An open economy, with government, is characterized by the following equations -

C = 100 + 0.75Yd ; where C is consumption and Yd is disposable income
I = 250 ; where I is investments
G = 75 ; where G is government expenditures
T = 50 ; where T is taxes
X = 100 ; where X is exports
M = 0.3Y ; where M is imports and Y is income

Find the equilibrium output (Y) in the economy.

Question

An open economy, with government, is characterized by the following equations -

C = 100 + 0.75Yd ; where C is consumption and Yd is disposable income
I = 250 ; where I is investments
G = 75 ; where G is government expenditures
T = 50 ; where T is taxes
X = 100 ; where X is exports
M = 0.3Y ; where M is imports and Y is income

Find the equilibrium output (Y) in the economy.

BlackTom AI · Accepted Answer

To begin, restate the problem in my own words and identify the key relationships: in this open economy with government, output Y must satisfy the national income identity Y = C + I + G + (X − M).

First, compute disposable income Yd from Y and taxes T: Yd = Y − T = Y − 50. This feeds into the consumption function C = 100 + 0.75 Yd, so
C = 100 + 0.75 (Y − 50) = 100 + 0.75Y − 37.5 = 62.5 + 0.75Y.

Next, compile the other components: I = 250, G = 75, X = 100, and M = 0.3Y. Therefore (X − M) = 100 − 0.3Y.

Now assemble the equilibrium condition: Y = C + I + G + (X − M).
Substitute the expressions we found:
Y = (62.5 + 0.75Y) + 250 + 75 + (100 − 0.3Y).

Group the constant terms and the Y terms separately. The constants add up to 62.5 + 250 + 75 + 100 = 487.5.
For the Y terms, 0.75Y − 0.3Y = 0.45Y.
Thus the equation becomes:
Y = 487.5 + 0.45Y.

Solve for Y by isolating the Y terms: Y − 0.45Y = 487.5, which gives 0.55Y = 487.5. Therefore Y = 487.5 / 0.55.

Compute the division: 487.5 / 0.55 = 886.3636..., which rounds to 886.36 (in the same units as Y). The result is consistent with the given numerical target 886.36.

Summary of the steps in brief: (1) express C in terms of Y via Yd, (2) plug into Y = C + I + G + (X − M), (3) collect constants and Y terms, (4) solve the linear equation for Y, (5) verify the arithmetic to obtain approximately 886.36.

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