D Problem Set 4

D.1 Another Overlapping Generations model

Consider the overlapping generations model again, with one small twist: agents care only about old age consumption, instead about caring about young and old age consumption equally. In other words, their utility function is given by: \[U=u(c_{t+1}^{o}).\] People still work when young, earn wage \(w_t\), the production function is Cobb-Douglas, etc.

  1. Why can the utility function be left unspecified for computing the level of saving?

  2. Derive the law of motion for the capital stock.

  3. What value of the saving rate \(s\) would imply this law of motion in the Solow model?

  4. Provide a condition on \(\alpha\) such that the capital stock is below the Golden Rule level.

  5. If \(\alpha = 1/3\), where is the capital stock compared to the Golden Rule level? Why is this not optimal? What should be done in that case?

  6. Compare this result from the one in the course, when agents cared both about young and old age consumption. Why is the result from question 5 intuitive economically, even before one writes down any equation?

D.2 The Neoclassical Labor Market Model

Consider the neoclassical labor market model of lecture 6. Assume that preferences and the production function are as in lecture 6: \[U(c, l)=c-B\frac{l^{1+\epsilon}}{1+\epsilon}, \qquad f(l)=Al^{1-\alpha}.\] Denote the wage by \(w\), and the price of consumption by \(p\).

  1. Derive the Labor Demand curve.

  2. Assume that \(\alpha=1/3\) and \(A=2\). Using your favorite spreadsheet software, plot this demand curve in a \((l, w/p)\) plane - that is, putting \(l\) on the x-axis and \(w/p\) on the y-axis.

  3. Take logs of both sides. What does the demand curve look like in a \((\log(l), \log(w/p))\) plane? What is the slope of the demand curve equal to? If \(\alpha\) is higher, is the demand curve steeper or flatter? What shifts the demand curve to the left or to the right?

  4. Derive the Labor Supply curve.

  5. Assume that \(\epsilon=5\) and \(B=2\). Using your favorite spreadsheet software, plot this supply curve in a \((l, w/p)\) plane - that is, putting \(l\) on the x-axis and \(w/p\) on the y-axis. Add the supply curve to the demand curve of question 2.

  6. Take logs of both sides. What does the supply curve look like in a \((\log(l), \log(w/p))\) plane? What is the slope of the supply curve equal to? If \(\epsilon\) is higher, is the supply curve steeper or flatter? What shifts the supply curve to the left, or to the right?

  7. Assume that productivity \(A\) decreases by 5%, to \(A=1.9\). What is the effect on the quantity of employment, and on the real wage? If \(\alpha\) is higher, is that effect larger or smaller? What is the economic intuition?

  8. Assume that leisure becomes relatively more attractive relative to working (think of Facebook, Netflix, etc.), so that \(B\) increases by 10% (the disutility of work increases). What is the effect on the quantity of employment, and on the real wage? If \(\epsilon\) is higher, is that effect larger or smaller? What is the economic intuition for this?

D.3 The “Keynesian” Labor Market Model

Consider the neoclassical labor market model of the previous problem.

  1. Assume that productivity \(A\) decreases by 5%, but that real wages \(w/p\) are rigid. Compute the change in the quantity of employment following a fall in productivity.

  2. Compare the effect with question 7 in the previous problem. Explain.

  3. Assume that leisure becomes relatively more attractive relative to working, so that \(B\) increases by 10%. Compute the change in the quantity of employment following a increase in leisure attractiveness.

  4. Compare the effect with question 8 in the previous problem. Explain.

D.4 The Bathtub model

Consider the bathtub model. Assume a monthly job separation rate equal to \(s=1\)%, and a monthly job finding rate equal to \(f=20\)%. Assume that the labor force is given by \(L=159\) million.

  1. Derive the steady-state unemployment rate. How many people are unemployed in the steady-state? How many people lose their jobs every month? How many people find a job every month?

  2. Assume that the economy starts with an unemployment rate equal to \(u_0=8\)%. Using your favorite spreadsheet software, show the evolution of the unemployment rate over time. How long before the unemployment rate reaches 5%?

  3. If \(s=2\)% instead, which job finding rate \(f\) gives the same steady-state unemployment rate?

  4. Assuming the separation rate and the job finding rate are given from question 3, answer question 2 again.

  5. Explain why an economy with more churning (that is, faster reallocation) - think of the US versus Europe - has a faster recovery in terms of unemployment after a recession. Note: A recession could be coming from a temporary increase in the job separation rate, or a temporary decrease in the job finding rate, which then goes back to its original value.