Answer :
The equation that describes the IS curve is: Y = 16.67r + 9256.86. The equation that describes the LM curve is: Y = 500i + 27607.84. The equation of AD curve is: Y = -0.53r + 12869.76. The equation that describes the FE curve is: N = 572. The real interest rate, r = 695.92.
(a) Calculation of general equilibrium level of real wage, employment, and output:
Ns = 32 + 10(1 - t)w is the labor supply curve.
Cd = 650 + 0.8(Y - T) - 100r describes the desired consumption.
Id = 650 - 100r describes the desired investment.
T = 40 + 0.5Y is the government tax.
G = 97.6 is the government purchase.
L = 0.5Y - 250i is the real money demand function.
Nd = Ns is the labor demand function.
The production function is: F = A(10N - 0.005N²) where A = 2.
The nominal money supply is fixed at 27700, and the expected inflation rate is πe = 2%.
Use the following formulas to calculate the general equilibrium level of real wage, employment, and output:
Nd = F
w = (1 - t)w
N = Nd w / (1 - t)
Y = F - wN
Cd = 650 + 0.8(Y - T) - 100r
Id = 650 - 100r
T = 40 + 0.5Y
G = 97.6
L = 0.5Y - 250i
Find the values of w, N, and Y using the above formulas:
Substituting N = Nd in Nd = Ns: 10N - 0.01N² = 32 + 10(1 - t)w
Solving for N: -0.01N² + 10N - 10tw + 32 = 0
Using the quadratic formula,
N = (10t - 10w ± √[(10w - 10t)² - 4(-0.01)(32)]) / (-0.02)
N = 500 + 250t - 5w or N = 800 + 250t - 5w
We know that: N = Nd w / (1 - t)
Substituting N = 500 + 250t - 5w in N = Nd w / (1 - t),
500 + 250t - 5w = 32 + 10(1 - t)w / (1 - t)
Solving for w: w = 22 - 0.08t + 0.4Y / 1 - 0.4
Using the above value of w in N = Nd w / (1 - t),
N = 14360 / (25 - 2t)
Substituting N = 14360 / (25 - 2t) in F = A(10N - 0.005N²),
F = 7168000 / (25 - 2t)
Now, we can find the values of Cd and Id using the given formulas:
Cd = 650 + 0.8(Y - T) - 100r
Id = 650 - 100r
Substituting T = 40 + 0.5Y in Cd,
Cd = 330 + 0.3Y - 100r
Substituting Y = F in Cd,
Cd = 330 + 0.3F - 100r
Substituting Y = F in Id,
Id = 650 - 100r
Find the value of r using the following formula:
Cd = Id
330 + 0.3F - 100r = 650 - 100r
Solving for r: r = -1,000
Substituting r = -1,000, F = 43,840
The general equilibrium level of real wage, employment, and output are:
w = 20.28
N = 572
Y = 19,120
(b) Calculation of the equation that describes the IS curve:
The IS curve represents all combinations of the interest rate and output that ensure the goods market is in equilibrium.
The IS curve can be obtained by setting the labor demand function Nd equal to the sum of consumption, investment, government purchases, and net exports:
Nd = Cd + Id + G - T
Cd = 330 + 0.3F - 100r
Id = 650 - 100r
T = 40 + 0.5Y
G = 97.6
Substituting the above values in Nd = Cd + Id + G - T,
10N - 0.01N² = 0.6Y - 30r + 468.8
Solving for Y: Y = 16.67r + 9256.86
So, the equation that describes the IS curve is: Y = 16.67r + 9256.86
(c) Calculation of the real interest rate, consumption and investment in the general equilibrium:
Substituting Y = 19,120 in Y = 16.67r + 9256.86,
r = 695.92
Substituting r = 695.92 in Cd = 330 + 0.3F - 100r,
Cd = -58,174.4
Substituting r = 695.92 in Id = 650 - 100r,
Id = -65,092
The real interest rate, consumption, and investment in the general equilibrium are:
r = 695.92
Cd = -58,174.4
Id = -65,092
(d) Calculation of the equation that describes the LM curve:
The LM curve represents all combinations of the interest rate and output that ensure the money market is in equilibrium.
The LM curve can be obtained by equating the real money supply to the real money demand function:
L = mP
Substituting L = 0.5Y - 250i, m = 27700 / (1 + πe), and P = 1 in L = mP,
0.5Y - 250i = 13803.92
Solving for Y: Y = 500i + 27607.84
So, the equation that describes the LM curve is: Y = 500i + 27607.84
(e) Calculation of the equation that describes the AD curve:
The AD curve represents all combinations of the price level and output that ensure the goods market and money market are in equilibrium.
The AD curve can be obtained by substituting Y = 16.67r + 9256.86 and Y = 500i + 27607.84 in the following formula:
Y = C(Y - T) + I(r) + G + NX(eP)
C(Y - T) = 330 + 0.3(Y - 40 - 0.5Y)
I(r) = 650 - 100r
G = 97.6
NX(eP) = 0
Substituting the above values,
Y = -0.53r + 12869.76
(f) Calculation of the price level in the general equilibrium:
The price level can be obtained by substituting Y = 19,120 in Y = -0.53r + 12869.76.
Substituting Y = 19,120,
P = 134
Bonus: Calculation of the equation that describes the FE curve:
The FE curve represents all combinations of the interest rate and output that ensure the goods market, money market, and labor market are in equilibrium.
The FE curve can be obtained by equating the labor demand function Nd to the labor supply function Ns:
Nd = Ns
Nd = 10N - 0.01N²
Ns = 32 + 10(1 - t)w
Substituting w = 20.28, t = 0.5, and Y = 19,120 in Ns,
Ns = 42.4 + 8Y / 3 - 2t
Substituting Nd = Ns,
0.01N² - 10N + 8Y / 3 - 2t + 42.4 = 0
Using the quadratic formula,
N = 572
So, the equation that describes the FE curve is: N = 572
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