(2 points) 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
means r = 10%) 􀜥􀵌170􀵅0.6�􀜻􀵆􀜶� 􀜶 􀵌 200 􀜫􀵌100􀵆4� 􀜩􀵌350 􀵬􀜯 􀜲
􀜯􀜲􀵰􀯦􀯗 􀵌􀵌􀜮􀜯􀜲􀴥􀴤􀵌􀵌07.7355􀜻 􀵆6�
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.
a. The IS Curve: To derive the IS curve, we start by equating aggregate demand (Y) to aggregate supply (C + I + G) in the economy.
Y = C + I + G
Given the consumption equation: C = 170 + 0.6(Y - T)
And the planned investment: I = 200 - 100r
And government spending: G = 350
Substituting these values into the aggregate demand equation, we get:
Y = 170 + 0.6(Y - T) + 200 - 100r + 350
Simplifying the equation, we have:
Y = 720 - 0.6T - 100r
This equation represents the IS curve, where real output (Y) is a function of taxes (T) and the interest rate (r).
b. The LM Curve: To derive the LM curve, we start with the money market equilibrium equation.
L(Y, r) = M
Given the money demand function: L(Y, r) = 0.2Y - 30r
And the money supply: M = 200
Setting the money market equilibrium equation equal to the money supply, we get:
0.2Y - 30r = 200
Simplifying the equation, we have:
Y = 1500 + 150r
This equation represents the LM curve, where real output (Y) is a function of the interest rate (r).
c. Expressing the IS and LM equations in terms of r:
IS curve: Y = 720 - 0.6T - 100r
LM curve: Y = 1500 + 150r
Now we can plot these curves on a graph with Y on the vertical axis and r on the horizontal axis. The slope of the IS curve is -100, and the slope of the LM curve is 150.
d. Calculating equilibrium levels:
To find the equilibrium levels of real output, we need to find the point where the IS and LM curves intersect. This occurs when Y and r are such that the IS equation and the LM equation are simultaneously satisfied.
By substituting the equations, we have:
720 - 0.6T - 100r = 1500 + 150r
Simplifying the equation, we get:
250r = -780 - 0.6T
From this equation, we can solve for the equilibrium level of r. Once we have r, we can substitute it back into either the IS or LM equation to find the equilibrium level of Y.
To calculate the equilibrium levels of planned investment (I) and consumption (C), we can substitute the equilibrium values of Y and r into their respective equations.
e. To calculate the government budget surplus at the equilibrium level of real output Y, we need to subtract government spending (G) from tax revenue (T). The government budget surplus (S) is given by:
S = T - G
Substituting the given value of G, we can calculate the government budget surplus.
f. If government spending (G) increases by 36 to 386, the IS equation would change to:
Y = 720 - 0.6T - 100r + 36
Simplifying the equation, we have:
Y = 756 - 0.6T - 100r
We can then find the new equilibrium levels of real output (Y) and the interest rate (r) by solving for their values using the new equation for IS and the original equation for LM.