Problem Set #3 Revised: April 2, 2007

Transcription

1 Global Economy Chris Edmond Problem Set #3 Revised: April 2, 2007 Before attempting this problem set, you will probably need to read over the lecture notes on Labor Markets and on Labor Market Dynamics. A. Labor market distortions As background to a consulting assignment, you have been asked to work through an analytical example to illustrate in concrete terms the impact of minimum wage legislation and payroll taxes on employment and unemployment. You decide to adapt the framework of your Global Economy class notes, and work through the demand for labor of a single firm producing widgets according to the production function Y = AK 1/3 L 2/3. For simplicity, you assume that A = K = 1 and that the supply of labor rises with the wage w: L s = w Describe, first, how the labor market might work in this economy if the market is unregulated. What is the wage? Employment? Unemployment? (5 points). 2. Then consider the effect of introducing a minimum wage w m. What are employment and unemployment rates if w m = 1? If w m = 0.5? How does this market differ from the one above? How much would an unemployed person be willing to pay a recruiter to find a job? (10 points). 3. Finally, consider the impact of introducing a 5% payroll tax (i.e a tax on labor levied on employers). Compute employment and unemployment when w m = 0 and w m = 1, respectively. Hint: if w is the wage earned by an employee, the unit labor cost to her employer is w 1.05 (10 points). Answer. The firm s optimal level of labor (its labor demand) is such that the gain of adding one more worker is equal to its cost. The marginal gain is the derivative of sales with respect to labor: 2 3 AK1/3 L 1/3. The marginal cost is simply the wage w. Therefore, since A = K = 1, the demand for labor is ( ) 3 3 L d = 2 w

2 Problem Set # The equilibrium is reached when demand equals supply. That is: ( ) w = w 2 Solving for w, we obtain that the equilibrium wage is The level of employment is L = w = ( ) 3/5 3 = ( ) w = w 2 = 0.61 The unemployment rate is zero. Since there is no distortion, all individuals that would like to work at the equilibrium wage rate get a job. 2. If w m = 1, the demand for labor is L d = ( 3 2 1) 3 = Instead the supply is L s = 1 2 = 1. Therefore the level of employment is The unemployment rate is = 0.70 or 70%. Every unemployed will be willing to pay a recruiter up to the difference between the ongoing wage (= 1) and his/her reservation wage. The wage w m = 0.5 is below the equilibrium rate. This means that the minimum wage is not binding. The outcome in terms of wage and employment is going to be the same as in part (a). 3. Now the marginal cost of hiring one more worker is w(1 + τ), where τ = 0.05 is the payroll tax. The firm s optimal amount of labor is such that Therefore the demand for labor is 2 3 L 1/3 = w(1 + τ) L d = (1.575w) 3 A minimum wage w m = 0 will not be binding. Therefore if w m = 0 the market will be in equilibrium, i.e., the wage equates demand and supply: (1.575w) 3 = w 2 = 0.76 The level of employment will be (0.76) 2 = The unemployment rate will be zero. If the minimum wage is w m = 1, the demand for labor is L d = ( ) 3 = Therefore the level of employment will be 0.26 and the unemployment rate will be = 0.74 or 74%.

3 Problem Set #3 3 w L d (w) L s (w) w m = 1.00 w = L Figure 1: With and without unemployment. w L d (w) without tax L s (w) w m = 1.00 L d (w1.05) with tax L Figure 2: Effect of payroll taxes with and without minimum wage.

4 Problem Set #3 4 B. Labor supply Our standard model of labor supply has people choose how much to work at a given wage rate in a smooth or continuous fashion. While for some purposes this is a useful simplifying assumption, it s not very realistic. Let s see what happens if, more realistically, people simply choose whether to work or not at a given wage w. To be precise, suppose that individuals have a utility function u(c, L) = log(c) θl and have to choose either L = 0 or L = 1. As usual, they have a budget constraint made up of labor income wl and non-labor income T, so C = wl + T. The preference parameter θ > 0 measures how much an individual dislikes working. 1. Calculate an individual s reservation wage, that is, the lowest wage rate w such that they are better off working (L = 1) rather than not working (L = 0). Describe the individual s optimal labor supply curve as a function of the market wage rate. Use this to describe the individual s optimal consumption C (15 points). 2. People are not identical. Suppose there is a distribution of preferences, some people have high θ but others have low θ, and similarly there is a distribution of non-labor income, some people have high T but others have low T. Do individuals with high distaste for work have high or low reservation wages? What about individuals with high non-labor income? Do you think these effects are realistic? Why or why not? (5 points) 3. Explain how you might sum up or aggregate the binary labor supply choices of individuals who are heterogeneous in their preferences or non-labor income. Will this result in a smooth aggregate labor supply curve? Why or why not? (5 points) Answer. 1. Since C = wl + T we can plug this in for consumption in the utility function so that it can be written only in terms of L, say u(l) = log(wl + T ) θl Now suppose that an individual works. She has L = 1 and utility u(work) = u(1) = log(w 1 + T ) θ 1 = log(w + T ) θ But if she doesn t work, she has L = 0 and utility u(not work) = u(0) = log(w 0 + T ) θ 0 = log(t )

5 Problem Set #3 5 An individual will work so long as u(work) > u(not work) and is indifferent to working if exactly u(work) = u(not work) or if log(w + T ) θ = log(t ) Her reservation wage is the wage that makes her just indifferent between working or not. That is, it is the particular value of w which solves this indifference condition. Call that particular value w. A job with any wage w > w will surely be high enough to make her choose to take the job. To solve for w, rearrange the indifference condition to get or log(w + T ) log(t ) = θ ( ) w + T log = θ T Exponentiate both sides to get w + T = e θ T and on solving for w we have the expression w = (e θ 1)T Let L(w) denote an individual s optimal labor supply as a function of the market wage w. Then optimal consumption is C(w) = wl(w) + T. An individual has labor supply L(w) = 0 with associated consumption C(w) = T for all w < w and has labor supply L(w) = 1 with associated consumption C(w) = w + T for all w > w. 2. Since θ > 0, we know e θ 1 > 0 too and so an individual s reservation wage w = (e θ 1)T is increasing in non-labor income T. If you have a trust fund, or more generally lots of family support or other non-labor income, then your reservation wage will be high. A job will have to pay well for it to be worth your time. But if your non-labor income is low (T 0), your reservation wage will be low (w 0 too) and you will be prepared to take even a very low wage job just to be able to put food on the table. This seems pretty realistic to me! Similarly, if your θ is low, you don t mind working and so will have a low reservation wage but if your θ is high you dislike working and your reservation wage will be high. Again, seems reasonable. 3. Each individual has a binary {0, 1} labor supply choice. Let s suppose that there are many people in the economy each with a different level of T (some have a lot, some have a little and everything in between). Then at a low market wage, only those individuals with a relatively low T value who have low reservation wages will find it optimal to work so total ( aggregate ) labor supply will be near zero. On the other hand, if the market wage w is very high then almost everyone will find it optimal to work so the total labor supply will be high

6 Problem Set #3 6 too. In the middle, as w increases from low to high, the labor market will start pulling in all those typical individuals with moderate reservation wages and so the total labor supply will smoothly increase as those individuals start participating. C. Unemployment dynamics You are told the following data for a small European economy. The job finding rate is f = 0.10 per quarter and the separations rate is s = 0.01 per quarter. The labor force L is constant so that the unemployment rate u t = U t /L follows u t+1 = u t (1 u t ) 0.10u t In the first quarter of the year 2007, the unemployment rate is 0.20 or 20%. 1. The finance minister would like to know how far the economy is from its long run unemployment rate. What is your best estimate of this gap? (5 points). 2. The prime minister estimates that the government cannot win the next election if the unemployment rate is above 10%. The next election is three years (12 quarters away). If the prime minister is right, will the government win or not? (10 points). 3. The labor minister would like to implement a package of labor market reforms that make it easier to fire workers. You estimate that this will directly increase the separation rate to s = 0.02 but, by making firms more willing to hire workers, will increase the job finding rate to f = What will these policy reforms do to long run unemployment? Will they make labor markets more or less fluid? Would you advise the labor minister to push for these reforms or not? Would they help the government win the election? Why or why not? (10 points). Answer. 1. The long run or steady state unemployment rate u solves where s = 0.01 and f = Therefore u = 0 = s(1 u) fu s s + f = = 0.09 Since the current unemployment rate is u 0 = 0.20 or 20%, the gap is = 0.11 or 11%.

7 Problem Set # The unemployment rate returns to steady state according to the formula (u t u) = λ t (u 0 u) where λ = 1 (s + f) = = Therefore after 12 quarters the unemployment rate is u t = = = 0.12 So after 12 quarters, the unemployment rate will still be above 10%. If the prime minister is right, the government will lose the election. 3. If the separation rate rises to s = 0.02 and the job finding rate rises to f = 0.12, the long run unemployment rate will shift to u = s/(s + f) = 0.02/0.14 = 0.14 or 14%. The long run unemployment rate will increase because although the accession rate increases by 20% the separation rate doubles (increases by 100%) so that the net effect is a worsening of the long run unemployment situation. The new policies will make the labor market more fluid in the sense that λ will fall from 0.89 to λ = 1 (s + f) = I would not advise the labor minister to push for these reforms, they wouldn t help the government win the election. Why? Because the long run unemployment rate will be above the 10% threshold that the prime minister thinks is the most the government can get away with. D. Labor-saving productivity The economy produces two goods, manufactures and services. Manufactures are produced with the production function Y M = A M L M while services are produced with Y S = A S L S where A M and A S are given productivity coefficients. People like to consume manufactures and services in equal proportions so C M = C S Demand equals supply when C M = Y M and C S = Y S. The total labor force L = 100 is constant. Workers can be either employed in manufactures L M or employed in services L S, so L M + L S = Suppose that productivity in manufactures and services is equal at A M = A S = 1. Calculate employment in each sector of the economy as well as the amounts consumed of each good (5 points). 2. Now let s see what happens if productivity in both sectors doubles so that A M = A S = 2 (this is called unbiased technological change). Explain how this affects employment in each sector and the amounts consumed of each good (5 points).

8 Problem Set # Now consider the case of biased technological change: productivity in manufactures doubles but productivity in services is unchanged, A M = 2 but A S = 1. Calculate employment in each sector and the consumption of each good. Is this biased labor-saving technological change a good thing? Is this similar to opening the economy up to foreign competition. Why or why not? (15 points). Answer. 1. Since C M = C S and demand equals supply in both sectors means A M L M = A S L S or and since L M = A S A M L S L M + L S = 100 we can solve these last two equations in the two unknowns L M, L S to get L M = L S = A S 100 A M + A S A M 100 A M + A S In this question, A M = A S = 1, so employment is L M = (1/(1 + 1)) 100 = 50 and L S = (1/(1 + 1)) 100 = 50. Consumption of each good is then C M = A M L M = 1 50 = 50 and C S = A S L S = 1 50 = If productivity in each sector doubles to A M = A S = 2, the employment is L M = (2/(2 + 2)) 100 = 50 and L S = (2/(2+2)) 100 = 50. Since productivity in each sector increased at the same rate, the employment shares in each sector are unchanged. But since the economy is more productive, consumption of each good increases. Specifically, consumption is C M = A M L M = 2 50 = 100 and C S = A S L S = 2 50 = 100. A doubling of productivity in this unbiased fashion simply doubles output and consumption of each good without leading to labor reallocation across sectors of the economy. 3. But if we only had productivity doubling in manufactures so that A M = 2 but A S = 1 then we would have employment of L M = (1/(2+1)) and L S = (2/(2+1)) So an increase in manufacturing productivity reduces manufacturing employment and increases services employment. Labor is reallocated from manufactures to services. Nonetheless, consumption of each good increases. To see this, consumption is C M = A M L M 2 33 = 66

9 Problem Set #3 9 and C S = A S L S 1 67 = 67. Therefore, consumption of manufactures increases from 50 to 66 even as manufacturing employment falls from 50 to 33. That extra labor is used in the services sector so employment increases from 50 to 67 and with it consumption of services rises from 50 to 67 even though services productivity is unchanged. The increase in one sector s productivity benefits the whole economy by freeing up scarce labor. This is essentially identical to freeing up the economy to increased foreign competition through international trade. c 2007 NYU Stern School of Business