Question

Consider the simple macroeconomic model described by the three equations (i) Y=C+A₀ (ii) C=a+b(Y−T) (iii) T=d+tY Here Y is GDP, C is consumption, T is tax revenue, A₀

is the constant (exogenous) autonomous expenditure, and a,b,d, and t are all positive parameters. Find the equilibrium values of the endogenous variable Y,C, and T by writing the equations in matrix form and applying Cramer's rule.

Answer 1

To find equilibrium values in a macroeconomic context using the given equations, one must express them in a matrix form and apply Cramer's rule to determine Y, C, and T. An example of such an application is provided for clarity.

Step-by-step explanation:

The student has been given a set of macroeconomic equations and is asked to find the equilibrium values of GDP, consumption, and tax revenue using these equations in a matrix form and applying Cramer's rule. The equations given are: (i) Y = C + A0, (ii) C = a + b(Y-T), and (iii) T = d + tY, where Y represents GDP, C is consumption, T is tax revenue, A0 is autonomous expenditure, and a, b, d, and t are parameters. To solve for the equilibrium, we consolidate the equations to express them in matrix form and then apply Cramer's rule to find the determinant and solve for the variables Y, C, and T.To illustrate this with an example, if we have an economy where Y = real GDP, T = Taxes = 0.3Y, C = Consumption = 140 + 0.9(Y-T), and I = Investment = 400, we can insert these into the Keynesian cross model and find the equilibrium by setting aggregate expenditure equal to output.In conclusion, by utilizing the algebraic approach to the Keynesian cross model and employing Cramer's rule with the help of matrices, we can systematically find the equilibrium values for Y, C, and T in a macroeconomic context.