Question

Consider a closed economy IS-LM model. The goods and the money market equations are respectively given as follows:
$ 𝑌 = 90 + 0.8𝑌_𝑑 − 100𝑖 + 𝐺$
$𝑀_𝑠 = 750 + 0.2𝑌 − 260𝑖$
where, 𝑌 = national income; $𝑌_𝑑$ = disposable income; 𝑇 = total tax given by 𝑇 = 5 + 0.2𝑌; 𝑖 = interest rate; 𝐺 = government expenditure = 300; $ 𝑀_𝑠$ = constant money supply = 950.
The value of 𝑇 at equilibrium 𝑌 is _______. (rounded off to the nearest integer).

Answer 1

Correct Answer: 215

Solving for Equilibrium National Income, Interest Rate, and Taxes

Step 1: Solve the IS Curve

Substituting \( Y_d = Y - T \) and \( T = 5 + 0.2Y \) into the IS equation: 

\[ Y = 90 + 0.8(Y - (5 + 0.2Y)) - 100i + 300 \]

Expanding the equation:

\[ Y = 90 + 0.8(Y - 5 - 0.2Y) - 100i + 300 \]

\[ Y = 90 + 0.8(0.8Y - 5) - 100i + 300 \]

\[ Y = 90 + 0.64Y - 4 - 100i + 300 \]

\[ Y = 386 + 0.64Y - 100i \]

Rearrange to isolate \( Y \):

\[ Y - 0.64Y = 386 - 100i \]

\[ 0.36Y = 386 - 100i \]

\[ Y = \frac{386 - 100i}{0.36} \]

\[ Y = 1072.22 - 277.78i \]

Step 2: Solve the LM Curve

Substituting \( M_s = 950 \) into the LM equation:

\[ 950 = 750 + 0.2Y - 260i \]

Rearrange:

\[ 950 - 750 = 0.2Y - 260i \]

\[ 200 = 0.2Y - 260i \]

Substituting \( Y = 1072.22 - 277.78i \):

\[ 200 = 0.2(1072.22 - 277.78i) - 260i \]

Expanding:

\[ 200 = 214.44 - 55.56i - 260i \]

\[ 200 = 214.44 - 315.56i \]

Solving for \( i \):

\[ 200 - 214.44 = -315.56i \]

\[ -14.44 = -315.56i \]

\[ i = \frac{14.44}{315.56} \approx 0.0457 \]

Step 3: Calculate the Value of \( T \)

Substituting \( i \approx 0.0457 \) into \( Y \):

\[ Y = 1072.22 - 277.78(0.0457) \]

\[ Y = 1072.22 - 12.7 \approx 1059.52 \]

Substituting \( Y = 1059.52 \) into the tax equation:

\[ T = 5 + 0.2(1059.52) \]

\[ T = 5 + 211.9 \approx 215 \]

Final Answer:

The value of \( T \) is 215.