) quantity, Q: 500 price, P: 50
O. Afonso, P. B. Vasconcelos
Computational Economics
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First computer program
demand and supply curves 100 demand supply
90 80 70
price, P
60 50 40 30 20 10 0
0
100
200
300
400
500 600 quantity, Q
700
800
900
1000
Supply and demand diagram
O. Afonso, P. B. Vasconcelos
Computational Economics
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First computer program
To show the consumer surplus and producer surplus, consider the following code. %% Supply−demand model : s u r p l u s and w e l f a r e % Implemented by : P . B . Vasconcelos and O. Afonso hold on ; % f r e e z e s t h e generated f i g u r e %% consumer s u r p l u s Pmax = Qd_bar / a ; Cs = Qe∗ (Pmax−Pe ) / 2 ; % area : ( 0 , Pe ) , ( 0 ,Pmax ) , ( Qe, Pe ) f i l l ( [ 0 , 0 , Qe ] , [ Pe , Pmax , Pe ] , ’ c ’ ) ; annotation ( ’ textbox ’ , [ 0 . 1 5 , 0 . 5 5 , 0 . 1 , 0 . 1 ] , . . . ’ S t r i n g ’ , ’ Consumer S u r p l u s ’ , ’ EdgeColor ’ , ’ none ’ ) fprintf ( ’ Consumer s u r p l u s , Cs : %g \ n ’ , Cs ) ;
O. Afonso, P. B. Vasconcelos
Computational Economics
15 / 38
First computer program
%% producer s u r p l u s : Pmin = −Qs_bar / b ; i f Qs_bar>=0 Ps = Pe∗Qs_bar +(Qe−Qs_bar ) ∗Pe / 2 ; % area : ( 0 , 0 ) , ( 0 , Pe ) , ( Qe, Pe ) , ( Qs_bar , 0 ) f i l l ( [ 0 , 0 , Qe, Qs_bar ] , [ 0 , Pe , Pe , 0 ] , ’ y ’ ) a n n o t a t i o n ( ’ t e x t b o x ’ , [ 0 . 1 5 0.35 0 . 1 0 . 1 ] , . . . ’ S t r i n g ’ , ’ Producer S u r p l u s ’ , ’ EdgeColor ’ , ’ none ’ ) else Ps = Qe∗ ( Pe−Pmin ) / 2 ; % area : ( 0 ,Pmim ) , ( 0 , Pe ) , ( Qe, Pe ) f i l l ( [ 0 , 0 , Qe ] , [ Pmin , Pe , Pe ] , ’ y ’ ) a n n o t a t i o n ( ’ t e x t b o x ’ , [ 0 . 1 5 0.35 0 . 1 0 . 1 ] , . . . ’ S t r i n g ’ , ’ Producer S u r p l u s ’ , ’ EdgeColor ’ , ’ none ’ ) end fprintf ( ’ Producer s u r p l u s , Ps : %g \ n ’ , Ps ) ;
The output is as follows. Consumer surplus, Cs: Producer surplus, Ps:
O. Afonso, P. B. Vasconcelos
12500 18750
Computational Economics
16 / 38
First computer program
demand and supply curves 100 demand supply
90 80 70 Consumer Surplus
price, P
60 50 40
Producer Surplus
30 20 10 0
0
100
200
300
400
500 600 quantity, Q
700
800
900
1000
Consumer and producer surplus
O. Afonso, P. B. Vasconcelos
Computational Economics
17 / 38
Economic model with international-trade policy
Two big countries, Home (H) and Foreign (F ), which produce and consume the final good, are considered. For simplification, the good can be costlessly transported, the exchange rate between currencies is fixed and production occurs under perfect competition. Home demand, Qd,H = Q d,H − aH PH ; Home supply, Qs,H = Q s,H + bH PH ; Foreign demand, Qd,F = Q d,F − aF PF ; Foreign supply, Qs,F = Q s,F + bF PF . Trade between countries emerges if, without trade (autarky), prices are different in both countries. Let us assume that the price is higher in H, which under free trade implies that producers in F begin to export it to H.
O. Afonso, P. B. Vasconcelos
Computational Economics
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Economic model with international-trade policy
The exports raises the price in F and lowers its price in H until the elimination of the difference. When prices become equal, there is no additional advantage to trade more quantity, and the prices and quantities traded stabilise. To find the world price, Pw and the quantity traded, Qw , two new curves are defined: Home import demand curve, MH = Qd,H − Qs,H , which represents the excess of what H consumes over what H produces; and Foreign export supply curve, EF = Qs,F − Qd,F , which is the excess of what F produces over what F consumes.
O. Afonso, P. B. Vasconcelos
Computational Economics
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Economic model with international-trade policy
MH intercepts the price axis at Pe in H in which import demand equals zero; from then on, MH is downward sloping because as price increases, the quantity of imports demanded declines. EF also intercepts the price axis at Pe in F in which export supply equals zero; from then on, EF is upward sloping. Hence, the world equilibrium occurs when MH equals EF , which occurs at price Pw . This framework can also be used to determine the effects on markets and agents’ welfare due to the imposition of some instrument of trade policy; e.g., a tariff on imports or an export subsidy.
O. Afonso, P. B. Vasconcelos
Computational Economics
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Economic model with international-trade policy
Effects of a tariff on imports A tariff is a tax on imports and can be either specific (a fixed sum per unit) or ad valorem (a proportion of the value imported). The effects of a specific tariff t per unit are now analysed, starting by considering a scenario without tariff. With the tariff there will be an excess demand in H and an excess supply in F . Thus, the H price will rise and it will fall in F until the price difference is t.
In H, producers supply more, while consumers demand less, so that fewer imports are demanded. Hence, the producer surplus increases and the consumer surplus falls. In F , the lower price leads to a smaller export supply. Thus, the producer surplus decreases and the consumer surplus increases.
The increase in the H price is less than the amount of the tariff, because part of the tariff is reflected in a decline in F price and thus is not passed on to H consumers. The government’s welfare is affected: it gains from collecting tariff revenue. O. Afonso, P. B. Vasconcelos
Computational Economics
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Economic model with international-trade policy
Effects of a tariff on imports Since the welfare gains and losses accrue to different agents, the overall cost–benefit evaluation of a tariff depends on how much consumers, producers and government are affected. In H, a tariff distorts the incentives of producers and consumers by inducing them to act as if imports were more expensive than they are: consumers reduce consumption, which imposes a consumption distortion loss, and producers expand production, which generates a production distortion loss; a gain in the terms of trade emerge because the tariff lowers foreign export prices.
In F , consumers gain, and the reverse occurs with foreign producers: the net welfare in F is negative, corresponding to the loss in trade terms. In terms of the world, the gain in trade terms in H is offset by the loss in trade terms in F and the distortions in H will prevail.
O. Afonso, P. B. Vasconcelos
Computational Economics
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Economic model with international-trade policy
Effects of an export subsidies An export subsidy is a payment to an agent that ships a good abroad, which can also be either specific (a fixed sum per unit) or ad valorem (a proportion of the value exported). A governmental export subsidy affects the price (it is exactly the reverse of those of a tariff): the price rises in the exporting country, but because the price falls in the importing country, the price increase is less than the subsidy. In the exporting country, consumers loss, producers gain, and the government loses because it must expend money on the subsidy. In net terms, the welfare of the exporting country is penalised.
In the Foreign importing country, consumers gain, and the reverse occurs with foreign producers. The net welfare in Foreign country is positive, corresponding to the gain in trade terms.
In terms of the world, now the loss in trade terms in H is offset by the gain in trade terms in F and the distortions will prevail. O. Afonso, P. B. Vasconcelos
Computational Economics
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Numerical solution
The models here are linear with respect to the endogenous variables and can therefore be defined by a system of linear equations, a11 x1 + a12 x2 + · · · + a1n xn = d1 a21 x1 + a22 x2 + · · · + a2n xn = d2 .. . an1 x1 + an2 x2 + ... + ann xn = dn or, in matrix form, Ax = d.
O. Afonso, P. B. Vasconcelos
Computational Economics
(4)
24 / 38
Numerical solution
Gaussian elimination The basic idea is to transform the original system in an equivalent simpler one using operations that do not change the solution. These are called elementary row operations: interchanging two equations; multiplying any equation by a nonzero scalar; adding a multiple of one equation to another.
By performing these operations, zero entries can be inserted into matrix A in order to produce an upper triangular matrix U. Then, system Ux = d (note that d is also changed with these operations) can be easily solved by back-substitution: xn = d n /unn ,
xi = d i −
n X
uij xj /uii ,
i = n − 1, · · · , 1.
(5)
j=i+1
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Computational Economics
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Numerical solution
Gaussian elimination in practice
MATLAB/Octave: the solution for Ax = d only requires the command A\d (backward slash), which performs Gauss elimination. The algorithm under backward slash: performs partial pivoting to ensure stability, and manages data efficiently in today’s computers’ hierarchical memory.
The solution of systems of linear equations is one of the most frequent tasks in numerical computing. Gaussian elimination is a direct method; alternatively there are the so-called iterative methods, which are more suited to large dimensional systems and/or sparse matrices.
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Computational Economics
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Computational implementation
Three cases are considered: autarky (in which a country is self-sufficient), free international trade and restricted trade imposed by the tariff or the subsidy. Baseline values: Q d,H = 100, aH = 20, Q s,H = 20, bH = 20, Q d,F = 80, aF = 20, Q s,F = 40 and bF = 20.
O. Afonso, P. B. Vasconcelos
Computational Economics
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Computational implementation
Presentation and solution for each country %% Supply−demand model w i t h t r a d e % Implemented by : P . B . Vasconcelos and O. Afonso disp ( ’−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−− ’ ) ; disp ( ’ Supply−demand model w i t h i n t e r n a t i o n a l t r a d e p o l i c y ’); disp ( ’−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−− ’ ) ; %% parameters aH = 1 0 ; bH = 5 ; aF = 2 0 ; bF = 2 5 ; %% exogenous v a r i a b l e s Qd_barH = 1000; Qs_barH
= 250; Qd_barF = 1400; Qs_barF
= 500;
%% model % s o l u t i o n f o r each c o u n t r y AH = [ 1 aH ; 1 −bH ] ; dH = [ Qd_barH ; Qs_barH ] ; xH = AH\ dH ; AF = [ 1 aF ; 1 −bF ] ; dF = [ Qd_barF ; Qs_barF ] ; xF = AF \ dF ; disp ( ’ computed endogenous v a r i a b l e s ( a u t a r k y ) : ’ ) fprintf ( ’ q u a n t i t y (Home) , QH: %7.2 f \ n ’ , xH ( 1 ) ) ; fprintf ( ’ p r i c e (Home) , PH: %7.2 f \ n ’ , xH ( 2 ) ) ; fprintf ( ’ q u a n t i t y ( F o r e i g n ) , QF: %7.2 f \ n ’ , xF ( 1 ) ) ; fprintf ( ’ p r i c e ( F o r e i g n ) , PF : %7.2 f \ n ’ , xF ( 2 ) ) ; O. Afonso, P. B. Vasconcelos
Computational Economics
28 / 38
Computational implementation
Home import demand and Foreign export supply
% Home I m p o r t Demand and F o r e i g n E x p o r t Supply % HID = QdH−QsH ; FES = QsF−QdF aW = aH+bH ; bW = bF+aF ; Qd_barW = Qd_barH−Qs_barH ; Qs_barW = Qs_barF−Qd_barF ; A = [ 1 aW ; 1 −bW ] ; d = [ Qd_barW ; Qs_barW ] ; x = A \ d ; QW = x ( 1 ) ; PW = x ( 2 ) ; disp ( ’ computed endogenous v a r i a b l e s ( f r e e i n t e r n a t i o n a l fprintf ( ’ q u a n t i t y ( World ) , QW: %7.2 f \ n ’ , fprintf ( ’ p r i c e ( World ) , PW: %7.2 f \ n ’ , QdHW = Qd_barH−aH∗x ( 2 ) ; QsHW = Qs_barH+bH∗x ( 2 ) ; QdFW = Qd_barF−aF∗x ( 2 ) ; QsFW = Qs_barF+bF∗x ( 2 ) ; fprintf ( ’ Home eq . demand a t PW, QdHW: %7.2 f \ n ’ , fprintf ( ’ Home eq . s u p p l y a t PW, QsHW: %7.2 f \ n ’ , fprintf ( ’ F o r e i g n eq . demand a t PW, QdFW: %7.2 f \ n ’ , fprintf ( ’ F o r e i g n eq . s u p p l y a t PW, QsFW: %7.2 f \ n ’ ,
O. Afonso, P. B. Vasconcelos
Computational Economics
trade ) : ’ ) QW) ; PW) ;
QdHW) ; QsHW) ; QdFW) ; QsFW) ;
29 / 38
Computational implementation
Plot the solution %% p l o t s Qmax = max ( [ xH ( 1 ) , xF ( 1 ) , x ( 1 ) ] ) ; Q = 0 : 1 . 5 ∗Qmax ; Pmax = max ( [ xH ( 2 ) , xF ( 2 ) , x ( 2 ) ] ) ; % Home c o u n t r y subplot ( 1 , 3 , 1 ) ; p l o t (Q, ( Qd_barH−Q) / aH , ’ b ’ ,Q, ( Q−Qs_barH ) / bH , ’ r−− ’ ) ; t i t l e ( ’Home market ’ ) ; xlabel ( ’ quantity , Q ’ ) ; ylabel ( ’ price , P ’ ) ; y l i m ( [ 0 , 2 ∗ Pmax ] ) ; x l i m ( [ 0 , 1 . 2 ∗Qmax ] ) % World subplot ( 1 , 3 , 2 ) ; p l o t (Q, ( Qd_barW−Q) /aW, ’ b ’ ,Q, ( Q−Qs_barW ) /bW, ’ r−− ’ ) ; t i t l e ( ’ World market ’ ) ; xlabel ( ’ quantity , Q ’ ) ; ylabel ( ’ price , P ’ ) ; y l i m ( [ 0 , 2 ∗ Pmax ] ) ; x l i m ( [ 0 , 1 . 2 ∗Qmax ] ) % foreign country subplot ( 1 , 3 , 3 ) ; p l o t (Q, ( Qd_barF−Q) / aF , ’ b ’ ,Q, ( Q−Qs_barF ) / bF , ’ r−− ’ ) ; t i t l e ( ’ F o r e i g n market ’ ) ; xlabel ( ’ quantity , Q ’ ) ; ylabel ( ’ price , P ’ ) ; y l i m ( [ 0 , 2 ∗ Pmax ] ) ; x l i m ( [ 0 , 1 . 2 ∗Qmax ] ) O. Afonso, P. B. Vasconcelos
Computational Economics
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Numerical results and simulation
--------------------------------------------------------Supply-demand model with international trade policy --------------------------------------------------------computed endogenous variables (autarky): quantity (Home), QH: 500.00 price (Home), PH: 50.00 quantity (Foreign), QF: 1000.00 price (Foreign), PF: 20.00 computed endogenous variables (free international trade): quantity (World), QW: 337.50 price (World), PW: 27.50 Home eq. demand at PW, QdHW: 725.00 Home eq. supply at PW, QsHW: 387.50 Foreign eq. demand at PW, QdFW: 850.00 Foreign eq. supply at PW, QsFW: 1187.50
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Computational Economics
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Numerical results and simulation
World market
Foreign market 100
90
90
90
80
80
80
70
70
70
60
60
60
50
price, P
100
price, P
price, P
Home market 100
50
50
40
40
40
30
30
30
20
20
20
10
10
10
0
0
500 1000 quantity, Q
0
0
500 1000 quantity, Q
0
0
500 1000 quantity, Q
Supply and demand curves for all markets O. Afonso, P. B. Vasconcelos
Computational Economics
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Numerical results and simulation
Free trade vs restricted trade: tariff H, as an importing country, imposes a (specific) tariff to limit imports to 250. --------------------------------------------------------Supply-demand model with international trade policy tariff --------------------------------------------------------H limits imports to 250 units computed endogenous variables (tariff): Home price, PHt: 33.33 Foreign price, PFt: 25.56 tariff: 7.78 Home quantity demanded, QdHt: 666.67 Home quantity supplied, QsHt: 416.67 Foreign quantity demanded, QdFt: 888.89 Foreign quantity demanded, QsFt: 1138.89 computed welfare variation: Home, WH_var: 230.90 Foreign, WF_var: -571.18
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Computational Economics
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Numerical results and simulation
World market
Foreign market 55
50
50
50
45
45
45
40
40
40
35
price, P
55
price, P
price, P
Home market 55
35
35
30
30
30
25
25
25
20
20
20
15
0
500 1000 quantity, Q
15
0
500 1000 quantity, Q
15 500
1000 1500 quantity, Q
Supply and demand curves, with tariff, for all markets O. Afonso, P. B. Vasconcelos
Computational Economics
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Numerical results and simulation
Free trade vs restricted trade: export subsidies F , as an exporting country, imposes a (specific) subsidy in order to export 400 units. --------------------------------------------------------Supply-demand model with international trade policy subsidy --------------------------------------------------------F applies a subsidy to exports to trade of 400 units computed endogenous variables (subsidy): Home price, PHs: 23.33 Foreign price, PFs: 28.89 subsidy: 5.56 Home quantity demanded, QdHs: 766.67 Home quantity supplied, QsHs: 366.67 Foreign quantity demanded, QdFs: 822.22 Foreign quantity demanded, QsFs: 1222.22 computed welfare variation: Home, WH_var: 512.15 Foreign, WF_var: -1710.07 O. Afonso, P. B. Vasconcelos
Computational Economics
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Numerical results and simulation
World market
Foreign market
70
70
60
60
60
50
50
50
price, P
70
price, P
price, P
Home market
40
40
40
30
30
30
20
20
20
0
500 1000 quantity, Q
0
500 1000 quantity, Q
500
1000 1500 quantity, Q
Supply and demand curves, with subsidy, for all markets O. Afonso, P. B. Vasconcelos
Computational Economics
36 / 38
Highlights
The supply–demand model provides a practical and simple explanation of how markets work. In competitive markets, the model allows for the determination of price and quantity traded, along with the social welfare. The framework is extended to consider international trade policy; the effects of a tariff to imports and of a subsidy to exports are analysed. The Gaussian elimination method for the solution of a system of linear equations is introduced.
O. Afonso, P. B. Vasconcelos
Computational Economics
37 / 38
Main references
A. Mas-Colell, M. D. Whinston, and J. R. Green Microeconomic theory Oxford University Press (1995) G. Dahlquist and Å Björck Numerical methods in scientific computing Society for Industrial Mathematics (2008) J. W. Demmel Applied Numerical Linear Algebra Society for Industrial Mathematics (1997)
O. Afonso, P. B. Vasconcelos
R. Krugman, M. Obstfeld, and M. J. Melitz International Economic: Theory and Practice Prentice Hall, 9th Edition (2011) J. M. Perloff Microeconomics: Theory and Applications with Calculus Prentice Hall, 3rd edition (2013) H. R. Varian Microeconomic Analysis W. W. Norton & Company, 3rd edition (1992)
Computational Economics
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Computational Economics: a concise introduction
O. Afonso, P. B. Vasconcelos
Computational Economics
1 / 38
Overview 1
Introduction
2
Economic model (in autarky)
3
First computer program
4
Economic model with international-trade policy
5
Numerical solution
6
Computational implementation
7
Numerical results and simulation
8
Highlights
9
Main references
O. Afonso, P. B. Vasconcelos
Computational Economics
2 / 38
Introduction
The supply–demand model is used to understand the determination of the price and of the quantity of a good sold on the market. In a context with a high degree of competition, buyers (sellers) bid against each other and thereby raise (lower) the price. The equilibrium is a point at which all the bidding has been done and nobody has an incentive to offer higher (or accept lower) prices. The closed–economy scenario is extended to an open one to accommodate international trade policy. Market effects as well as social welfare are analysed. MATLAB/Octave is used to solve the model in the different cases.
O. Afonso, P. B. Vasconcelos
Computational Economics
3 / 38
Economic model (in autarky)
The model is composed of endogenous and exogenous variables, parameters and functional forms. The solution provides the market equilibrium price and quantity. It is straightforward to evaluate the impacts of changes in exogenous factors that affect demand or supply and to measure those impacts on social welfare. The equations that characterise the model are: demand, Qd = Q d − aP; supply, Qs = Q s + bP.
O. Afonso, P. B. Vasconcelos
Computational Economics
4 / 38
Economic model (in autarky)
Endogenous variables: quantity demanded, Qd ; quantity offered, Qs ; price of the good, P.
Exogenous variables: independent/autonomous quantity demanded, Q d ; independent/autonomous quantity offered, Q s .
Parameters: a > 0 is the sensitivity of the demand to price; b > 0 is the sensitivity of the supply to price.
O. Afonso, P. B. Vasconcelos
Computational Economics
5 / 38
Economic model (in autarky)
Demand Qd is the total amount of a good that buyers would choose to purchase at a given price P as well as other (exogenous) variables Q d , such as income and wealth, prices of substitutes and complements, population, preferences (tastes) and expectations of future prices Qd = Q d − aP.
(1)
The Law of Demand states that when the price rises, the quantity demanded falls (everything else remains constant); Changes or shifts in demand occur when one of the determinants of demand other than price changes; Representing (P, Qd ), respectively, on the x–axis and y–axis, it can be stated that: demand position depends on Q d ; ∂Qd < 0 (negative slope). ∂P
O. Afonso, P. B. Vasconcelos
Computational Economics
6 / 38
Economic model (in autarky)
Supply Qs is the total amount of a good that sellers would choose to produce and sell given the price P, as well as other (exogenous) variables Q s , such as prices of factors of production, prices of alternative products the firm could produce, technology, productive capacity and expectations of future prices Qs = Q s + bP.
(2)
The Law of Supply states that when the price rises, the quantity supplied also rises (everything else remains constant). Changes or shifts in supply occur when one of the determinants of supply other than price changes. Supply position depends on Q s ; ∂Qs ∂P
> 0 (positive slope).
O. Afonso, P. B. Vasconcelos
Computational Economics
7 / 38
Economic model (in autarky)
Demand and supply curves
The equilibrium is reached when the supply and demand curves cross Qd = Qs , determining the equilibrium price Pe and the equilibrium quantity Qe Pe =
Q d −Q s a+b
(3)
Q = Q − aP = Q + bP . e s e e d If price is below Pe , then there is ‘excess demand’ or ‘shortage’, Qd > Qs , and the quantity that actually occurs will be Qs . If price is above Pe , then there is ‘excess supply’ or ‘surplus’, Qd < Qs , and the quantity that actually occurs will be Qd .
O. Afonso, P. B. Vasconcelos
Computational Economics
8 / 38
Economic model (in autarky)
Demand and supply curves
Thus, four basic laws of supply and demand are stressed. If demand increases (demand curve shifts to the right) and supply remains unchanged, a shortage occurs, leading to a higher equilibrium price. If demand decreases (demand curve shifts to the left) and supply remains unchanged, a surplus occurs, leading to a lower equilibrium price. If demand remains unchanged and supply increases (supply curve shifts to the right), a surplus occurs, leading to a lower equilibrium price. If demand remains unchanged and supply decreases (supply curve shifts to the left), a shortage occurs, leading to a higher equilibrium price.
O. Afonso, P. B. Vasconcelos
Computational Economics
9 / 38
Economic model (in autarky)
Social welfare Consumer surplus, Cs , is the consumers’ utility gain when what they are willing to pay for the good is higher than what they really have to pay; graphically, for a linear demand function, it is the area given by: Cs =
Qe (Pmax − Pe ) , 2
where Pmax ≡ Qad is the price in which the quantity demanded is zero. Producer surplus, Ps , is the gain that producers’ obtain when the price of the good that they sell in the market is higher than what they would be willing to sell; graphically, for a linear supply function, it is the area of: s )Pe Pe Q s + (Qe −Q , if Q s ≥ 0 2 Ps = Qe (Pe −Pmin ) , otherwise, 2 where Pmin ≡ − Qbs . The social welfare, Sw , is the sum of the consumer surplus and the producer surplus: Sw = Cs + Ps . O. Afonso, P. B. Vasconcelos
Computational Economics
10 / 38
First computer program
Consider Q d = 1000, Q s = 250, a = 10 and b = 5. %% Supply−demand model % Implemented by : P . B . Vasconcelos and O. Afonso disp ( ’−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−− ’ ) ; disp ( ’ Supply−demand model i n a u t a r k y ’); disp ( ’−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−− ’ ) ; %% parameters a = 1 0 ; % s e n s i t i v i t y o f t h e demand t o p r i c e b = 5; % s e n s i t i v i t y of the supply to p r i c e %% exogenous v a r i a b l e s Qd_bar = 1000; % independent / autonomous q u a n t i t y demanded Qs_bar = 250; % independent / autonomous q u a n t i t y o f f e r e d
O. Afonso, P. B. Vasconcelos
Computational Economics
11 / 38
First computer program
%% % % %
endogenous v a r i a b l e s Qd, q u a n t i t y demanded Qs , q u a n t i t y o f f e r e d P, price
%% model f p r i n t f ( ’Qd = %g − %g∗P \ n ’ , Qd_bar , a ) % demand f p r i n t f ( ’ Qs = %g + %g∗P \ n ’ , Qs_bar , b ) % s u p p l y %% compute t h e endogenous v a r i a b l e s % solving a n a l i t i c a l l y Pe = ( Qd_bar−Qs_bar ) / ( a+b ) ; Qe = Qd_bar−a∗Pe ; disp ( ’ computed endogenous v a r i a b l e s : ( e q u i l i b r i u m p o i n t ) ’ ) fprintf ( ’ q u a n t i t y , Q: %g \ n ’ , Qe) ; fprintf ( ’ price , P: %g \ n ’ , Pe ) ; %% show t h e curves ( i n t h i s case t h e l i n e s ) Q = 0:2∗Qe ; p l o t (Q, ( Qd_bar−Q) / a , ’ b ’ ,Q, ( Q−Qs_bar ) / b , ’ r−− ’ ) ; t i t l e ( ’ demand and s u p p l y curves ’ ) ; legend ( ’ demand ’ , ’ s u p p l y ’ ) ; xlabel ( ’ quantity , Q ’ ) ; ylabel ( ’ price , P ’ ) ; y l i m ( [ 0 , Qd_bar / a ] ) ; x l i m ( [ 0 , 2 ∗Qe ] )
O. Afonso, P. B. Vasconcelos
Computational Economics
12 / 38
First computer program
First numerical results and simulations
--------------------------------------------------------Supply-demand model in autarky --------------------------------------------------------Qd = 1000 - 10*P Qs = 250 + 5*P computed endogenous variables: (equilibrium point) quantity, Q: 500 price, P: 50
O. Afonso, P. B. Vasconcelos
Computational Economics
13 / 38
First computer program
demand and supply curves 100 demand supply
90 80 70
price, P
60 50 40 30 20 10 0
0
100
200
300
400
500 600 quantity, Q
700
800
900
1000
Supply and demand diagram
O. Afonso, P. B. Vasconcelos
Computational Economics
14 / 38
First computer program
To show the consumer surplus and producer surplus, consider the following code. %% Supply−demand model : s u r p l u s and w e l f a r e % Implemented by : P . B . Vasconcelos and O. Afonso hold on ; % f r e e z e s t h e generated f i g u r e %% consumer s u r p l u s Pmax = Qd_bar / a ; Cs = Qe∗ (Pmax−Pe ) / 2 ; % area : ( 0 , Pe ) , ( 0 ,Pmax ) , ( Qe, Pe ) f i l l ( [ 0 , 0 , Qe ] , [ Pe , Pmax , Pe ] , ’ c ’ ) ; annotation ( ’ textbox ’ , [ 0 . 1 5 , 0 . 5 5 , 0 . 1 , 0 . 1 ] , . . . ’ S t r i n g ’ , ’ Consumer S u r p l u s ’ , ’ EdgeColor ’ , ’ none ’ ) fprintf ( ’ Consumer s u r p l u s , Cs : %g \ n ’ , Cs ) ;
O. Afonso, P. B. Vasconcelos
Computational Economics
15 / 38
First computer program
%% producer s u r p l u s : Pmin = −Qs_bar / b ; i f Qs_bar>=0 Ps = Pe∗Qs_bar +(Qe−Qs_bar ) ∗Pe / 2 ; % area : ( 0 , 0 ) , ( 0 , Pe ) , ( Qe, Pe ) , ( Qs_bar , 0 ) f i l l ( [ 0 , 0 , Qe, Qs_bar ] , [ 0 , Pe , Pe , 0 ] , ’ y ’ ) a n n o t a t i o n ( ’ t e x t b o x ’ , [ 0 . 1 5 0.35 0 . 1 0 . 1 ] , . . . ’ S t r i n g ’ , ’ Producer S u r p l u s ’ , ’ EdgeColor ’ , ’ none ’ ) else Ps = Qe∗ ( Pe−Pmin ) / 2 ; % area : ( 0 ,Pmim ) , ( 0 , Pe ) , ( Qe, Pe ) f i l l ( [ 0 , 0 , Qe ] , [ Pmin , Pe , Pe ] , ’ y ’ ) a n n o t a t i o n ( ’ t e x t b o x ’ , [ 0 . 1 5 0.35 0 . 1 0 . 1 ] , . . . ’ S t r i n g ’ , ’ Producer S u r p l u s ’ , ’ EdgeColor ’ , ’ none ’ ) end fprintf ( ’ Producer s u r p l u s , Ps : %g \ n ’ , Ps ) ;
The output is as follows. Consumer surplus, Cs: Producer surplus, Ps:
O. Afonso, P. B. Vasconcelos
12500 18750
Computational Economics
16 / 38
First computer program
demand and supply curves 100 demand supply
90 80 70 Consumer Surplus
price, P
60 50 40
Producer Surplus
30 20 10 0
0
100
200
300
400
500 600 quantity, Q
700
800
900
1000
Consumer and producer surplus
O. Afonso, P. B. Vasconcelos
Computational Economics
17 / 38
Economic model with international-trade policy
Two big countries, Home (H) and Foreign (F ), which produce and consume the final good, are considered. For simplification, the good can be costlessly transported, the exchange rate between currencies is fixed and production occurs under perfect competition. Home demand, Qd,H = Q d,H − aH PH ; Home supply, Qs,H = Q s,H + bH PH ; Foreign demand, Qd,F = Q d,F − aF PF ; Foreign supply, Qs,F = Q s,F + bF PF . Trade between countries emerges if, without trade (autarky), prices are different in both countries. Let us assume that the price is higher in H, which under free trade implies that producers in F begin to export it to H.
O. Afonso, P. B. Vasconcelos
Computational Economics
18 / 38
Economic model with international-trade policy
The exports raises the price in F and lowers its price in H until the elimination of the difference. When prices become equal, there is no additional advantage to trade more quantity, and the prices and quantities traded stabilise. To find the world price, Pw and the quantity traded, Qw , two new curves are defined: Home import demand curve, MH = Qd,H − Qs,H , which represents the excess of what H consumes over what H produces; and Foreign export supply curve, EF = Qs,F − Qd,F , which is the excess of what F produces over what F consumes.
O. Afonso, P. B. Vasconcelos
Computational Economics
19 / 38
Economic model with international-trade policy
MH intercepts the price axis at Pe in H in which import demand equals zero; from then on, MH is downward sloping because as price increases, the quantity of imports demanded declines. EF also intercepts the price axis at Pe in F in which export supply equals zero; from then on, EF is upward sloping. Hence, the world equilibrium occurs when MH equals EF , which occurs at price Pw . This framework can also be used to determine the effects on markets and agents’ welfare due to the imposition of some instrument of trade policy; e.g., a tariff on imports or an export subsidy.
O. Afonso, P. B. Vasconcelos
Computational Economics
20 / 38
Economic model with international-trade policy
Effects of a tariff on imports A tariff is a tax on imports and can be either specific (a fixed sum per unit) or ad valorem (a proportion of the value imported). The effects of a specific tariff t per unit are now analysed, starting by considering a scenario without tariff. With the tariff there will be an excess demand in H and an excess supply in F . Thus, the H price will rise and it will fall in F until the price difference is t.
In H, producers supply more, while consumers demand less, so that fewer imports are demanded. Hence, the producer surplus increases and the consumer surplus falls. In F , the lower price leads to a smaller export supply. Thus, the producer surplus decreases and the consumer surplus increases.
The increase in the H price is less than the amount of the tariff, because part of the tariff is reflected in a decline in F price and thus is not passed on to H consumers. The government’s welfare is affected: it gains from collecting tariff revenue. O. Afonso, P. B. Vasconcelos
Computational Economics
21 / 38
Economic model with international-trade policy
Effects of a tariff on imports Since the welfare gains and losses accrue to different agents, the overall cost–benefit evaluation of a tariff depends on how much consumers, producers and government are affected. In H, a tariff distorts the incentives of producers and consumers by inducing them to act as if imports were more expensive than they are: consumers reduce consumption, which imposes a consumption distortion loss, and producers expand production, which generates a production distortion loss; a gain in the terms of trade emerge because the tariff lowers foreign export prices.
In F , consumers gain, and the reverse occurs with foreign producers: the net welfare in F is negative, corresponding to the loss in trade terms. In terms of the world, the gain in trade terms in H is offset by the loss in trade terms in F and the distortions in H will prevail.
O. Afonso, P. B. Vasconcelos
Computational Economics
22 / 38
Economic model with international-trade policy
Effects of an export subsidies An export subsidy is a payment to an agent that ships a good abroad, which can also be either specific (a fixed sum per unit) or ad valorem (a proportion of the value exported). A governmental export subsidy affects the price (it is exactly the reverse of those of a tariff): the price rises in the exporting country, but because the price falls in the importing country, the price increase is less than the subsidy. In the exporting country, consumers loss, producers gain, and the government loses because it must expend money on the subsidy. In net terms, the welfare of the exporting country is penalised.
In the Foreign importing country, consumers gain, and the reverse occurs with foreign producers. The net welfare in Foreign country is positive, corresponding to the gain in trade terms.
In terms of the world, now the loss in trade terms in H is offset by the gain in trade terms in F and the distortions will prevail. O. Afonso, P. B. Vasconcelos
Computational Economics
23 / 38
Numerical solution
The models here are linear with respect to the endogenous variables and can therefore be defined by a system of linear equations, a11 x1 + a12 x2 + · · · + a1n xn = d1 a21 x1 + a22 x2 + · · · + a2n xn = d2 .. . an1 x1 + an2 x2 + ... + ann xn = dn or, in matrix form, Ax = d.
O. Afonso, P. B. Vasconcelos
Computational Economics
(4)
24 / 38
Numerical solution
Gaussian elimination The basic idea is to transform the original system in an equivalent simpler one using operations that do not change the solution. These are called elementary row operations: interchanging two equations; multiplying any equation by a nonzero scalar; adding a multiple of one equation to another.
By performing these operations, zero entries can be inserted into matrix A in order to produce an upper triangular matrix U. Then, system Ux = d (note that d is also changed with these operations) can be easily solved by back-substitution: xn = d n /unn ,
xi = d i −
n X
uij xj /uii ,
i = n − 1, · · · , 1.
(5)
j=i+1
O. Afonso, P. B. Vasconcelos
Computational Economics
25 / 38
Numerical solution
Gaussian elimination in practice
MATLAB/Octave: the solution for Ax = d only requires the command A\d (backward slash), which performs Gauss elimination. The algorithm under backward slash: performs partial pivoting to ensure stability, and manages data efficiently in today’s computers’ hierarchical memory.
The solution of systems of linear equations is one of the most frequent tasks in numerical computing. Gaussian elimination is a direct method; alternatively there are the so-called iterative methods, which are more suited to large dimensional systems and/or sparse matrices.
O. Afonso, P. B. Vasconcelos
Computational Economics
26 / 38
Computational implementation
Three cases are considered: autarky (in which a country is self-sufficient), free international trade and restricted trade imposed by the tariff or the subsidy. Baseline values: Q d,H = 100, aH = 20, Q s,H = 20, bH = 20, Q d,F = 80, aF = 20, Q s,F = 40 and bF = 20.
O. Afonso, P. B. Vasconcelos
Computational Economics
27 / 38
Computational implementation
Presentation and solution for each country %% Supply−demand model w i t h t r a d e % Implemented by : P . B . Vasconcelos and O. Afonso disp ( ’−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−− ’ ) ; disp ( ’ Supply−demand model w i t h i n t e r n a t i o n a l t r a d e p o l i c y ’); disp ( ’−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−− ’ ) ; %% parameters aH = 1 0 ; bH = 5 ; aF = 2 0 ; bF = 2 5 ; %% exogenous v a r i a b l e s Qd_barH = 1000; Qs_barH
= 250; Qd_barF = 1400; Qs_barF
= 500;
%% model % s o l u t i o n f o r each c o u n t r y AH = [ 1 aH ; 1 −bH ] ; dH = [ Qd_barH ; Qs_barH ] ; xH = AH\ dH ; AF = [ 1 aF ; 1 −bF ] ; dF = [ Qd_barF ; Qs_barF ] ; xF = AF \ dF ; disp ( ’ computed endogenous v a r i a b l e s ( a u t a r k y ) : ’ ) fprintf ( ’ q u a n t i t y (Home) , QH: %7.2 f \ n ’ , xH ( 1 ) ) ; fprintf ( ’ p r i c e (Home) , PH: %7.2 f \ n ’ , xH ( 2 ) ) ; fprintf ( ’ q u a n t i t y ( F o r e i g n ) , QF: %7.2 f \ n ’ , xF ( 1 ) ) ; fprintf ( ’ p r i c e ( F o r e i g n ) , PF : %7.2 f \ n ’ , xF ( 2 ) ) ; O. Afonso, P. B. Vasconcelos
Computational Economics
28 / 38
Computational implementation
Home import demand and Foreign export supply
% Home I m p o r t Demand and F o r e i g n E x p o r t Supply % HID = QdH−QsH ; FES = QsF−QdF aW = aH+bH ; bW = bF+aF ; Qd_barW = Qd_barH−Qs_barH ; Qs_barW = Qs_barF−Qd_barF ; A = [ 1 aW ; 1 −bW ] ; d = [ Qd_barW ; Qs_barW ] ; x = A \ d ; QW = x ( 1 ) ; PW = x ( 2 ) ; disp ( ’ computed endogenous v a r i a b l e s ( f r e e i n t e r n a t i o n a l fprintf ( ’ q u a n t i t y ( World ) , QW: %7.2 f \ n ’ , fprintf ( ’ p r i c e ( World ) , PW: %7.2 f \ n ’ , QdHW = Qd_barH−aH∗x ( 2 ) ; QsHW = Qs_barH+bH∗x ( 2 ) ; QdFW = Qd_barF−aF∗x ( 2 ) ; QsFW = Qs_barF+bF∗x ( 2 ) ; fprintf ( ’ Home eq . demand a t PW, QdHW: %7.2 f \ n ’ , fprintf ( ’ Home eq . s u p p l y a t PW, QsHW: %7.2 f \ n ’ , fprintf ( ’ F o r e i g n eq . demand a t PW, QdFW: %7.2 f \ n ’ , fprintf ( ’ F o r e i g n eq . s u p p l y a t PW, QsFW: %7.2 f \ n ’ ,
O. Afonso, P. B. Vasconcelos
Computational Economics
trade ) : ’ ) QW) ; PW) ;
QdHW) ; QsHW) ; QdFW) ; QsFW) ;
29 / 38
Computational implementation
Plot the solution %% p l o t s Qmax = max ( [ xH ( 1 ) , xF ( 1 ) , x ( 1 ) ] ) ; Q = 0 : 1 . 5 ∗Qmax ; Pmax = max ( [ xH ( 2 ) , xF ( 2 ) , x ( 2 ) ] ) ; % Home c o u n t r y subplot ( 1 , 3 , 1 ) ; p l o t (Q, ( Qd_barH−Q) / aH , ’ b ’ ,Q, ( Q−Qs_barH ) / bH , ’ r−− ’ ) ; t i t l e ( ’Home market ’ ) ; xlabel ( ’ quantity , Q ’ ) ; ylabel ( ’ price , P ’ ) ; y l i m ( [ 0 , 2 ∗ Pmax ] ) ; x l i m ( [ 0 , 1 . 2 ∗Qmax ] ) % World subplot ( 1 , 3 , 2 ) ; p l o t (Q, ( Qd_barW−Q) /aW, ’ b ’ ,Q, ( Q−Qs_barW ) /bW, ’ r−− ’ ) ; t i t l e ( ’ World market ’ ) ; xlabel ( ’ quantity , Q ’ ) ; ylabel ( ’ price , P ’ ) ; y l i m ( [ 0 , 2 ∗ Pmax ] ) ; x l i m ( [ 0 , 1 . 2 ∗Qmax ] ) % foreign country subplot ( 1 , 3 , 3 ) ; p l o t (Q, ( Qd_barF−Q) / aF , ’ b ’ ,Q, ( Q−Qs_barF ) / bF , ’ r−− ’ ) ; t i t l e ( ’ F o r e i g n market ’ ) ; xlabel ( ’ quantity , Q ’ ) ; ylabel ( ’ price , P ’ ) ; y l i m ( [ 0 , 2 ∗ Pmax ] ) ; x l i m ( [ 0 , 1 . 2 ∗Qmax ] ) O. Afonso, P. B. Vasconcelos
Computational Economics
30 / 38
Numerical results and simulation
--------------------------------------------------------Supply-demand model with international trade policy --------------------------------------------------------computed endogenous variables (autarky): quantity (Home), QH: 500.00 price (Home), PH: 50.00 quantity (Foreign), QF: 1000.00 price (Foreign), PF: 20.00 computed endogenous variables (free international trade): quantity (World), QW: 337.50 price (World), PW: 27.50 Home eq. demand at PW, QdHW: 725.00 Home eq. supply at PW, QsHW: 387.50 Foreign eq. demand at PW, QdFW: 850.00 Foreign eq. supply at PW, QsFW: 1187.50
O. Afonso, P. B. Vasconcelos
Computational Economics
31 / 38
Numerical results and simulation
World market
Foreign market 100
90
90
90
80
80
80
70
70
70
60
60
60
50
price, P
100
price, P
price, P
Home market 100
50
50
40
40
40
30
30
30
20
20
20
10
10
10
0
0
500 1000 quantity, Q
0
0
500 1000 quantity, Q
0
0
500 1000 quantity, Q
Supply and demand curves for all markets O. Afonso, P. B. Vasconcelos
Computational Economics
32 / 38
Numerical results and simulation
Free trade vs restricted trade: tariff H, as an importing country, imposes a (specific) tariff to limit imports to 250. --------------------------------------------------------Supply-demand model with international trade policy tariff --------------------------------------------------------H limits imports to 250 units computed endogenous variables (tariff): Home price, PHt: 33.33 Foreign price, PFt: 25.56 tariff: 7.78 Home quantity demanded, QdHt: 666.67 Home quantity supplied, QsHt: 416.67 Foreign quantity demanded, QdFt: 888.89 Foreign quantity demanded, QsFt: 1138.89 computed welfare variation: Home, WH_var: 230.90 Foreign, WF_var: -571.18
O. Afonso, P. B. Vasconcelos
Computational Economics
33 / 38
Numerical results and simulation
World market
Foreign market 55
50
50
50
45
45
45
40
40
40
35
price, P
55
price, P
price, P
Home market 55
35
35
30
30
30
25
25
25
20
20
20
15
0
500 1000 quantity, Q
15
0
500 1000 quantity, Q
15 500
1000 1500 quantity, Q
Supply and demand curves, with tariff, for all markets O. Afonso, P. B. Vasconcelos
Computational Economics
34 / 38
Numerical results and simulation
Free trade vs restricted trade: export subsidies F , as an exporting country, imposes a (specific) subsidy in order to export 400 units. --------------------------------------------------------Supply-demand model with international trade policy subsidy --------------------------------------------------------F applies a subsidy to exports to trade of 400 units computed endogenous variables (subsidy): Home price, PHs: 23.33 Foreign price, PFs: 28.89 subsidy: 5.56 Home quantity demanded, QdHs: 766.67 Home quantity supplied, QsHs: 366.67 Foreign quantity demanded, QdFs: 822.22 Foreign quantity demanded, QsFs: 1222.22 computed welfare variation: Home, WH_var: 512.15 Foreign, WF_var: -1710.07 O. Afonso, P. B. Vasconcelos
Computational Economics
35 / 38
Numerical results and simulation
World market
Foreign market
70
70
60
60
60
50
50
50
price, P
70
price, P
price, P
Home market
40
40
40
30
30
30
20
20
20
0
500 1000 quantity, Q
0
500 1000 quantity, Q
500
1000 1500 quantity, Q
Supply and demand curves, with subsidy, for all markets O. Afonso, P. B. Vasconcelos
Computational Economics
36 / 38
Highlights
The supply–demand model provides a practical and simple explanation of how markets work. In competitive markets, the model allows for the determination of price and quantity traded, along with the social welfare. The framework is extended to consider international trade policy; the effects of a tariff to imports and of a subsidy to exports are analysed. The Gaussian elimination method for the solution of a system of linear equations is introduced.
O. Afonso, P. B. Vasconcelos
Computational Economics
37 / 38
Main references
A. Mas-Colell, M. D. Whinston, and J. R. Green Microeconomic theory Oxford University Press (1995) G. Dahlquist and Å Björck Numerical methods in scientific computing Society for Industrial Mathematics (2008) J. W. Demmel Applied Numerical Linear Algebra Society for Industrial Mathematics (1997)
O. Afonso, P. B. Vasconcelos
R. Krugman, M. Obstfeld, and M. J. Melitz International Economic: Theory and Practice Prentice Hall, 9th Edition (2011) J. M. Perloff Microeconomics: Theory and Applications with Calculus Prentice Hall, 3rd edition (2013) H. R. Varian Microeconomic Analysis W. W. Norton & Company, 3rd edition (1992)
Computational Economics
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