Chapter 7 Model Apps
The Labor Market, Wages, and Unemployment
Explore the models of Chapter 7 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 7: The Labor Market, Wages, and Unemployment
See also: Chapter 7 Data Apps, the chapter’s data without code.
Supply and demand in the labor market
Fig. 3: The labor market
Move the slider to shift labor demand up or down; the dashed gray lines are the starting curves, and the title reads the new equilibrium.
\[ w = \bar{a}_d - \bar{b}_d L, \qquad w = \bar{a}_s + \bar{b}_s L \]
\[ L^* = \frac{\bar{a}_d - \bar{a}_s}{\bar{b}_d + \bar{b}_s} \]
\[ w^* = \frac{\bar{b}_s\bar{a}_d + \bar{b}_d\bar{a}_s}{\bar{b}_d + \bar{b}_s} \]
where \(L\) is employment, \(w\) the wage, \(\bar{a}_d\) the demand intercept, \(\bar{a}_s = 10\) the supply intercept and \(\bar{b}_d = \bar{b}_s = 1\) the slopes.
Takeaway. A higher demand intercept means firms value every worker more, so the demand curve shifts up along the supply curve and the wage and employment both rise; a lower one lowers both, and at 100 nothing changes. A shift in demand always moves the wage and employment in the same direction, which is how the data reveal a demand shock.
Source. Adapted from Fig. 7.3 in Jones (2024), Macroeconomics, 6th ed.
Fig. 4: An income tax at rate τ
Move the slider to change the tax rate; watch the economy move from A to B and the green bar, the tax wedge, grow.
\[ (1-\tau)\,w = \bar{a}_s + \bar{b}_s L \]
\[ L_\tau = \frac{(1-\tau)\bar{a}_d - \bar{a}_s}{(1-\tau)\bar{b}_d + \bar{b}_s} \]
\[ w_\tau = \bar{a}_d - \bar{b}_d L_\tau \]
where \(\tau\) is the tax rate, \(w\) the wage firms pay and \((1-\tau)w\) the wage workers keep.
Takeaway. Workers decide how much to work from the wage they keep, so the tax shifts labor supply up and to the left, and the economy moves from A to B: employment falls, the wage firms pay rises and the wage workers keep falls. The gap between the two wages, the tax wedge, widens as the rate rises; at a zero rate there is no wedge and nothing changes.
Source. Adapted from Fig. 7.4 in Jones (2024), Macroeconomics, 6th ed.
Fig. 5: A reduction in labor demand
Move the slider to change the fall in labor demand; the dashed line is the old demand curve, and the title reads the new equilibrium.
\[ w = (\bar{a}_d - \Delta) - \bar{b}_d L \]
\[ \Delta L^* = -\frac{\Delta}{\bar{b}_d + \bar{b}_s} \]
\[ \Delta w^* = -\frac{\bar{b}_s\,\Delta}{\bar{b}_d + \bar{b}_s} \]
where \(\Delta\) is the fall in labor demand and \(\bar{b}_d = \bar{b}_s = 1\) the slopes of the two curves.
Takeaway. When firms want fewer workers at every wage, the demand curve shifts down along the supply curve and the economy moves from A to B: the wage and employment both fall. With equal slopes the shock is split evenly between the two, a larger fall in demand lowers both by more, and with no fall nothing changes.
Source. Adapted from Fig. 7.5 in Jones (2024), Macroeconomics, 6th ed.
Fig. 6: A reduction in labor demand with wage rigidity
Move the slider to change the fall in labor demand; the green bar shows who wants a job at the old wage but cannot find one.
\[ L'' = \frac{\bar{a}_d - \Delta - \bar{w}}{\bar{b}_d} = L^* - \frac{\Delta}{\bar{b}_d} \]
\[ L^* - L'' = \frac{\Delta}{\bar{b}_d} \]
where \(\bar{w} = w^*\) is the old wage, which does not fall, and \(\Delta\) the fall in labor demand.
Takeaway. With the wage stuck at its old level, firms move down the new demand curve to C, so the whole shock falls on employment: with these slopes it drops twice as much as with a flexible wage. At the old wage just as many people still want to work, so the gap between them and the jobs on offer is unemployment that a flexible wage would not create; with no fall in demand there is none.
Source. Adapted from Fig. 7.6 in Jones (2024), Macroeconomics, 6th ed.
Fig. 6, the bathtub model: unemployment settles at its natural rate
Move the slider to change the job finding rate; watch how fast unemployment falls and where it settles, at the dashed line.
\[ \Delta U_{t+1} = \bar{s}E_t - \bar{f}U_t \]
\[ u^* = \frac{\bar{s}}{\bar{f} + \bar{s}} \]
\[ u_t = u^* + (1 - \bar{s} - \bar{f})^t\,(u_0 - u^*) \]
where \(U\) and \(E\) are the unemployed and the employed, \(\bar{s} = 0.01\) the monthly job separation rate, \(\bar{f}\) the job finding rate and \(u_0\) = 10 percent the starting rate.
Takeaway. Unemployment stops changing when the jobs lost each month equal the jobs found, at the natural rate. A higher job finding rate widens the drain: unemployment falls faster from its starting point and settles at a lower rate, while a lower one leaves it falling slowly toward a higher rate.
Source. Adapted from the bathtub model of unemployment (Section 7.4) in Jones (2024), Macroeconomics, 6th ed. Our own simulation and interactive chart, with our own starting point, layout and code; the textbook draws no figure for it.
The rising return to education
Fig. 8, present value: what 45 years of wages are worth today
Hover over a bar to read what one year’s 100,000-dollar wage, paid that many years from now, is worth today.
\[ pdv = \frac{\text{future value}}{(1 + R)^T} \]
\[ pdv = \text{payment}\times\frac{1 - \left(\frac{1}{1+R}\right)^n}{1 - \frac{1}{1+R}} \]
where \(R\) = 3 percent is the interest rate, \(T\) the number of years until a payment arrives and \(n\) = 45 the number of yearly payments.
Takeaway. A dollar paid in the future is worth less than a dollar today, because a smaller sum saved today would grow into it with interest, so each year’s wage is divided by \((1+R)^T\) and the bars shrink with distance. Added up, 45 years of pay are worth only about 25 years’ pay today: the value of a worker’s human capital.
Source. Our own calculation with equation (10) of the chapter and illustrative numbers (a wage of 100,000 dollars, \(R\) = 3 percent); the chart uses no data. Inspired by the present-value discussion in Section 7.6 of Jones (2024), Macroeconomics, 6th ed. To compare with real earnings by education, see the U.S. Bureau of Labor Statistics, Education pays.
Fig. 9: Understanding the rising return to education
Move the slider to shift college demand out, with supply shifted out by 10; watch the college wage and the premium in the title.
\[ \Delta L^* = \frac{x_d + x_s}{\bar{b}_d + \bar{b}_s} \]
\[ \Delta w^* = \frac{\bar{b}_s\,x_d - \bar{b}_d\,x_s}{\bar{b}_d + \bar{b}_s} \]
where \(x_d\) and \(x_s = 10\) are the outward shifts in college demand and supply, and \(\bar{b}_d = 0.4\) and \(\bar{b}_s = 0.6\) the slopes of the college curves.
Takeaway. Both shifts raise college employment, but they pull the college wage in opposite directions: it rises only if demand shifts out by more than two thirds of the supply shift, the ratio of the slopes. With a small demand shift the extra supply wins and the premium falls; with a larger one demand wins, and more graduates go together with a higher premium, as in the data since 1980.
Source. Adapted from Fig. 7.9 in Jones (2024), Macroeconomics, 6th ed.