Question

question 10 (1 point)
in a closed economy with the following information:

• consumption function: ( c = 50 + 0.6(y - t) )
• government spending (( g )): 15
• investment (( i )): 15
• taxes (( t )): 2

calculate the equilibrium level of income (( y )) and the corresponding level of consumption (( c )).
( \bigcirc ) ( y = 200, c = 170 )
( \bigcirc ) ( y = 197, c = 167 )
( \bigcirc ) ( y = 180, c = 160 )
( \bigcirc ) ( y = 170, c = 150 )

Explanation:

Step1: Recall the equilibrium condition for a closed economy

In a closed economy, the equilibrium level of income \( Y \) is given by the sum of consumption \( C \), investment \( I \), and government spending \( G \). So the formula is \( Y = C + I + G \).

We know the consumption function is \( C = 50 + 0.6(Y - T) \), \( I = 15 \), \( G = 15 \), and \( T = 2 \).

Step2: Substitute the consumption function into the equilibrium equation

Substitute \( C = 50 + 0.6(Y - T) \) into \( Y = C + I + G \):

\( Y = 50 + 0.6(Y - T) + I + G \)

Now plug in the values \( T = 2 \), \( I = 15 \), \( G = 15 \):

\( Y = 50 + 0.6(Y - 2) + 15 + 15 \)

Step3: Simplify the equation

First, expand the term \( 0.6(Y - 2) \):

\( Y = 50 + 0.6Y - 1.2 + 15 + 15 \)

Now combine the constant terms: \( 50 - 1.2 + 15 + 15 = 50 + 15 + 15 - 1.2 = 80 - 1.2 = 78.8 \)? Wait, no, wait: 50 -1.2 is 48.8, then 48.8 +15 is 63.8, then 63.8 +15 is 78.8? Wait, that can't be right. Wait, maybe I made a mistake in the numbers. Wait, the problem says \( I = 15 \), \( G = 15 \), \( T = 2 \), \( C = 50 + 0.6(Y - T) \). Wait, maybe the government spending is 15? Wait, let's re - calculate the constants:

\( 50+15 + 15-1.2=50 + 30-1.2 = 78.8\)? No, that seems off. Wait, maybe the government spending is 15? Wait, let's do the algebra again.

\( Y=50 + 0.6Y-1.2 + 15+15\)

Combine like terms:

The constant terms: \( 50-1.2 + 15+15=50 + 30-1.2 = 78.8\)

The \( Y \) terms: \( Y-0.6Y = 0.4Y\)

So we have \( 0.4Y=78.8\)

Then \( Y=\frac{78.8}{0.4}=197\)

Now calculate \( C \):

\( C = 50+0.6(Y - T)\), substitute \( Y = 197\) and \( T = 2\)

\( C=50 + 0.6(197 - 2)=50+0.6\times195=50 + 117 = 167\)

Answer:

\( Y = 197 \), \( C = 167 \) (corresponding to the option \( Y = 197 \), \( C = 167 \))