Question

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Let the production function be F = A(10N − 0.005N2 ), where A = 2. The labor supply curve is Ns = 32 + 10(1 − t)w, where w is the real wage and t = 0.5 is the tax rate. Let r be the real interest rate, the desired consumption and desired investment can be described by Cd = 650 + 0.8(Y − T) − 100r and Id = 650 − 100r, respectively. Government tax is T = 40 + 0.5Y and government purchase is G = 97.6. Really money demand function is L = 0.5Y − 250i, where i is the nominal interest rate. Assume nominal money supply is fixed at 27700, and the expected inflation rate πe = 2%. (a) Calculate the general equilibrium level of real wage, employment and output. (b) Find the equation that describes the IS curve. (c) Calculate the real interest rate, consumption and investment in the general equilibrium. (d) Find the equation that describes the LM curve. (e) Find the equation that describes the AD curve. (f) Calculate the price level in the general equilibrium. Bonus Find the equation that describes the FE curve.

Answer 1

(a) To calculate the general equilibrium level of real wage, employment and output, the given functions are:Production function: F = A(10N − 0.005N² ), where A = 2Labor supply curve: Ns = 32 + 10(1 − t)w, where t = 0.5Desired consumption function: Cd = 650 + 0.8(Y − T) − 100rDesired investment function: Id = 650 − 100rGovernment tax: T = 40 + 0.5YGovernment purchase: G = 97.6Real money demand function: L = 0.5Y − 250iNominal money supply: M = 27700Expected inflation rate: πe = 2%Now, the General equilibrium level of real wage, employment, and output are:Real wage:From labor supply equation Ns = 32 + 10(1 − t)w, we have:w = (Ns - 32)/10 + t/2Using t = 0.5 and A = 2, the production function can be written as:F = 2(10N − 0.005N² )F = 20N - 0.01N²Marginal product of labor (MPL) is given by:MPL = dF/dN = 20 - 0.02NFrom MPL = w, we have:w = 20 - 0.02NPutting the value of w in the above equation: (Ns - 32)/10 + t/2 = 20 - 0.02NNs - 320 = 200N - N²/50N² - 200N + 320 = 0N² - 400N + 6400 = 0N = 200 ± 20√6Putting N = 200 + 20√6 in (Ns - 32)/10 + t/2 = 20 - 0.02N, we get:w = $ 2.18Employment:From labor supply equation Ns = 32 + 10(1 − t)w, we have:Ns = 32 + 10(1 − t)wPutting t = 0.5 and w = $2.18, we get:Ns = 50.9Output:Output is given by:Y = F(Y) = 2(10N − 0.005N² )Putting N = 200 + 20√6, we get:Y = $ 10,752.12Therefore, the general equilibrium level of real wage, employment, and output is $2.18, 50.9, and $10,752.12, respectively.(b) The IS curve equation is given by:Y = Cd + Id + G − TFrom Cd = 650 + 0.8(Y − T) − 100r and Id = 650 − 100r, we have:Cd + Id = 1300 − 100rSubstituting T = 40 + 0.5Y and G = 97.6 in the above equation, we get:Y = 347.36 − 0.25r(c) Real interest rate:From the IS curve equation:Y = 347.36 − 0.25rPutting Y = 10,752.12, we have:r = 1,657.5Consumption:From Cd = 650 + 0.8(Y − T) − 100rPutting T = 40 + 0.5Y and r = 1,657.5, we get:Cd = $ 8,538.14Investment:From Id = 650 − 100rPutting r = 1,657.5, we get:Id = $ 48,250Real equilibrium can be calculated by the intersection of IS and LM curves, which is the solution for the pair of equations:Y = 347.36 − 0.25r .....(1)L = 0.5Y − 250i ......(2)Using L = M/P and M = 27700, we can rewrite equation (2) as:i = 0.002Y − 0.0048Substituting the value of i in equation (1), we get:Y = 383.64 − 0.125YPutting Y = 10,752.12, we get:P = $ 80.28(d) The equation that describes the LM curve is given by:M/P = L, where L = 0.5Y − 250iPutting i = 0.002Y − 0.0048, we get:L = 0.5Y − 125(0.002Y − 0.0048)L = 0.5Y − 0.25The equation that describes the LM curve is Y = 2,000P.(e) The equation that describes the AD curve is given by:AD = C + I + G + NX = Cd + Id + G + NX = 1300 − 100r + 97.6 + 0 = 1397.6 − 100r(f) The equation that describes the FE curve is given by:Y = YFEYFE is the level of output where the aggregate demand (AD) is equal to the potential output (Y). Therefore, putting AD = Y and solving for Y gives the equation for FE curve:1397.6 − 100r = A(10N − 0.005N² )Putting r = 1,657.5, we get:YFE = $ 10,749.5The equation that describes the FE curve is Y = $ 10,749.5.