2. Assume that GDP (Y) is 5,000. Consumption (C) is given by the equation
C = 1,200 + 0.3(Y-T) – 50r where r is the real interest rate. Investment (I) is given by the equation I = 1,200 – 50r. Taxes (T) are 1,000 and government spending (G) is 1,500.
a. What are the equilibrium values of C, I and r?
b. What are the values of private saving, public saving and national saving?
c. Now assume there is a technological innovation that makes business want to invest more. It raises the investment equation to I = 2,100 – 50 r. What are the new equilibrium values of C, I, and r?
d. What are the values of private saving, public saving and national saving?
C=15+0.50Y+0.0.5W
I=100
W=1,200
Calculate the values of equilibrium Y,C,and S
a. To find the equilibrium values of C, I, and r, we need to set the total desired spending (aggregate expenditure) equal to the total output (GDP). In equilibrium, Y = C + I + G.
Given:
Y = 5,000
C = 1,200 + 0.3(Y - T) - 50r
I = 1,200 - 50r
T = 1,000
G = 1,500
Substituting these values into the equation for equilibrium, we get:
5,000 = (1,200 + 0.3(Y - 1,000) - 50r) + (1,200 - 50r) + 1,500
Simplifying the equation, we have:
5,000 = 2,400 + 0.3(Y - 1,000) - 100r
Let's rearrange the equation to solve for Y:
0.3(Y - 1,000) - 100r = 5,000 - 2,400
0.3Y - 300 - 100r = 2,600
Combining the like terms:
0.3Y - 100r = 2,600 + 300
0.3Y - 100r = 2,900
Now let's solve for Y by isolating it:
0.3Y = 100r + 2,900
Y = (100r + 2,900) / 0.3
We have equation (1) for Y.
Next, let's solve for C using the given equation:
C = 1,200 + 0.3(Y - T) - 50r
Substitute equation (1) for Y into the equation for C:
C = 1,200 + 0.3[((100r + 2,900) / 0.3) - 1,000] - 50r
Simplify the equation:
C = 1,200 + (100r + 2,900) - 1,000 - 50r
C = 3,100 + 50r
We have equation (2) for C.
Lastly, let's solve for I using the given equation:
I = 1,200 - 50r
We already have equation (2) for C, so we can substitute it into the equation for I:
I = 1,200 - 50r
We have equation (3) for I.
Now, to find the values of C, I, and r, we need to solve equations (1), (2), and (3) simultaneously by substituting and solving.
b. To find the values of private saving, public saving, and national saving, we use the following equations:
Private Saving (Sprivate) = (Y - T - C)
Public Saving (Spublic) = (T - G)
National Saving (S) = (Sprivate + Spublic)
Now, let's calculate the values:
Private Saving (Sprivate) = (5,000 - 1,000 - C) [Substitute the given values for Y and T]
Public Saving (Spublic) = (1,000 - 1,500) = -500 [Substitute the given values for T and G]
National Saving (S) = (Sprivate + Spublic)
c. Now, assuming a technological innovation that increases investment, we have a new equation for I: I = 2,100 - 50r.
We need to find the new equilibrium values of C, I, and r. We will solve them using the same method as in part a.
Given:
Y = 5,000
C = 1,200 + 0.3(Y - T) - 50r
I = 2,100 - 50r
T = 1,000
G = 1,500
Substituting these values into the equation for equilibrium (Y = C + I + G), we get:
5,000 = (1,200 + 0.3(Y - 1,000) - 50r) + (2,100 - 50r) + 1,500
Simplify the equation:
5,000 = 3,300 + 0.3(Y - 1,000) - 100r
Let's rearrange the equation to solve for Y:
0.3(Y - 1,000) - 100r = 5,000 - 3,300
0.3Y - 300 - 100r = 1,700
Combining the like terms:
0.3Y - 100r = 2,000
Now let's solve for Y by isolating it:
0.3Y = 100r + 2,000
Y = (100r + 2,000) / 0.3
We have equation (4) for Y.
Next, let's solve for C using the given equation again:
C = 1,200 + 0.3(Y - T) - 50r
Substitute equation (4) for Y into the equation for C:
C = 1,200 + 0.3[((100r + 2,000) / 0.3) - 1,000] - 50r
Simplify the equation:
C = 1,200 + (100r + 2,000) - 1,000 - 50r
C = 2,200 + 50r
We have equation (5) for C.
Lastly, let's solve for I using the new given equation:
I = 2,100 - 50r
We already have equation (5) for C, so we can substitute it into the equation for I:
I = 2,100 - 50r
We have equation (6) for I.
Now, to find the new values of C, I, and r, we need to solve equations (4), (5), and (6) simultaneously by substituting and solving.
d. To find the new values of private saving, public saving, and national saving, we use the same equations as in part b:
Private Saving (Sprivate) = (Y - T - C)
Public Saving (Spublic) = (T - G)
National Saving (S) = (Sprivate + Spublic)
Now, let's calculate the new values based on the new equilibrium values of C, I, and r.