Outline of Chapter 4

1
Outline of Chapter 4

• The Representative Consumer
• Labor Supply
• The Representative Firm
• Production Function
• The Marginal Productivity of labor and Capital
• Supply Shocks
• Equilibrium in the Classical Labor Market
• Full Employment
• Factors that change equilibrium

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• The Supply of Labor
• Determined by individuals (or families)
• Choice problem market work vs. non-market
  activities (leisure, home production)
• Individual compares the cost of working to the
  benefits of working. The decision depends on the
  marginal cost of an extra hour of work versus the
  marginal benefit of working the extra hour.
• Benefit
• Cost
• Tradeoff is analogous to the firms problem
  balance the benefit (MRP) to hiring an extra hour
  of work versus the cost (real wage)
• The representative consumer chooses between a
  consumption bundle, C, and leisure, L.

3

• C has many goods in it
• Why do we use a representative consumer if people
  are different?
• Microfoundations for the Labor Supply Curve
• Optimization subject to constraints
• Start by specifying a utility function (or
  preferences) U(C,L)
• We can compare two bundles (C1,L1) and (C2,L2)
• If U(C1,L1) gt U(C2,L2) then bundle 1 is
  strictly preferred to bundle 2
• If U(C1,L1) lt U(C2,L2) then bundle 2 is strictly
  preferred to bundle 1
• If U(C1,L1) U(C2,L2) then the consumer is
  indifferent between the two bundles

4

• Important Properties of Preferences
• 1. More is preferred to less
• 2. Consumer prefers diversity in consumption/
  leisure bundle
• These properties can be interpreted with
  indifference curves or restrictions on the
  derivatives of the utility function
• Restrictions on derivatives
• Indifference curve a curve which connects a set
  of points which represent consumption bundles
  among which consumer is indifferent (Graph)
• (C1,L1) and (C2,L2) bundles are on the same
  indifference curve (I1), so the consumer is
  indifferent between the bundles
• all bundles on I2 are preferred to those on I1
• Indifference curves slope downward. Why?
• An indifference curve is convex (bowed towards
  the origin). Why?

5

• We now have the representative consumers
  preferences, we need the consumers constraints
  and assumptions about the economic environment.
• Behavioral assumption the consumer takes prices
  as given and acts as if his/her actions have no
  effect on prices (this is perfectly valid if
  consumer is small relative to the market)
•  The consumers constraints
• The consumer buys good C in the market at price P
• The consumer sells labor at price W
•  The consumer has a total of H hours available
  for work/leisure
• NS represents the labor supplied to the market
  we will determine what this is
• Consumers time constraint NS L H

6

• Consumers nominal budget constraint
• C WNS ? T
• C real consumption
• W real wage
• ? real profits from owning the representative
  firm could be any other non-wage income
• T real lump sum taxes an assumption for now
• The budget constraint says that nominal
  consumption equals nominal disposable income
• We can rewrite the budget constraint substituting
  in the time constraint
• C W(H-L) ? T
• Or
• C WL WH ? T
• This says that real consumption plus leisure
  purchased equals the value of time available plus
  disposable income

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• The optimal consumption/leisure bundle when T gt ?
• Budget constraint
• The intercept of the x-axis is from setting C0
  in the budget constraint and solving for L
• The optimal choice is on the highest obtainable
  indifference curve (dont choose inside the
  feasible set)
• Graph
• Optimal choice of C L bundle when ? gt T
• The budget constraint now has a kink
• We can also solve this model using constrained
  optimization (with a Lagrangian)
• The effect of an increase in Non-wage income
• for the case ? gtT, the case ? lt T is similar
• ?-T would increase due to an increase in profits
  or a reduction in the lump-sum tax
• Graph

8

• With the initial the budget constraint (C1,L1) is
  the optimal consumption bundle
• non-wage income increases (the real wage hasnt
  increased), this is from increased profits or
  lower taxes (given our current assumptions)
• the new budget constraint is parallel to the old
  budget constraint (along the sloped part of the
  line) since the real wage hasnt changed
• The result is that both consumption and leisure
  increase because they are normal goods.
• Changes in the real wage rate
• An increase in the real wage (graph)
• Point (C1,L1) is one possible outcome. We know
  that C must increase because C is a normal good.
• What will happen to leisure?

9

• Deriving the labor Supply Curve
• Using utility, which defines the shape and
  position of the indifference curves, face the
  consumer with a sequence of real wages (holding
  non-wage income fixed) and determine the amount
  of leisure/labor chosen at each real wage by
  looking at the tangency point of the indifference
  curve and the budget constraint.
• repeat this exercise for all possible real wages
• trace out leisure/labor choice on a graph using
  NS(W) H L(W)
• where the function L(W) is determined by the
  indifference curve/budget constraint tangency
  points (graph)
• The shape of the labor supply curve depends on
  whether or not income or substitution effects
  dominate
• If the substitution effect dominates, then ?

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• Shifts in the Labor Supply Curve
• Changes in non-wage disposable income
• Changes profits (?)
• Changes in taxes (T)
• Wealth
• Change in the expected future real wage
  (intertemporal decision)

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The Production Function

• The production function relates inputs to
  outputs. It answers the question Given a
  quantity of inputs how much output can be
  produced?
• Inputs (factors of production)
• The production function for 1 period
  Y X ? F(K,N)
• Y is output
• Z is total factor productivity (TFP), Z is
  unobservable
• K is the capital stock
• N is labor, which is measured by total hours
  worked
• F(.,.) represents a mathematical function that
  describes how the inputs are combined to produce
  the output, which is then scaled by total factor
  productivity
• Short run versus long run

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• The Cobb-Douglas Production function
• ?? is the share parameter. It equals the share
  of income that goes to capital from the
  production process. ? must be between 0 and 1.
• 1-? is the share of income that goes to labor.
• Estimating Z
• Economists think that ? is around 0.36. This
  number is derived by looking at labor income data
  and observing that about 64 of income flows to
  labor. This share appears to be relatively
  constant over time.

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• The economy produces with the same production
  function in every time period. So we can write
  the production function using time subscripts
• Productivity is volatile
• Productivity shocks may drive fluctuations in
  output.
• Can variability in output be caused by changes in
  labor and capital?
• Is this production function and measure of
  productivity reasonable?

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• Some caveats to the production function
• The production function we are using is only an
  abstraction of reality. For example, we could
  include land and other materials explicitly in
  the production function. Example
• We dont have to fully utilize the capital stock
  if we dont want to. The reason why producers may
  not fully utilize their capital stock is that
• the capital will depreciate faster the more
  fully it is used,
• Spare capacity allows firms to respond to good
  economic conditions.
• In the US average utilization of the capital
  stock is around 80.

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• Either of these variations will lead to a
  different measure of productivity.
• In the case of variable utilization the
  variability will be smaller.
• Why?
• Calculating productivity
• Year Output Capital Labor Z Z Growth
• 1988 5865 6348 115
• 1989 6062 6502 117.3
• 1990 6136 6650 118.8
• 1991 6079 6743 117.7
• 1992 6244 6830 118.5

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Properties of the Production Function

• Constant Returns to Scale
• Constant Returns to Scale if we multiply both
  inputs by a constant, then output increases by
  the same multiple
• Example C is a constant, if we multiply both the
  labor and capital inputs by C the production
  function would be
• Z ? F(C?K,C?N), while output would now be C
  ?Y.
• Cobb-Douglas
• Why is constant returns to scale important? The
  scale of the economy does not matter. The
  production function we wrote down is
  representative of the production ability of the
  entire economy. So, despite the fact that in
  reality there are many firms, from a
  macroeconomic perspective, and for analysis of
  the macroeconomy, we can use the representative
  firm

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• Output increases when either capital or labor
  increases The marginal products are always
  positive.
• The marginal product of an input is the amount
  that output increases with a unit increase in one
  factor of production
• The marginal product of labor decreases as the
  quantity of labor input increases
• diminishing marginal product of labor
• Example
• The marginal product of labor is the slope of the
  production function, holding capital fixed and
  increasing the labor input (graph)
• Slope
• Numerical Example

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• the marginal product of the 1st unit of labor is
• The marginal product of the 2nd unit of labor is
  
• The marginal product of the 3rd unit of labor
  is
• Output increases with each additional unit of
  labor, but at a decreasing rate
• We could also use calculus to calculate the
  marginal product It is the derivative of the
  production function with respect to N.
• The marginal product of labor function
• The marginal product of the labor input increases
  as the amount of capital input increases
• Example, Graph, Numerical Example

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• 3 and 4 can be redone for the marginal product of
  capital.
• The production function for Capital/Output is
  more sharply curved than for Labor/Output, why?
• Graph
• Productivity shocks (supply shocks), or changes
  in A, rotates the production function curve and
  shifts both marginal product curves
• Changes in productivity can be either positive or
  negative
• Graphically
• Reasons why we get productivity shocks

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Labor Demand

• Labor demand is the amount of labor demanded by
  firms (or the one representative firm). Labor
  demand is determined solely by looking at the
  firms optimization problem
• In the short run the amount of capital is
  considered fixed. It cannot be altered within the
  current period. Capital takes time to build and
  install. Labor can be altered within the period
  and hence is called a variable input. This fixed
  level of capital will be denoted .
• The firm looks at the marginal product of labor
  (MPN) and the value of that additional output
  Marginal Revenue Product (MRPN)
• MRPN MPN ? Price of the Output

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• The firm also looks at the cost of the variable
  input W ? N, and of an additional unit of labor
  W
• The firms optimization problem is to maximize
  profits Y W ? N
• real revenue function Z ? F(K,N)
• W ? N is called the variable cost function
• Combing the two we get the profit function
• ? Z ? F(K,N)-W ? N
• Graph
• Optimal labor choice is N
• At N the slope of the revenue function equals
  the slope of the variable cost function.
• Note that the slope of the revenue function is
  just the slope of the production function, which
  is just the marginal product of labor

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• The slope of the variable cost function is W
• Therefore at N the firm is setting MPN W
• Why?
• We can also solve the profit maximization problem
  by taking the first order condition and setting
  it equal to 0
• Changes in the real wage result in movements
  along the labor demand curve.
• A lower real wage means that more labor is
  demanded. If W falls from W to W then hiring
  more workers make sense. The value of the
  additional output is less but the cost is less
  also.
• Graph
• Shifts in the labor demand curve anything that
  shifts the marginal product of labor. Example?
• A shift in the labor demand curve results in a
  change in the amount of labor demanded for any
  given real wage rate.

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• Labor Market equilibrium
• Draw labor supply and labor demand curves and
  find intersection (graph)
• Assumption Firms consumers take real wage as
  given when solving their decision problems. The
  real wage is determined as the equilibrium in the
  market
• Critical assumption the real wage adjusts
  instantly to maintain equilibrium
• N represents full employment
• Changes in N result from shifts in either the
  labor supply or the labor demand curves, or both.
  
• Example Increase in productivity
• Full employment output
• Y Z?F(K,N)
• Note that shocks to total factor productivity
  have two effects on equilibrium output