. (30 points) Consider an economy described as follows: Y=C+I+G Y=9,000 G=2,000 T=1,000 C=2000+1/2(Y-T) I=1,600-100r

To find the equilibrium level of output in this economy, we need to solve for the equilibrium level of output (Y) where aggregate demand equals aggregate supply.<br /><br />Given the equations:<br />$Y = C + I + G$<br />$Y = 9,000$<br />$G = 2,000$<br />$T = 1,000$<br />$C = 2,000 + \frac{1}{2}(Y - T)$<br />$I = 1,600 - 100r$<br /><br />First, let's substitute the given values into the consumption function:<br />$C = 2,000 + \frac{1}{2}(Y - 1,000)$<br /><br />Now, we can substitute the consumption function into the aggregate demand equation:<br />$Y = C + I + G$<br />$Y = (2,000 + \frac{1}{2}(Y - 1,000)) + (1,600 - 100r) + 2,000$<br /><br />Simplify the equation:<br />$Y = 2,000 + \frac{1}{2}Y - 500 + 1,600 - 100r + 2,000$<br />$Y = \frac{1}{2}Y + 4,100 - 100r$<br /><br />To solve for Y, we need to isolate it on one side of the equation:<br />$\frac{1}{2}Y = Y - 4,100 + 100r$<br />$\frac{1}{2}Y = 100r - 4,100$<br />$Y = 200r - 8,200$<br /><br />Now, we need to find the equilibrium level of output by setting aggregate demand equal to aggregate supply:<br />$Y = C + I + G$<br />$9,000 = (2,000 + \frac{1}{2}(Y - 1,000)) + (1,600 - 100r) + 2,000$<br /><br />Substitute the expression for Y:<br />$9,000 = (2,000 + \frac{1}{2}(200r - 8,200 - 1,000)) + (1,600 - 100r) + 2,000$<br />$9,000 = (2,000 + \frac{1}{2}(200r - 9,200)) + (1,600 - 100r) + 2,000$<br />$9,000 = 2,000 + 100r - 4,600 + 1,600 - 100r + 2,000$<br />$9,000 = 4,000 - 2,600$<br />$9,000 = 1,400$<br /><br />Since the equation does not make sense, it seems there is an error in the given information or the calculations. Please double-check the provided values and equations.