Suppose that the following equations describe an economy (C, I, G, T, and Y
are measured in billions of dollars and r is measured in percent; for example, r = 10
C=170+0.6(Y-T),T=200,I=100-4r,G=350
(M/P)d=L=0.75Y-6r, (M/P)s=735
a. Derive the equation for the IS curve (Hint: It is easier to solve for real output Y here)
b. Derive the equation for the LM curve (Hint: Again, it is easier to solve for real output
Y here)
c. Now express both the IS and LM equations in terms of r. Draw both curves and
calculate their slopes.
d. Use the equations from Parts a and b to calculate the equilibrium levels of real output Y,
the interest rate r, planned investment I, and consumption C.
e. At the equilibrium level of real output Y, calculate the value of the government budget
surplus.
f. Suppose that G increases by 36 to 386. Derive the new IS and LM equations and draw
these curves on the graph you drew for Part c.
. Suppose that the following equations describe an economy. (C, I,G, T and Y are
measured in billions of dollars, and r is measured as a percent; for example, r = 10 =
10%):
C = 170 + 0.6(Y − T)
T = 200
I = 100 − 4r
G = 350
(M/P)d
= L = 0.75Y − 6r
Ms
/P = M / P = 735
(a) Derive the equation for the IS curve. (Hint: It is easier to solve for Y here).
Solution: Y = 1250 – 10r
a. To derive the equation for the IS curve, we need to solve for real output, Y. We have the following equations:
C = 170 + 0.6(Y - T)
T = 200
I = 100 - 4r
G = 350
We know that total output, Y, is equal to C + I + G. Substituting the given equations, we get:
Y = (170 + 0.6(Y - 200)) + (100 - 4r) + 350
Now, simplify the equation:
Y = 170 + 0.6Y - 120 + 100 - 4r + 350
Combine like terms:
0.4Y = 500 - 4r
Divide both sides by 0.4:
Y = 1250 - 10r
So, the equation for the IS curve is Y = 1250 - 10r.
b. To derive the equation for the LM curve, we use the equation for the money market equilibrium:
(M/P)d = L = 0.75Y - 6r
(M/P)s = 735
Since (M/P)d = (M/P)s at equilibrium, we get:
0.75Y - 6r = 735
Solve for Y:
Y = (735 + 6r) / 0.75
So, the equation for the LM curve is Y = 980 + 8r.
c. The IS and LM equations in terms of r are:
IS: Y = 1250 - 10r
LM: Y = 980 + 8r
Let's plot these curves on a graph:
IS curve:
- Y-axis: Y
- X-axis: r
Using the equation Y = 1250 - 10r, plot points for different values of r and connect them to form the IS curve.
LM curve:
- Y-axis: Y
- X-axis: r
Using the equation Y = 980 + 8r, plot points for different values of r and connect them to form the LM curve.
The slope of the IS curve is -10, and the slope of the LM curve is 8.
d. To calculate the equilibrium levels, we set the IS and LM equations equal to each other:
1250 - 10r = 980 + 8r
Solve for r:
18r = 270
r = 15
Substitute the value of r into the IS equation to find Y:
Y = 1250 - 10(15) = 1100
Planned investment I can be calculated using the given equation:
I = 100 - 4r = 100 - 4(15) = 40
Consumption C can be calculated using the given equation:
C = 170 + 0.6(Y - T) = 170 + 0.6(1100 - 200) = 790
Therefore, the equilibrium levels are:
Real output Y = 1100 billion dollars
Interest rate r = 15%
Planned investment I = 40 billion dollars
Consumption C = 790 billion dollars
e. To calculate the government budget surplus, we need to compare government expenditure (G) and government income (T).
Government income (T) is given as 200 billion dollars.
Government expenditure (G) is given as 350 billion dollars.
Government budget surplus = T - G = 200 - 350 = -150 billion dollars
Therefore, the government budget surplus is -$150 billion dollars.
f. If government expenditure (G) increases by 36 to 386, we can derive the new IS and LM equations.
The updated IS equation:
C = 170 + 0.6(Y - T)
T = 200
I = 100 - 4r
G = 386
Using the same steps as in part a, we can solve for Y and get the new IS equation:
Y = 1502.5 - 10r
The updated LM equation remains the same:
Y = 980 + 8r
Plotting the new IS and LM curves on the same graph as part c will show the changes as a result of the increase in government expenditure.
a. To derive the IS curve, we need to find an equilibrium condition for the goods market. The goods market equilibrium occurs when aggregate expenditure (AE) equals real output (Y).
AE is the sum of consumption (C), investment (I), and government spending (G). Let's substitute the given equations into the AE equation:
AE = C + I + G
= (170+0.6(Y-T)) + (100-4r) + 350
Now, we need to find the equilibrium condition where AE equals Y:
AE = Y
Substituting the expressions for C, I, and AE, we have:
170+0.6(Y-T) + 100-4r + 350 = Y
Now, solve for Y in terms of r and T:
Y = (170 + 100 + 350 + 0.6(Y-T)) - 4r
Y = 620 + 0.6Y - 0.6T - 4r
Rearranging the equation to solve for Y:
Y - 0.6Y = 620 - 0.6T - 4r
0.4Y = 620 - 0.6T - 4r
Y = (620 - 0.6T - 4r) / 0.4
This is the equation for the IS curve.
b. To derive the LM curve, we need to find an equilibrium condition for the money market. The money market equilibrium occurs when money demand (Md) equals money supply (Ms).
Md is represented by the liquidity preference function, which is L. Substituting the given equation for L, we have:
Md = L = 0.75Y - 6r
Ms is given as 735.
Now, we need to find the equilibrium condition where Md equals Ms:
Md = Ms
0.75Y - 6r = 735
Solving for Y in terms of r, we have:
0.75Y = 6r + 735
Y = (6r + 735) / 0.75
This is the equation for the LM curve.
c. Now, let's express both the IS and LM equations in terms of r.
IS curve: Y = (620 - 0.6T - 4r) / 0.4
LM curve: Y = (6r + 735) / 0.75
To draw both curves, plot Y on the vertical axis and r on the horizontal axis. Calculate the slopes of the curves by taking the derivatives of Y with respect to r.
The slope of the IS curve can be found by taking the derivative of Y with respect to r:
dY/dR = -4/0.4 = -10
The slope of the LM curve can be found by taking the derivative of Y with respect to r:
dY/dR = 6/0.75 = 8
d. To calculate the equilibrium levels of real output Y, interest rate r, planned investment I, and consumption C, we need to solve the IS and LM equations simultaneously.
Equilibrium occurs when the IS and LM curves intersect. So, set the IS and LM equations equal to each other:
(620 - 0.6T - 4r) / 0.4 = (6r + 735) / 0.75
Now, solve this equation to find the values of Y and r at equilibrium.
e. To calculate the government budget surplus at the equilibrium level of real output Y, subtract government spending (G) from the sum of taxes (T) and transfer payments (TR):
Government budget surplus = (T + TR) - G
Substitute the given equation for G into the formula above and evaluate it at the equilibrium level of real output Y.
f. To derive the new IS and LM equations when government spending (G) increases to 386, substitute this new value into the original IS equation from part a. Solve this equation to find the new Y in terms of r.
Now, substitute the new value of Y into the original LM equation from part b to find the new equation for the LM curve.
Plot the new IS and LM curves on the graph from part c to visualize the effect of the increase in government spending on the equilibrium.