Source: Challenger layoff data.
| Number of Episodes | Percent of companies |
|---|---|
| 0 | 77.6% |
| 1 | 10.3% |
| 2 | 4.3% |
| 3 | 2.6% |
| 4 | 1.7% |
| 5 | 1.1% |
| 6 | 0.6% |
| 7 | 0.6% |
| 8 | 0.4% |
| 9 | 0.2% |
| 10+ | 0.6% |
Based on certain model parameters (shown in Appendix section 11), this figure graphs two functions: (1) the model equilibrium and (2) the counterfactual case. Function (1) is labelled “Workers don’t observe z”. Function (2) is labelled “Workers observe z”. The x-axis is labelled “z”, which denotes the firm’s business condition parameter. There are three tick marks on the x-axis: looking from left to right, the first is “z lowerbar”, the second is “z tilde”, and the third is “z upperbar”. The y-axis is labelled “N”, which denotes the firm’s employment choice (number of workers). Both axes start at zero and have no specified end point. Function (2) starts at the origin and, looking from left to right, it increases in a continuous and concave manner until it reaches the point labelled “z tilde” on the x-axis. At point “z tilde”, the function jumps up by a small amount. To the right of “z tilde”, Function (2) continues to increase in a continuous and concave manner. To summarize, Function (2) is concave and monotonically increasing, with a small discontinuity at “z tilde”. Function (1) also starts at the origin. From the origin to the point “z lowerbar”, it overlaps precisely with Function (2). At “z lowerbar”, Function (1) jumps up to a new point on the y-axis. This point is labelled “N_P”, the level of employment with pooling. Between “z lowerbar” and “z upperbar”, Function (1) is flat and lies above Function (2). To the right of “z upperbar”, Function (1) overlaps precisely with Function (2) and is thus increasing and concave for all values of z above “z upperbar”.
This figure is based on Function (1) that was described on Figure 2. The function is labelled “Workers don’t observe z”. As in Figure 2, the x-axis labelled “z”, which denotes the firm’s business condition parameter. The y-axis is labelled “N”, which denotes the firm’s employment choice (number of workers). Both axes start at zero and have no specified end point. Also, as in Figure 2, the x-axis has three tick marks: looking from left to right, the first is “z lowerbar”, the second is “z tilde”, and the third is “z upperbar”. The main difference from Figure 2 is that four points on the function are highlighted. Looking from right to left, Point A corresponds to a value of z that is above all of the tick marks (above “z upperbar”). Point B lies between “z tilde” and “z upperbar”. Point C lies between “z lowerbar” and “z tilde”; Point D lies between the origin and “z lowerbar”. As in Figure 2, the function is increasing and concave from the origin to “z lowerbar”, then flat at the level N=N_P for values of z between “z lowerbar” and “z upperbar”, and then increasing and concave above “z upperbar”. The current graph, Figure 3, also has an arrowhead from point A to point B and from point C to point D. The arrows are meant to indicate the path that a firm would take to adjust employment when faced with negative shocks to z.
This figure is a three-dimensional graph. The x-axis is denoted z_{t-1}, the y-axis is denoted “Ratio”, and the additional axis that represents the depth of the graph is denoted z_t-z_{t-1}, which represents the size of the shock. The range of the axes are approximately: [17,22] for the x-axis, [0,1] for the y-axis, and [-6,0] for the additional axis. The main variable of interest is the “Ratio” that lies along the y-axis, whereas the other variables are shown because they both affect the Ratio values. Looking from left to right along the x-axis, the surface area of the graph is flat for values of z_{t-1} between 17 and 18; this corresponds to a Ratio of zero. As z_{t-1} rises, the Ratio rises in a continuous manner, reaching one at the point z_{t-1}=19 and z_t-z_{t-1}=-1. Actually, lower (more negative) values of the shock size (z_t-z_{t-1}) lower the Ratio slightly. In other words, if the shock size is -4 when z_{t-1}=19, then the Ratio is only around 0.5; it increases until reaching one near the values of z_{t-1}=20 and z_t-z_{t-1}=-4. To sum up, the three-dimensional graph shows that the Ratio starts from zero and increases in a slow and continuous manner as both z_{t-1} and z_t-z_{t-1} increase. The ratio reaches a maximum value of one, which is depicted as a flat surface when z_{t-1} is larger and when the shock (which takes negative values) is closer to zero.
This figure is a three-dimensional graph. The axes denote the same variables as in Figure 4: the x-axis is denoted z_{t-1}, the y-axis is denoted “Ratio”, and the additional axis that represents the depth of the graph is denoted “z_t-z{t-1}”. The range of the axes are approximately: [11,15] for the x-axis, [1,11] for the y-axis, and [-6,-2] for the additional axis. The main variable of interest is the “Ratio” that lies along the y-axis, whereas the other variables are shown because they both affect the Ratio values. Looking from left to right along the x-axis, for values of z_{t-1} less than 12, Ratio=1 and the surface area of the graph is very flat. Once the value of z_{t-1} reaches approximately 12, the graph jumps up, corresponding to a Ratio value around 2. At this point, values of the shock (shown on the additional axis) also matter: when the shock is closer to zero, the Ratio is above ten, whereas for more negative shocks (in the range of [-5,-6]), the Ratio is around 2. Some parts of the graph are not covered by the surface area of the plotted function because the graph focuses only on shocks that are large enough to bring the firm type below “z lowerbar”. To sum up, the surface area of the graph shows a ratio of 1 for all values of z_{t-1} below 12, regardless of the size of the shock. For larger values of z_{t-1}, the Ratio appears to be at least twice as high, sometimes more than ten times as high when combined with a value of “z_t-z{t-1}” close to zero.
Based on certain model parameters (shown in Appendix section 11), this figure graphs two profit functions. Function (1) assumes that the firm chooses its “genuine employment level”, N*, and workers react with low effort, e_L. Function (2) assumes that that the firm chooses the pooling level of employment, N_P, and workers react with high effort, e_H. The x-axis is labelled “z”, which denotes the firm’s business condition parameter. This axis starts at zero and ends at “z tilde”. There is one tick mark that lies roughly in the middle of the x-axis, denoted “z lowerbar”. The y-axis is labelled “Profit”. It has a tick mark at the value “-X(N_P)”, which denotes a strictly negative value. The y-axis begins at an unspecified value below -X(N_P) and ends at an unspecified value considerably greater than zero (there also is a horizontal line through the middle of the graph denoting the Profit=0 level). Function (1) starts at the point (0,0) and is continuous, increasing, and convex. Function (2) starts at the point (0, -X(N_P)) and is continuous, increasing, and linear. The most important aspect of the graph is the comparison between Function (1) and Function (2). Function (1) is always positive, while Function (2) is initially negative for values of z close to zero. Both functions are increasing in z, but Function (2) increases more steeply than Function (1). Although Function (1) lies above Function (2) for z less than “z lowerbar”, the two functions cross at the point denoted “z lowerbar”. For values of z above “z lowerbar”, Function (2) lies above Function (1); the gap between Function (2) and Function (1) gets larger as z increases.
Based on certain model parameters (shown in Appendix section 11), this figure graphs two functions that correspond to the two equilibrium conditions of the model. Function (1) represents the “Beliefs” condition and Function (2) represents the “Zero profit” condition. The x-axis is labelled “z upperbar” and the y-axis is labelled “z lowerbar”. Both axes correspond to values of z (the firm’s business condition parameter), however, the x-axis covers smaller values of z: [0, “z tilde”] and the y-axis covers larger values of z: [“z tilde”, “z hat”]. To give an illustration of the existence proof, three lines highlight the range of possible values for z lowerbar and z upperbar. First, there is a horizontal line that highlights all values of “z lowerbar” along the x-axis, with open circles at the end points; the open circles are at (0, z tilde) and at (z tilde, z tilde). Second, there is a vertical line that highlights all values of “z upperbar” that correspond to z lowerbar=z tilde, with an open circle at (z tilde, z tilde). These first two lines are meant to represent all possible endpoints for the zero profit condition. Finally, there is a third line that highlights possible endpoints of the beliefs condition. This is a vertical line that lies along the y-axis, with an open circle at the origin (i.e. at (0, z tilde)). An additional point, (z tilde, z tilde), corresponds to the other endpoint of the beliefs condition and is also highlighted. Broadly speaking, the three lines outline the graph, forming a U-shape with sharp corners; the final point is in the lower right-hand corner of the graph. Turning back to the functions, Function (2) starts from the middle of the x-axis (approximately) and increases in a linear manner until it reaches the vertical line on the far right. Function (1) instead starts from the middle of the y-axis (approximately) and decreases in a concave manner until it reaches the point in the lower right-hand corner. These two functions cross somewhere near the middle of the graph; the point at which they cross is the solution for [z lowerbar, z upperbar] in the numerical example (and is indicated as such in the graph).