Productivity, Output, and Employment
Productivity, Output, and Employment 8 September 2009
1
Reading • Chapter 3 of Abel/Bernanke/Croushore
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The Production Function • What determines the quantity of goods and services than an economy produce in period of time? 1. Quantities of various inputs — capital equipment, labor, raw materials, land, and energy. 2. The technology of production — the level of efficiency with which the factor inputs are combined. Technology encompasses not only technical efficiency, but also factors such as management practices, the business environment, property rights and the like. • The production function represents the relationship between output and inputs: Y = AF (K, N) , where K = the capital stock, N = the number of labor hours hired, A = total factor productivity, or simply productivity. • Two main properties of production functions: 1. Slopes upward: more of any input produces more output. 2. Slope becomes flatter as input rises: diminishing marginal product as input increases. • The marginal product of capital (MPK), ∂Y /∂K > 0 but diminishes as capital input increases; i.e., ∂ 2 Y /∂K 2 < 0. 1
Figure 1: Output as a function of capital input
Figure 2: The marginal product of capital
2
Figure 3: The marginal product of labor • The marginal product of labor (MPN), ∂Y /∂N > 0 but diminishes as labor input increases; i.e., ∂ 2 Y /∂N 2 < 0. • Supply shocks, or productivity shocks, refer to abrupt and usually unexpected changes to productivity. It is easiest to model supply shocks by changes to total factor productivity, A. Supply shocks can be positive or negative. Examples: weather, inventions and innovations, government regulations, oil prices, states of the financial market. • A positive supply shock usually not only raises output but also the marginal products, and likewise for a negative supply shock.
3
Labor demand • Assumptions: 1. Hold capital stock fixed; i.e., a short-run analysis. 2. Workers are all alike. 3. Markets are competitive. • A given firm maximizes: short—run profit = P × AF (K, N ) − W × N, 3
Figure 4: Supply shocks yielding a first order condition: P×
∂Y − W = 0; ∂N
i.e., W = w, P where w is known as the real wage — units of output the nominal wage W can buy. MPN =
• The (horizontal) sum of individual firms’ labor demand constitutes the aggregate labor demand curve. • In so far as a positive (negative) supply shock raises (lowers) the M P N for each N , the labor demand curve shifts up (down) in response. If capital and labor input are complements (substitutes); i.e., ∂2Y > ( 0 and ∂u/∂l > 0, respectively, but diminish; i.e., ∂ 2 u/∂c2 < 0 and ∂ 2 u/∂l2 < 0. 3. Good consumption and leisure are complements; i.e., ∂ 2 u/∂c∂l > 0. 4. The household has initial (nominal) wealth B and h hours in the period to divide between hours of market work n at wage W and leisure l. • Utility maximization subject to budget constraint: max u (c, l) , c,l
5
Figure 6: Shocks to labor demand subject to l = h − n,
(1)
B + W × n = P × c. Rewrite the second constraint: B + w × n = c. P
(2)
Substitute (1) and (2) into the objective function, u max n
µ
¶
B + w × n, h − n . P
Taking first order condition, ∂u ∂u w− = 0. ∂c ∂l Rearranging, ∂u/∂l = w; (3) ∂u/∂c i.e., the MRS between leisure and good consumption be equal to the real wage. 6
• The RHS of the condition w denotes how many units of c the household has to give up to raise l by a unit; i.e., the relative price (in terms of good consumption) of leisure. To interpret the LHS, suppose that good consumption changes by an amount dc and leisure changes by an amount dl. Then utility changes by du =
∂u ∂u dc + dl. ∂c ∂l
If the change in utility is zero, ∂u/∂l dc =− . dl ∂u/∂c The ratio on the LHS of (3) is how many units of c the household is willing to give up to raise l by a unit. • Graphical analysis: substitute (1) into (2) , B + w × (h − l) = c, P or that
B + wh = c + wl, P with a slope given by ∂c/∂l = −w.
Labor—leisure choice
7
(4)
Figure 7: Income effect of an increase in non—labor income • An increase in non—labor income B/P : the budget set shifts up. For each l, there is a larger c, where the slope remains equal to w. If good consumption and leisure are both normal goods, the expansion of the budge set should lead to an increase in c and l. An increase in l is a decrease in n. We say that there is a negative income effect of an increase in non—labor income on labor supply. • An increase in real wage w. For each l, there is an increase in c. The budget set shifts up. The expansion tends to induce an increase in l and a corresponding decrease in n. We say that there is a negative income effect of an increase in w on labor supply. But the slope also steepens. The condition (3) says that there should be an increase in ∂u/∂l . M RS = ∂u/∂c The MU of l increases when l decreases; the MU of c decreases when c increases. There is a substitution effect at work where as l gets relatively more expensive, the household substitutes away from it. But the decrease in l is an increase in n. We say that there is a positive substitution effect of an increase in w on labor supply. • In general, it cannot be determined whether an increase in w would lead to an increase or decrease in n. The income effect points to a decrease in n. The substitution effect points to an increase in n. In case the increase in w is for a very short interval of time (a temporary increase), the effect on wealth is 8
Figure 8: Income and substituion effect of an increase in w negligible, whereby the substitution effect tends to dominate, resulting in an increase in labor supply. The longer the high wage is expected to last, the stronger the income effect; thus labor supply will increase by less or more likely to become decreasing in w. Empirical evidence seems to support the theory’s prediction. Adding the individual labor supply gives rise to an aggregate labor supply curve. We assume that the change in w is less than permanent, and that the substitution effect dominates. • Factors that shift the aggregate labor supply curve: 1. changes in wealth 2. changes in expected future real wage 3. changes in working-age population (higher birth rate, immigration) 4. changes in labor force participation (increased female labor participation, elimination of mandatory retirement)
9
Figure 9: Aggregate labor supply
5
Labor market equilibrium and full-employment output • In equilibrium, the labor market clears, where the market real wage adjusts to equate demand and supply. An important assumption is that the adjustment is rapid, so that the labor market stays in equilibrium almost always. • Full—employment output (also known as potential output) is the level of output when labor input is equal to the level determined in labor market equilibrium, ³
´
Y = AF K, N . At a point in time or where the period under consideration is short enough, the economy’s capital stock is more or less fixed, as determined by the history of investment. Full-employment output can change when the economy is subject to a supply shock. For instance, a temporary adverse supply shock would lead to a decline in full-employment output directly through the decline in total factor productivity. There is also an indirect effect through the decrease in equilibrium employment in the labor market.
10
Figure 10: Labor market equilibrium
Figure 11: The effect of an adverse supply shock
11
6
Unemployment • The adverse supply shock brings about a lower equilibrium employment. And the presumption is that the adjustment in w necessary for the labor market to move from points A and B is very rapid, so that the labor market is deemed to clear almost always. If an unemployed worker is one who is willing to work at the prevailing market wage but is not able to find a job, a standard demand—supply analysis of the labor market cannot explain the existence of unemployment, as well as how it changes over time. • In reality unemployment is ubiquitous, to say the least. Indeed in many places, the unemployment rate can be substantially above zero even when the economy is expanding at full pace. To explain the existence of unemployment, one approach is to introduce some “frictions” into the labor market. That is, we need a model whereby persons in the labor force who lack jobs take time to find them, and businesses with unfilled vacancies take time to fill them. Thus a model of unemployment and unfilled job vacancies should be one in which the processes of persons searching for jobs and businesses for workers take up real resources, most notably time. If workers and jobs are homogeneous commodities–the assumption underlying the standard demand-supply analysis–the job search for workers and the worker search for firms perhaps would be trivial. The labor market may certainly be organized like a centralized exchange, such as the stock market, within which all buy and sell orders take place. The truth is in reality workers and jobs differ among one another in various dimensions. Clearly it is important for individual workers to find jobs that match their skills, and likewise for firms with unfilled vacancies. The labor market is by all means a decentralized market, in which trade takes place in distinct locales, and in which its participants may be required to engage in sometime lengthy search for suitable trading partners. Unemployment that is caused by search frictions in the labor market is known as frictional unemployment. • Alternatively, there are chronically unemployed workers: workers who are unemployed a large part of the time because the lack of marketable skills prevents them from finding long-term employment. Often this kind of unemployment is referred to as structural unemployment. • Another important source of structural unemployment is that at any one time, there are shrinking and expanding sectors in the economy. The reallocation of workers from the former to the latter can be lengthy. • It is conventional to refer the unemployment rate that prevails when actual unemployment is just equal to the sum of frictional and structural unemployment as the natural rate of unemployment, u. The employment that exceeds u is
12
known as cyclical unemployment, unemployment that is caused by short—run fluctuations in the economy. • There is a well—known empirical relationship between output and unemployment, called the Okun’s law, which says that a one percentage point change in the unemployment rate is associated with a 2% change in point. We write Y −Y = 2 (u − u) . Y That is, in reality, output seems to fluctuate substantially more than employment. In recession, apparently, not only that employment falls, the hours of work per worker also declines, causing average labor productivity to fall.
13
1
Reading • Chapter 3 of Abel/Bernanke/Croushore
2
The Production Function • What determines the quantity of goods and services than an economy produce in period of time? 1. Quantities of various inputs — capital equipment, labor, raw materials, land, and energy. 2. The technology of production — the level of efficiency with which the factor inputs are combined. Technology encompasses not only technical efficiency, but also factors such as management practices, the business environment, property rights and the like. • The production function represents the relationship between output and inputs: Y = AF (K, N) , where K = the capital stock, N = the number of labor hours hired, A = total factor productivity, or simply productivity. • Two main properties of production functions: 1. Slopes upward: more of any input produces more output. 2. Slope becomes flatter as input rises: diminishing marginal product as input increases. • The marginal product of capital (MPK), ∂Y /∂K > 0 but diminishes as capital input increases; i.e., ∂ 2 Y /∂K 2 < 0. 1
Figure 1: Output as a function of capital input
Figure 2: The marginal product of capital
2
Figure 3: The marginal product of labor • The marginal product of labor (MPN), ∂Y /∂N > 0 but diminishes as labor input increases; i.e., ∂ 2 Y /∂N 2 < 0. • Supply shocks, or productivity shocks, refer to abrupt and usually unexpected changes to productivity. It is easiest to model supply shocks by changes to total factor productivity, A. Supply shocks can be positive or negative. Examples: weather, inventions and innovations, government regulations, oil prices, states of the financial market. • A positive supply shock usually not only raises output but also the marginal products, and likewise for a negative supply shock.
3
Labor demand • Assumptions: 1. Hold capital stock fixed; i.e., a short-run analysis. 2. Workers are all alike. 3. Markets are competitive. • A given firm maximizes: short—run profit = P × AF (K, N ) − W × N, 3
Figure 4: Supply shocks yielding a first order condition: P×
∂Y − W = 0; ∂N
i.e., W = w, P where w is known as the real wage — units of output the nominal wage W can buy. MPN =
• The (horizontal) sum of individual firms’ labor demand constitutes the aggregate labor demand curve. • In so far as a positive (negative) supply shock raises (lowers) the M P N for each N , the labor demand curve shifts up (down) in response. If capital and labor input are complements (substitutes); i.e., ∂2Y > ( 0 and ∂u/∂l > 0, respectively, but diminish; i.e., ∂ 2 u/∂c2 < 0 and ∂ 2 u/∂l2 < 0. 3. Good consumption and leisure are complements; i.e., ∂ 2 u/∂c∂l > 0. 4. The household has initial (nominal) wealth B and h hours in the period to divide between hours of market work n at wage W and leisure l. • Utility maximization subject to budget constraint: max u (c, l) , c,l
5
Figure 6: Shocks to labor demand subject to l = h − n,
(1)
B + W × n = P × c. Rewrite the second constraint: B + w × n = c. P
(2)
Substitute (1) and (2) into the objective function, u max n
µ
¶
B + w × n, h − n . P
Taking first order condition, ∂u ∂u w− = 0. ∂c ∂l Rearranging, ∂u/∂l = w; (3) ∂u/∂c i.e., the MRS between leisure and good consumption be equal to the real wage. 6
• The RHS of the condition w denotes how many units of c the household has to give up to raise l by a unit; i.e., the relative price (in terms of good consumption) of leisure. To interpret the LHS, suppose that good consumption changes by an amount dc and leisure changes by an amount dl. Then utility changes by du =
∂u ∂u dc + dl. ∂c ∂l
If the change in utility is zero, ∂u/∂l dc =− . dl ∂u/∂c The ratio on the LHS of (3) is how many units of c the household is willing to give up to raise l by a unit. • Graphical analysis: substitute (1) into (2) , B + w × (h − l) = c, P or that
B + wh = c + wl, P with a slope given by ∂c/∂l = −w.
Labor—leisure choice
7
(4)
Figure 7: Income effect of an increase in non—labor income • An increase in non—labor income B/P : the budget set shifts up. For each l, there is a larger c, where the slope remains equal to w. If good consumption and leisure are both normal goods, the expansion of the budge set should lead to an increase in c and l. An increase in l is a decrease in n. We say that there is a negative income effect of an increase in non—labor income on labor supply. • An increase in real wage w. For each l, there is an increase in c. The budget set shifts up. The expansion tends to induce an increase in l and a corresponding decrease in n. We say that there is a negative income effect of an increase in w on labor supply. But the slope also steepens. The condition (3) says that there should be an increase in ∂u/∂l . M RS = ∂u/∂c The MU of l increases when l decreases; the MU of c decreases when c increases. There is a substitution effect at work where as l gets relatively more expensive, the household substitutes away from it. But the decrease in l is an increase in n. We say that there is a positive substitution effect of an increase in w on labor supply. • In general, it cannot be determined whether an increase in w would lead to an increase or decrease in n. The income effect points to a decrease in n. The substitution effect points to an increase in n. In case the increase in w is for a very short interval of time (a temporary increase), the effect on wealth is 8
Figure 8: Income and substituion effect of an increase in w negligible, whereby the substitution effect tends to dominate, resulting in an increase in labor supply. The longer the high wage is expected to last, the stronger the income effect; thus labor supply will increase by less or more likely to become decreasing in w. Empirical evidence seems to support the theory’s prediction. Adding the individual labor supply gives rise to an aggregate labor supply curve. We assume that the change in w is less than permanent, and that the substitution effect dominates. • Factors that shift the aggregate labor supply curve: 1. changes in wealth 2. changes in expected future real wage 3. changes in working-age population (higher birth rate, immigration) 4. changes in labor force participation (increased female labor participation, elimination of mandatory retirement)
9
Figure 9: Aggregate labor supply
5
Labor market equilibrium and full-employment output • In equilibrium, the labor market clears, where the market real wage adjusts to equate demand and supply. An important assumption is that the adjustment is rapid, so that the labor market stays in equilibrium almost always. • Full—employment output (also known as potential output) is the level of output when labor input is equal to the level determined in labor market equilibrium, ³
´
Y = AF K, N . At a point in time or where the period under consideration is short enough, the economy’s capital stock is more or less fixed, as determined by the history of investment. Full-employment output can change when the economy is subject to a supply shock. For instance, a temporary adverse supply shock would lead to a decline in full-employment output directly through the decline in total factor productivity. There is also an indirect effect through the decrease in equilibrium employment in the labor market.
10
Figure 10: Labor market equilibrium
Figure 11: The effect of an adverse supply shock
11
6
Unemployment • The adverse supply shock brings about a lower equilibrium employment. And the presumption is that the adjustment in w necessary for the labor market to move from points A and B is very rapid, so that the labor market is deemed to clear almost always. If an unemployed worker is one who is willing to work at the prevailing market wage but is not able to find a job, a standard demand—supply analysis of the labor market cannot explain the existence of unemployment, as well as how it changes over time. • In reality unemployment is ubiquitous, to say the least. Indeed in many places, the unemployment rate can be substantially above zero even when the economy is expanding at full pace. To explain the existence of unemployment, one approach is to introduce some “frictions” into the labor market. That is, we need a model whereby persons in the labor force who lack jobs take time to find them, and businesses with unfilled vacancies take time to fill them. Thus a model of unemployment and unfilled job vacancies should be one in which the processes of persons searching for jobs and businesses for workers take up real resources, most notably time. If workers and jobs are homogeneous commodities–the assumption underlying the standard demand-supply analysis–the job search for workers and the worker search for firms perhaps would be trivial. The labor market may certainly be organized like a centralized exchange, such as the stock market, within which all buy and sell orders take place. The truth is in reality workers and jobs differ among one another in various dimensions. Clearly it is important for individual workers to find jobs that match their skills, and likewise for firms with unfilled vacancies. The labor market is by all means a decentralized market, in which trade takes place in distinct locales, and in which its participants may be required to engage in sometime lengthy search for suitable trading partners. Unemployment that is caused by search frictions in the labor market is known as frictional unemployment. • Alternatively, there are chronically unemployed workers: workers who are unemployed a large part of the time because the lack of marketable skills prevents them from finding long-term employment. Often this kind of unemployment is referred to as structural unemployment. • Another important source of structural unemployment is that at any one time, there are shrinking and expanding sectors in the economy. The reallocation of workers from the former to the latter can be lengthy. • It is conventional to refer the unemployment rate that prevails when actual unemployment is just equal to the sum of frictional and structural unemployment as the natural rate of unemployment, u. The employment that exceeds u is
12
known as cyclical unemployment, unemployment that is caused by short—run fluctuations in the economy. • There is a well—known empirical relationship between output and unemployment, called the Okun’s law, which says that a one percentage point change in the unemployment rate is associated with a 2% change in point. We write Y −Y = 2 (u − u) . Y That is, in reality, output seems to fluctuate substantially more than employment. In recession, apparently, not only that employment falls, the hours of work per worker also declines, causing average labor productivity to fall.
13
