Chapter 11 Model Apps

The IS Curve

Explore the models of Chapter 11 without any code. Every chart below is live: move its slider to change one part of the model and watch the curves respond, and hover over a line to read its values.

← Back to Chapter 11: The IS Curve

See also: Chapter 11 Data Apps, the chapter’s data without code.

Deriving the IS curve

Fig. 1: Introducing the IS curve

Move the slider to change the real interest rate; the dot slides along the IS curve, and the title reads investment and output.

\[ \frac{I_t}{\bar{Y}_t} = \bar{a}_i - \bar{b}\,(R_t - \bar{r}) \]

\[ \tilde{Y}_t = \bar{a} - \bar{b}\,(R_t - \bar{r}) \]

where \(R_t\) is the real interest rate, \(\bar{r} = 5\) percent the marginal product of capital, \(\bar{a}_i\) the normal investment share, \(\bar{a} = 0\) and \(\bar{b} = 2\).

Takeaway. A higher real interest rate makes borrowing dearer, so firms invest less and output falls: each point on the rate lowers investment, and output, by \(\bar{b}\) points of potential output. At \(R_t = \bar{r}\) investment is at its normal share and output at potential; above it output is below potential, and below it output is above.

Source. Adapted from Fig. 11.1 in Jones (2024), Macroeconomics, 6th ed.

Fig. 2: The IS curve

Move the slider to change the sensitivity of investment; the dashed line is the baseline curve, and watch the new one turn around the dot.

\[ \tilde{Y}_t = \bar{a} - \bar{b}\,(R_t - \bar{r}) \]

\[ R_t = \bar{r} + \frac{\bar{a} - \tilde{Y}_t}{\bar{b}} \]

where \(\bar{a} = 0\) is the aggregate demand shock, \(\bar{r} = 5\) percent the marginal product of capital and \(\bar{b}\) the sensitivity of investment.

Takeaway. The sensitivity \(\bar{b}\) measures how strongly investment, and so output, reacts to the gap between the real interest rate and the marginal product of capital. A higher value turns the curve around the dot into a flatter line, so the same change in the rate moves output more; a lower value makes it steeper, and at 2 the curve is the baseline. At the dot, where \(R_t = \bar{r}\), the interest-rate term vanishes, so output stays at \(\bar{a}\) whatever the slope.

Source. Adapted from Fig. 11.2 in Jones (2024), Macroeconomics, 6th ed.

Using the IS curve

Fig. 3: An increase in the real interest rate

Move the slider to set the new interest rate; the economy moves along the curve from A to B, and the title reads output.

\[ \tilde{Y}_t = \bar{a} - \bar{b}\,(R_t - \bar{r}) \]

where \(\bar{a} = 0\), \(\bar{b} = 2\) and \(\bar{r} = 5\) percent, and the real interest rate rises from \(R = \bar{r}\) at A to \(R'\) at B.

Takeaway. Only the interest rate changes, so the IS curve stays where it is and the economy slides up along it, from A to B. Above 5 percent, borrowing costs more than new capital earns, so investment falls and each point on the rate lowers output by \(\bar{b} = 2\) points; at 5 percent the economy stays at A.

Source. Adapted from Fig. 11.3 in Jones (2024), Macroeconomics, 6th ed.

Fig. 4: An aggregate demand shock

Move the slider to set the new demand shock; the dashed line is the old curve, and watch output move from A to B.

\[ \tilde{Y}_t = \bar{a} - \bar{b}\,(R_t - \bar{r}) \]

\[ \bar{a} \equiv \bar{a}_c + \bar{a}_i + \bar{a}_g + \bar{a}_{ex} - \bar{a}_{im} - 1 \]

where \(\bar{a}\) collects the five demand shares and moves to the new shock \(\bar{a}'\), while the interest rate stays at \(R = \bar{r} = 5\) percent.

Takeaway. A demand shock changes spending at every interest rate, so it shifts the whole IS curve: a positive shock shifts it right, a negative one shifts it left, and at zero nothing moves. With the interest rate still at \(\bar{r}\), the interest-rate term is zero, so output moves one for one with the shock, from A to B.

Source. Adapted from Fig. 11.4 in Jones (2024), Macroeconomics, 6th ed.

Microfoundations: consumption and fiscal policy

Fig. 5: The life-cycle model of consumption

Move the slider to change the length of life; the dashed line is consumption, and the title reads its level and the saving years.

\[ \sum_{t=0}^{T-1}\frac{y^p}{(1+R)^t} = \sum_{t=0}^{T-1}\frac{y_t}{(1+R)^t} \]

\[ c_t = y^p = \frac{1}{T}\sum_{t=0}^{T-1} y_t \]

where \(y_t\) is income at age \(t\), \(T\) the length of life, \(y^p\) permanent income and \(c_t\) consumption, with no interest (\(R = 0\)).

Takeaway. Consumption is set at average lifetime income, so it stays flat while income rises and falls: people borrow or get help when young, save in middle age and live off their savings in retirement. A longer life adds retirement years with low income, so average income, and with it consumption, falls and people must save more in middle age; a shorter life does the reverse.

Source. Adapted from Fig. 11.5 in Jones (2024), Macroeconomics, 6th ed.

Worked exercise: three shocks to the IS curve

Fig. 7: A reduction in the real interest rate

Move the slider to set the new, lower interest rate; the economy moves down the curve from A to B, and the title reads output.

\[ \tilde{Y}_t = \bar{a} - \bar{b}\,(R_t - \bar{r}) \]

where \(\bar{a} = 0\), \(\bar{b} = 2\) and \(\bar{r} = 5\) percent, and the real interest rate falls from \(R = \bar{r}\) at A to \(R'\) at B.

Takeaway. A cut in the real interest rate is again a movement along the IS curve, this time down it. Below 5 percent, new capital earns more than it costs to borrow, so investment rises and each point of the cut raises output by \(\bar{b} = 2\) points; at 5 percent the economy stays at A. This is how monetary policy fights a recession.

Source. Adapted from Fig. 11.7 in Jones (2024) (the textbook’s answer to Exercise 2(a)), Macroeconomics, 6th ed.

Fig. 8: Pessimistic consumers

Move the slider to set the new demand shock; the dashed line is the old IS curve, and the title reads output at B.

\[ \tilde{Y}_t = \bar{a} - \bar{b}\,(R_t - \bar{r}) \]

\[ \bar{a} \equiv \bar{a}_c + \bar{a}_i + \bar{a}_g + \bar{a}_{ex} - \bar{a}_{im} - 1 \]

where pessimistic consumers lower the consumption share \(\bar{a}_c\), and so \(\bar{a}\), to the new shock \(\bar{a}'\), while the interest rate stays at \(R = \bar{r}\).

Takeaway. Pessimistic consumers spend a smaller share of potential output, which lowers the demand shock and shifts the IS curve left; at zero nothing moves. With the interest rate unchanged, output falls one for one with the shock: a recession caused by weaker demand, not by a fall in what the economy can produce.

Source. Adapted from Fig. 11.8 in Jones (2024) (the textbook’s answer to Exercise 2(b)), Macroeconomics, 6th ed.

Fig. 9: Improvements in information technology

Move the slider to set the new marginal product of capital; the dotted line marks its height, and watch output move from A to B.

\[ \tilde{Y}' = \bar{a} - \bar{b}\,(R - \bar{r}') \]

\[ \tilde{Y}' = \bar{b}\,(\bar{r}' - \bar{r}) \]

where \(\bar{r} = 5\) percent is the old marginal product of capital and \(\bar{r}'\) the new one, with \(\bar{a} = 0\) and \(R = \bar{r}\) held fixed.

Takeaway. Better technology makes new capital more productive, so the IS curve shifts up and now reaches potential output at the new, higher \(\bar{r}'\). The real interest rate has not changed, so new capital earns more than it costs to borrow, investment rises and output goes up by \(\bar{b} = 2\) points for each point of \(\bar{r}'\) above \(\bar{r}\); at 5 percent nothing changes.

Source. Adapted from Fig. 11.9 in Jones (2024) (the textbook’s answer to Exercise 2(c)), Macroeconomics, 6th ed.