Coursenotes_ECON301

xa r 3w r 3w 10px 20wr12 xb 4r 2w 4r 2w

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Unformatted text preview: t supply of good Y. At market equilibrium, the supply of good Y must equal the aggregate demand of good Y. QY = Y(r , w) = r + w 4(wr)1/2 MARKET FOR LABOUR Substituting this output supply, QY, into the producer demands for labour, we get... LX = QX(r / w)1/2 LX = r + w r1/2 1/2 4(wr) w1/2 LX = r + w 4w 4r + 2w 20(wr)1/2 KY = QY(w / r)1/2 w1/2 KY = r + w 4(wr)1/2 r1/2 KY = r + w 4r and aggregate demand for capital K=r+w 4r K=r+w 2r At capital market equilibrium, the Aggregate capital demand must match the aggregate capital supply. r+w 2r =1 +r+w 4r LY(r, w) = lY QY LY = r + w r1/2 4(wr)1/2 w1/2 LY = r + w 4w and aggregate demand for labour L=r+w +r+w 4w 4w L=r+w 2w At labour market equilibrium, the the aggregate labour demand must match the aggregate labour supply. r+w =1 2w If we solve for capital market equilibrium condition, we get the relationship between the price of labour and the price of capital. 2w = r + w w=r r/w=1 This is the equilibrium price solution that we need to figure out everything else. If we solve for capital market equilibrium condition, we get the following relationship between the price of labour and the price of capital. 2r = r + w r=w w/r=1 This is the equilibrium price solution that we need to figure out everything else. Let's figure out the rest of our equilibrium price vector... If we choose capital as the numeraire then r=1 256 r=w=1 PX = 2(wr)1/2 = 2r = 2 PY = 2(wr)1/2 = 2r = 2 Now let's figure out the consumer demands in equilibrium (both private goods for now)... XA = r + 3w = r + 3w 10PX 20(wr)1/2 X*A = 4 / 20 = 1 / 5 XB = 4r + 2w = 4r + 2w 10PX 20(wr)1/2 X*B = 6 / 20 = 3 / 10 So goods market demands are... X* = X*A + X*B = Y* = Y*A + Y*B = YA = r + 3w = r + 3w 10PY 20(wr)1/2 Y*A = 4 / 20 = 1 / 5 YB = 4r + 2w = 4r + 2w 10PY 20(wr)1/2 Y*B = 6 / 20 = 3 / 10 and just to check market clearing, this should be the market supply (i.e. amount produced by the producers) KX = r + w 4r KY = r + w 4r K=r+w 2r = = =1 LX = r + w = 4w LY = r + w = 4w L=r+w=1 2w (notice that this implies factor market clearing since facto...
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