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Unformatted text preview: Department of Economics Fall 2010 University of California, Berkeley Economics 100B Problem Set #4 Suggested Solutions Page 1 of 13 Problem Set #4 Suggested Solutions 1. (1.5 points total) a. ( point) Starting from the equation Y=AD and using the equations for the components of aggregate demand that were given in lecture on October 7, derive the equation for the equilibrium real interest rate. There are two ways to approach this problem. One takes the S=I approach, which is what you see in Box 7.3 in the textbook. Rather than replicate that approach here, lets instead type up the alternative approach, which is to start from Y = AD, substitute in the expressions for C, I, G, and NX, and then solve for r. You get the same result assuming you start with the same expressions for C, I, G, and NX but when you start from Y=AD, you have fewer negative signs floating around and thus fewer potential errors to make. Starting from Y = AD = C + I + G + NX Substitute in the expressions for C, I, G, and NX Y = [C + C Y (Y D )] + [I I r r] + G + [GX 0 + X f Y f + X IM IM Y Y] Expand the definitions of disposable income and of the real exchange rate Y = [C + C Y (Y T tY)] + [I I r r] + G + [GX 0 +X f Y f + X ( r [ r r f ] ) IM IM Y Y] Distribute the terms r and X Y = [C + C Y (Y T - tY)] + [I I r r] + G + [GX 0 +X f Y f + X X r r + X r r f IM IM Y Y] Factor out the Y in the consumption function Y = [C + C Y (1 t)Y C Y T ] + [I I r r] + G + [GX 0 +X f Y f + X X r r + X r r f IM IM Y Y] Now gather all the terms with r on the left and slide the Y to the right (remember the sign flips when you move something from one side of the equals sign to the other) I r r + X r r = -Y + C + C Y (1 t)Y C Y T 0 + I + G + GX 0 +X f Y f + X + X r r f IM IM Y Y Lets gather all the Y terms together, just for fun I r r + X r r = -Y + C Y (1 t)Y IM Y Y C Y T 0 + C 0 + I + G + GX 0 +X f Y f + X + X r r f IM Now factor out r on the left (I r + X r )r = -Y + C Y (1 t)Y IM Y Y C Y T 0 + C 0 + I + G + GX 0 +X f Y f + X + X r r f IM And divide by the term (I r + X r ), and were done!...
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