Chapter 3. Numerical Problems. A % increase in A % % %

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1 Chapter 3 umerical Problems 1 =AK.3.7 In order to find the growth of total factor productivity, we start by calculating the value of A in the production function. A = / K.3.7. We then calculate the growth rate of A as: Growth rate = (A t+1 - A t )/ A t x 100 %. We get the following values: A % increase in A % % % We calculate the marginal product of labor (MP) by seeing what are the changes in the level of output () when you add one unit of labor (increase by 1): 1 = AK.3.7 ex: 1 = A(1879) = = AK. 3(+1). 7 ex: 2 = A(1879) = 2286 MP=( 2-1)/[(+1)-] =( 2-1) ex: MP = =24 The exact method to compute the MP is to take the derivative of output with respect to ; d / d = 0.7A(K/ ) =.2(K+.5 ) =100; =.2K MP Slope = K

2 For each additional unit of capital, output increases by 0.2 units. The MPK is 0. 2 The slope of the production function line is 0.2. The MPK is constant regardless of the level of K. So there is no diminishing marginal productivity of capital. The production function is a straight line, that is it has a constant slope. K=100; = K When is 100, output is - 0.2( )= 22. When is 110, is So the MP for raising from 100 to 110 is ( )/ 10 = When is 120, is So the MP for raising from 110 to 120 is ( ) / 10 = There is diminishing marginal productivity of labor because the MP decreases as increases. In the graph this is shown as a decline in the slope of the production function as increases. 3. MP = ( 2-1)/[(+1)-] = ( 2-1) MRP = ( P 2 -P 1)/[(+1)-] = ( P 2 - P 1)= P( 2-1) = P MP MP MRP MRP (P=5) (P= 10) P =$5.

3 W=$38. The firm hires one worker, since MRP ($40) is greater than W ($38) at = 1. It does not hire the second worker, since MRP ($35) is less than W ($38) at = 2. Or in real terms: w=w/p=$38/$5=7.6. The firm hires one worker, since MP (8) is greater than W (7.6) at = 1. It does not hire the second worker, since MP (15) is less than W (7.6) at = 2. W = $27. The firm hires the second worker, since MRP ($35) is greater than W ($27) at = 2. It hires the third worker, since MRP ($30) is higher than W ($27) at = 3. It does not hire the fourth worker, since MRP ($25) is less than W ($27) at = 4. (3) W = $22. The firm hires four workers, since MRP ($25) is greater than W ($22) at = 4. It does not hire five workers, since MRP ($20) is less than W ($22) at = 5. c. Graph (1) shows the relationship between labor demand and the nominal wage. This graph is different from a labor demand curve because a labor demand curve shows the relationship between labor demand and the real wage. Graph (2) shows the labor demand curve. W w d. P = $10. W= $38. the firm hires five workers. MRP ($40) is greater than W ($38) at = 5. The firm does not hire six workers, since MRP ($30) is less than W ($38) at = 6. With five workers, output is 30 widgets, compared to 8 widgets in part a) when the firm hired only one worker. So the increase in the price of the product increases the firm's labor demand and output. e. If output doubles, MP doubles, and MRP doubles. The MRP is the same as it was in part d) when the price doubled. So the demand for labor is the same as it was in part d) (ote that the labor demand curve is not the same as in part d). But the output produced by five workers now doubles to 60 widgets. f. Since MRP = P x MP, then a doubling of either P or MP leads to a doubling of MRP. Since labor demand (or the amount of labor that the firm wants) is chosen by setting MRP equal to W, the choice is the same, whether P doubles or MP doubles. 4. MP = A(100 - ) A = 1. MP = The demand for labor is found by setting MP=w=W/P or MRP(=P.MP)=W. That is A(100 - )=w which implies =100-w/A.

4 Or P.A(100 - )=W which implies =100-W/(P.A) W = $10. w = W / P = $10 / $2 =5. Setting w = MP, 5 = 100 -, so =95. W = $20. w = W / P= $20 / $2 =10. Setting w = MP, 10= 100 -, so =90. These two points are plotted as line D a. The labor demand is described by 100 d =w which is the same as 100 w = d The equilibrium in the labor market is found by equating the labor demand with the labor supply. That is, in equilibrium the amount of labor demanded ( d ) is equal to the amount of labor supplied ( s ). In equilibrium d = s If labor supply = 95, then 100 w=95 then the equilibrium real wage is 5. A = 2. MP = 2(100 - ). (1) W = $10. w =W / P = $10 / $2 = 5. Setting w = MP, 5 = 2(100 - ), so 2 = 195, so =97.5. W = $20. w=w / P = $20 / $2 = 10. Setting w = MP, 10 = 2(100 - ), so 2=190, so = 95. These two points are plotted as line D b W D a D b If labor supply= 95, then the equilibrium real wage is If the lump-sum tax is increased, there's an income effect on labor supply, not a substitution effect (since the real wage isn't changed). An increase in the lump-sum tax reduces a worker's wealth, so labor supply increases. If T = 35, then S = w + (2 x 35) = w. Labor demand is given by w = MP = 309-2, or 2 = w, so = w/2. Setting labor supply equal to labor demand gives w/2 = w, so 62.5 = 12.5w, thus w = 62.5/12.5 = 5. With w = 5, = 92 + (12 x 5) = 152. c. Since the equilibrium real wage is below the minimum wage, the minimum wage is binding. With w = 7, = /2 = ote that S = 92 + (12 x 7) = 176, so S > and there is unemployment.

5 6. Since w = 4.5 K , -0.5 = 4.5 K 0.5 /w, so = K/w 2. When K = 25, = /w 2. If t = 0.0, then S = 100w 2. Setting labor demand equal to labor supply gives /w 2 = 100w 2, so w 4 = , or w = 1.5. Then S = 100 (1.5) 2 = 225. [Check: = /1.5 2 = 225.] = = 45(225) 0.5 = 675. The total after-tax wage income of workers is (1-t) w S = 1.5 x 225 = If t = 0.6, then S = 100 [(1-0.6) w] 2 = 16w 2. The marginal product of labor is MP = 22.5 / 0.5, so = 100 [(1-0.6) x 22.5 / 0.5 ] 2, so 2 = 8100, so = 90. Then = = 45(90) 0.5 = Then w = 22.5 / = The total after-tax wage income of workers is (1-t) w S = 0.4 x 2.37 x 90 = ote that there's a big decline in output and income, although the wage is higher. c. A minimum wage of 2 is binding if the tax rate is zero. Then = /2 2 = 126.6, S = 100 x 2 2 = 400. Unemployment is Income of workers is w = 2 x = 253.2, which is lower than without a minimum wage, because employment has declined so much. 8. umber who become unemployed: From not in the labor force: 2% of 68.8 million = million From employed: 1% of million = million Total = 2.71 million umber who become employed: From unemployed: 22% of 5.9 million = million From not in the labor force: 3 % of 68.8 million = million Total = million umber who become not in the labor force: From employed: 2% of million = million From unemployed: 13 % of 5.9 million =.767 million Total = million 9. Since (bar - ) / bar = 2.5(u ubar) bar - = 2.5(u - ubar) bar = [1-2.5(u-ubar)]bar, bar = /[1-2.5(u - ubar)]. Using the formula above, this table shows the value of, given values for u, ubar and. ear u bar The first calculation of bar / bar comes from calculating the percent change in bar from part The second calculation of bar / bar comes from using Eq. (3.6): / = bar / bar Au. ear bar bar / bar / u bar / bar

6 The two methods give fairly close answers. 10. a- Total hours worked per week = 1900 workers x 40 hours per worker = 76,000 hours per week. Total output per week = 76,000 total hours per week x 10 units of output per hour = 760,000 units of output. The unemployment rate is 100 unemployed / 2000 labor supply = 0.05, or 5 %. b- Employment falls 4 % from 1900 to: (1-0.04) x 1900 = The labor force falls 0.2 % from 2000 to: ( ) x 2000 = With a labor force of 1996 and employment of 1824, unemployment is = 172. The unemployment rate is 172 / 1996 = 0.086, or 8.6%. Hours worked per employed worker falls 2.5 % from 40 to: ( ) x 40 = 39. Total hours per week = 39 hours per worker x 1824 workers = 71,136. So total hours per week falls by (76,000-71,136) / 76,000 = =6.4%. Total output per week falls 1.4% for every 1 % drop in hours, so output falls by 6.4 x 1.4%=8.96%. Since output was 760,000, it now falls to 760,000 x ( ) = 691,904. The Okun's Law coefficient is the percent change in output divided by the increase in the unemployment rate = / ( ) = 2.49.

7 Analytical Problems K In the initial situation, capital K 1, and labor 1, produce output 1 ; when productivity rises they produce output Suppose that a small increase in capital to K 2 =K 1 +1 with labor left at, produces output 2 in the initial situation. Then it produces when productivity rises by 10 %. The marginal product of capital (MPK) in the initial situation is ( 2-1 ) / (K 2 - K 1 ) When productivity rises the new MPK is ( ) / (K 2 - K 1 ) = 1. 1 ( 2-1 ) / (K 2 - K 1 ). So the new MPK is 10 % higher than the old MPK. This argument holds for MP as well. If you substitute for K everywhere and follow the same steps, you will show that the new MP is 10 % higher than the old MP. c. es, it is possible for a beneficial productivity shock to leave the MPK and MP unchanged. This can happen only if the shock is additive, that is, if it shifted the whole production function upward, but did not affect its slope at any point. In graphs 3 and 4 this is shown as a shift up in the production function, leaving the slope unchanged K

8 3. As shown in the graph, when the real wage (w) is above its market-clearing level, labor supply ( s ) exceeds labor demand ( d ). The difference is the amount of unemployment U W w D S W bar D bar S With the real wage higher than the wage that clears the market at full employment, labor demand must be lower than it is at full employment, so employment is also lower. Therefore output is lower because of the real wage rigidity. 6. es, it is possible for the unemployment rate and the employment ratio to rise during the same month. For example, suppose the population falls, the labor force is constant, the number of unemployed rises, and the number of employed falls (but by less than the decline in population). Then the unemployment rate rises, since there are more unemployed but the same labor force, but the employment ratio rises, since population declines more than employment does. es, it is possible for the participation rate to fall at the same time that the employment ratio is rising. For example, suppose that population is constant, the labor force declines, employment rises, and unemployment falls. The participation rate falls, since there are fewer people in the labor force from the same population. The employment ratio is rising, since employment rises while population is constant. 7. Since Sally earns $150,000 per year, she is far above the cap, so the Social Security tax doesn't affect her after-tax wage (so there's no substitution effect) the higher tax only affects her income and thus has only an income effect. Since both proposals reduce Sally's income by the same amount, she'll increase her labor supply by the same amount under both proposals. Under proposal A, Fred's labor supply doesn't change because his tax rate stays the same and he remains below the cap. So there's neither an income effect nor a substitution effect. Under proposal B, the Social Security tax rate Fred faces would rise to 15% from 12.4%, so Fred's after-tax wage rate declines and there's both an income effect and a substitution effect. The income effect leads Fred to work more, since the higher tax leads to a reduction in Fred's income. The substitution effect leads Fred to reduce his supply of labor, since the after-tax wage is lower, so there's less reward to working. Whether Fred will supply more labor or less labor under proposal B thus depends on whether the substitution effect is stronger or weaker than the income effect.

9 Chapter 11 6 With an unemployment rate of 5 %, there are initially 5 million unemployed and 95 million employed. Since 1% of the employed become unemployed, 95 million x.01 = 950,000 move from employment to unemployment each month. Since 19 % of the unemployed become employed, 5 million x.19 = 950,000 from unemployment each month. Since the same number move from employed to unemployed as the number moving from unemployed to employed, the unemployment rate remains 5 % in February and March. ote: All amounts are in millions. April: Employed (E) to Unemployed (U): 95 x.03 = U to E is 5 x.19 =.95. So U = = 6.9. The unemployment rate is u = 6.9%. May: E to U: 93.1 x.01 =.931. U to E is 6.9 x.19 = So U = = 6.52 and u = 6.52%. June: E to U: x.01 = U to E is 6.52 x So U = and u = 6.216%. July: E to U: x. 0 1 = U to E is x.19 = So U = = and u = %.