Question:
Consider a closed economy with consumption function $ C = 2 + 0.5π$, where π is income. The government expenditure is 3 and investment function is $ I = 4 β 0.5r $ , where r is interest rate. Then, the slope of the IS curve will be
Consider a closed economy with consumption function $ C = 2 + 0.5π$, where π is income. The government expenditure is 3 and investment function is $ I = 4 β 0.5r $ , where r is interest rate. Then, the slope of the IS curve will be
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Updated On: Feb 10, 2025
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Solution and Explanation
The IS Curve and Its Slope
The IS curve represents the combinations of income (Y) and the interest rate (i) where the goods market is in equilibrium, meaning that aggregate demand equals aggregate supply. The equation for the IS curve is derived from the equilibrium condition in the goods market:
Y = C(Y β T) + I(i) + G
Where:
- C(Y β T) is the consumption function, which depends on disposable income (Y β T).
- I(i) is the investment function, which depends inversely on the interest rate (i).
- G is government spending.
Step 1: Understanding the Relationship Between Income (Y) and Interest Rate (i)
- Investment Sensitivity to Interest Rate: The investment function I(i) is negatively related to the interest rate i. When the interest rate increases, the cost of borrowing increases, leading to a decrease in investment. Conversely, when the interest rate decreases, investment rises.
- Output (Y) Dependence on Investment: Investment is a component of aggregate demand. Thus, any change in investment caused by changes in the interest rate will affect the level of output (Y).
This negative relationship between the interest rate and output forms the basis for the downward-sloping IS curve.
Step 2: Determining the Slope of the IS Curve
The slope of the IS curve depends on two key factors:
- Investment Sensitivity to the Interest Rate (Iβ²(i)):
- If investment is highly sensitive to changes in the interest rate, a small change in the interest rate will result in a large change in investment, leading to a flatter IS curve.
- Conversely, if investment is less sensitive to changes in the interest rate, the IS curve will be steeper.
- Multiplier Effect (1/(1 β MPC)):
- The marginal propensity to consume (MPC) affects the slope of the IS curve through the multiplier effect.
- A higher MPC increases the multiplier, amplifying the changes in output (Y) caused by changes in investment.
Mathematically, the slope of the IS curve can be expressed as:
Slope of IS = - (1 / (1 - MPC)) Γ (βI / βi)
- The negative sign reflects the inverse relationship between the interest rate and income (i β β Y β, or i β β Y β).
- The term βI / βi represents the sensitivity of investment to the interest rate.
Step 3: Conclusion
The IS curve is downward sloping because:
- Higher interest rates reduce investment, leading to lower aggregate demand and income.
- Lower interest rates increase investment, leading to higher aggregate demand and income.
The exact steepness (slope) of the IS curve depends on:
- How sensitive investment is to changes in the interest rate (Iβ²(i)).
- The size of the multiplier effect (1 / (1 - MPC)).
In most cases, the slope of the IS curve is negative, reflecting the inverse relationship between income and the interest rate, influenced by investment behavior and consumption dynamics.
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