Mathematics for Economists. Cramer s Rule = A j

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1 Mathematics for Economists Cramer s Rule Introduction When there is a system of equations and the number of unknown variables equals the number of equations the system has a solution. Let us consider a system of n linear equations for n unknowns Ax = b. Given that b is homogenous and equal to zero if the determinant of matrix A is equal to zero then the system has an infinite number of solutions. However if the determinant of matrix A is different to zero then the system will have a unique and trivial zero solution. When b is not equal to zero (non-homogenous), then if the determinant of matrix A is equal to zero then if the rank of A is equal to the rank of the augmented matrix of A then the system will have an infinite number of solutions. Alternatively under non-homogeneity if the rank of A is not equal to the augmented matrix of A then the system will be inconsistent. If b is not equal to zero (non-homogenous) and the determinant of matrix A is not equal to zero then the system will have a unique solution. In this section we explore the method known as Cramer's Solution through the following examples. Three commonly used ways to solve a system of linear equations 1. Inverse Matrix Method Since A 1 exists the solution x can be found by x = A 1 b 2. Gauss Method uses elementary row operations until the matrix A has been reduced to the identity matrix. Then the vector b that the elementary row operations have been performed on is the solution. This is because as we turn the matrix A into the identity matrix we are adjusting the vector b therefore what is left is the solution. 3. Cramer's Rule uses an equation to find all the element the solution vector x Cramer s Rule = A j A Cramer's Rule Cramer s rule is a method for solving a system of linear equations. This rule uses determinants to solve the system and is formally written as the following. x i = A i A x i is the i th unknown in a system of equation and A is the determinant of the coefficient matrix. Additionally A i is the determinant of a matrix formed from the original coefficient matrix by replacing the column of coefficients of x i with the column vector of constants as shown in the following examples. 1

2 Example 1 A System of Linear Equations Solve the system of linear equations for x 1 and x 2. 7x 1 + 5x 2 = 8 4x 1 x 2 = Write down the system in matrix form AX = B = x 1 x 2 = Find the determinant of A A = 7 ( 1) + 5 (4) = To Solve for x 1, replace column 1, the coefficients with x 1 with the vector of constants B. A 1 = Find the determinant of the new matrix A 1 A 1 = Now use the Cramer s Rule formula above x 1 = A 1 A = = Use the same procedure to get the result for x 2 A 2 = A 2 = 8 x 2 = A 2 A = 8 13 = Example 2: General Equilibrium Suppose the price of corn is related to the price of wheat and we are trying to find the equilibrium price for each market. Find the solution to the following system of equations. 10P w P c = 87 2P w + 30P c = 98 AX = B = p w pc = A = ( 1) ( 2) = 298 A 1 = A 1 = ( 1) 98 =

3 p w = A 1 A = = A 2 = 2 98 A 2 = (2) = 1154 p c = A 1 A = = Therefore we get P w = and p c = as the equilibrium prices of wheat and corn. Example 3: General Equilibrium Suppose that we now have three goods in a market and we want to find the general market equilibrium. We know the supply and demand relations for all three goods and they are described below. Good 1 S 1 = 8 5P 1 + 3P 2 + 2P 3 D 1 = 2 + 3P 1 + 2P 2 Good 2 S 2 = 6 + 4P 1 8P 2 + 2P 3 D 2 = P P 2 Good 3 S 1 = P 1 + 3P 2 7P 3 D 1 = 2 6P 3 Now equate the Demand and Supply equations for each good (D i = S i where i = 1,2,3) 8P 1 P 2 2P 3 6 3P 1 2P 2 2P 3 = 6 15P 1 3P 2 P P 1 6 AX = B = P 2 = P A = A = 8 (2 ( 1) 2 3) ( 1) ( 3) ( 1) ( 3) = A 1 = A 1 = 6 (2 ( 1) 2 3) ( 1)(6 ( 1) 2 0) + 2( ) = 153 P 1 = A 1 A = =

4 Therefore the price of good 1 P 1 in the general market equilibrium is $ We repeat the same logic to derive the general market equilibrium P 2 and P 3. Example 5 The IS-LM Model: System of Linear Equations The equilibrium income/output Y and interest rates r are solutions to the system of linear equations. The income. As there are two equations for a solution there cannot be more than two unknowns. When solving this system all the other variables will have values and the interest rate and output solve the system so they each side equate. The solutions depend on the given policy parameters of G and M s all the other factors are behavioural. The advantage of using a system of equations to describe the equilibrium is that you can see how the solutions to the system change as the parameters change. Suppose we have the following IS sy = I 0 ar + G = 0.4Y = r LM M s = my + M 0 hr = 300 = 0.88Y r Hint: We can rewrite these equations to make it easier to put them into matrix form. To order the matrix correctly simply vertically align like variables. In this case we have the variables Y in line then followed by the interest rate variables r then the constants. 0.4Y + 134r = Y 111r = 50 The left hand side of the equation will be the coefficient matrix and the right hand side will be the vector of constants. When converting into the matrix we remove the Y and r and put them into a solution vector because these two variables are the solution for the system. Coefficient Matrix A = , Solution Vector X = y and the Vector of r Constants B = Writing the system of equations in matrix form. AX = B = Y = 400 r 50 Now we use Cramer's Rule to solve the system of equations. A = A = 0.4 ( 111) = A 1 = A 1 = 400 ( 111) = Y = A 1 A = =

5 A 2 = A 2 = = Y = A 1 A = = Therefore, we have solved the system and in equilibrium the level of output is Y = and the interest rate is r =