Chapter 4: Consumer and Firm Behaviour - The Work-Leisure Decision and Profit Maximization

The Representative Consumer

  • Considers the behaviour of a single consumer who represents the whole economy.
  • Preferences - Consumption Good and Leisure.

Utility Function

U(C,l)U(C,l) where UU is utility function, CC is quantity of consumption, and ll is quantity of leisure, as a particular consumption bundle.

  • U(C1,l1)>U(C2,l2)U(C_{1},l_{1}) > U(C_{2},l_{2}) then bundle 1 is strictly preferred.
  • U(C1,l1)=U(C2,l2)U(C_{1},l_{1}) = U(C_{2},l_{2}) then consumer is indifferent.

Three Assumptions

  1. More is preferred to less.
  2. The consumer likes diversity in the consumption bundle.
  3. Consumption and leisure are normal goods. (purchased more if income is higher.)

Indifference Map

A family of indifference curves with points representing consumption bundles where the consumer is indifferent.

  • Indifference curves slope downward and are convex.
  • Marginal Rate of Substitution – The rate at which the consumer is willing to substitute leisure for consumption goods (or vice versa).
  • MRSl,CMRS_{l,C} = -[The slope of the indifference curve passing through (C,I)](C, I)]


Budget Constraint

  • Competitive behaviour dictates that the consumer is a price-taker and treats market prices as being given.

Barter Economy – All trade involves exchange of goods for goods. (Just Consumption Goods and Leisure Time)

Time Constraint

(h)=l+Ns(h) = l + N^{s}

  • hh - Time constraint
  • ll - Leisure time
  • NsN^{s} - Time working, also shown as (hl)(h-l)

Real Disposable Income

Wage income plus dividend income minus taxes.

  • w – Price of labour time sold by the consumer, or real wage.
  • The consumption good plays the role of numeraire in the economy.
  • π (Pi) - Quantity of dividend income.
  • T – Lump-sum tax paid by the consumer to the government. (Is not affected by consumer price).

The Budget Constraint

C=wNs+πTC=wN^{s}+\pi-T

  • Constraint = (wage x hours worked) + dividends - taxes
  • Rearranged as C+wl=wh+πTC+wl=wh+\pi-T
  • Right hand side is implicit disposable income and left hand side is implicit expenditure on consumption and leisure.
  • In slope-intercept form - C=wl+wh+πTC=-wl+wh+\pi-T
  • -w is slope of the budget constraint, wh+πTwh+\pi-T is vertical intercept (maximum quantity of consumption attainable)
  • To get horizontal intercept, solve for leisure time - h+(πT)/wh +\left(\pi – T \right)/w

If T<πT<\pi , then there will be a kink, where the consumer cannot consumer more than hh hours of leisure. This is because there is additional dividend income that could be thrown away at any time.

  • Examples will always deal with this case. (graph above


Consumer Optimization

  • Assumes the consumer is rational in preferences and budget constraint
  • Assumes that T>πT>\pi, and that the point where the Indifference Curve is tangent to the budget constraint is the optional consumption bundle. (Point HH)

  • When consumer optimizes, MRSl,C=wMRS_{l,C} = w
  • Marginal rate of substitution of leisure is equal to the real wage.
  • Relative Price of a good xx in terms of a good y is the number of units of y that trade for a unit of xx.
  • In competitive markets, consumer sets the MRSMRS equal to the relative price of one good in terms of another.
  • Points AA and BB are not optimal because a consumer prefers a mix of consumption and leisure.

Change in Real Dividends or Taxes

  • Changing πT\pi-T holds ww constant.
  • Pure Income Effect
  • An increase in π\pi could be caused by increase in firm productivity, which increases dividends.
  • A decrease in TT, a tax cut causes an increase in disposable income.
  • When π>T\pi>T, an increase in πT\pi-T is considered.
  • The budget line is shifted out, with a higher vertical intercept and parallel to the original BLBL.
  • Increase in income causes an increase in consumption less than the difference between the vertical intercepts because non-wage income increases, but wage income falls since the consumer works less.
  • Reduction in income does not completely offset the increase in non-wage income because consumption is a normal good.

Change in Real Wage - Substitution and Income Effects

  • Holds π\pi and TT constant to remove the pure income effect.
  • An increase in real wage ww causes the budget constraint to shift out, but the new constraint is steeper, and the kink remains fixed.

Substitution Effect – Increase in real wage causes a new budget constraint (JKD)(JKD), where the vertical intercept is higher. (Moves from FF to OO on graph)

  • Consumption increases and leisure decreases because of a real wage increase.
  • Since wage is higher, consumer works more to consumer more.

Income Effect – Real wage stays the same as budget constraint shifts out (JKD  to  EBD)(JKD\; to\; EBD) and non-wage income increases. (Moves from OO to HH on graph)

  • Both goods are normal, so consumption increases and leisure increases.
  • Since wage is higher, consumer consumes more and takes more time off.

An increase in real wage could lead to an increase or a decrease in labour supply NSN^{S}.

Labour Supply Curve – Shows how much labour the consumer wants to supply for each possible value for the real wage.

  • NS(w)=hl(w)N^{S}(w) = h – l(w)
  • Where hh = total hours available, and l(w)l(w) is a function that shows the amount of leisure desired in terms of wage ww.
  • Assumes that the substitution effect is larger than the income effect, meaning that the supply curve is upward sloping.

Perfect Complements

  • Representative consumers’ goods are desired to be consumed in fixed proportions, so C/lC/l is always equal to some constant “aa”.
  • C=alC=al, where a>0a>0 is a constant, where the optimal consumption bundle will lie. (The quantities of consumption and leisure must satisfy the budget constraint and this equation)
  • Solving these reveals -
  • L=(wh+πT)÷(a+w)L = (wh + \pi – T) \div (a + w)
  • C=a(wh+πT)÷(a+w)C = a(wh + \pi – T) \div (a+w)
  • Leisure and consumption increase with non-wage disposable income (πT)(\pi – T)  and they both increase with real wage ww increases.
  • If aa increases, the consumer prefers more consumption relative to leisure, so at optimum, there is more CC and less ll .
  • There are no substitution effects.

The Representative Firm

  • Firms demand labour and supply consumption goods, shown by a single representative firm.
  • Assumes that firms own productive capital and hire labour to produce consumption goods.
  • Production technology available is shown as - Y=zF(K,Nd)Y = zF(K, N^{d})
  • zz is total factor productivity (increasing will make KK and NdN^{d} more productive, causing higher output.
  • YY is output of consumption goods.
  • KK is quantity of capital input in the production process.
  • NdN^{d} is the quantity of labour input measured in total hours worked by employees.
  • FF is a function of KK and NdN^{d}
  • In SRSR , firms cannot vary quantity of plant and equipment KK but are flexible in hiring and laying off workers NdN^{d}

Marginal Product

Additional output that can be produced with one additional unit of that factor input, holding constant the quantities of other factor inputs.

  • Slope of the production function is the MPMP.
  • Varying the quantity of labour (fixed capital) causes the Marginal Product of Labour MPNMP_{N} to be the slope.
  • MPNMP_{N} declines with quantity of labour.
  • Varying the quantity of capital (fixed labour) causes the Marginal Product of Capital MPKMP_{K} to be the slope.
  • MPKMPK declines with quantity of capital.

Production Functions have 5 Key Properties

1 Constant Returns to Scale

  • At any constant x>0,xzF(K,Nd)=zF(xK,xNd).x > 0, xzF(K,N^{d}) = zF(xK,xN^{d}).
  • If all factor inputs are changed by a factor “xx” then output changes by the same factor. (i.e. x=2x=2, output doubles as well)
  • Alternatives
  • Increasing Returns to Scale – Larger firms are more efficient than small firms.
  • Decreasing Returns to Scale – Smaller firms are just as efficient as large firms.

2 Output increases when either capital input or labour input increases

  • Marginal products of labour and capital are both positive MPN>0MP_{N}>0 and MPK>0MP_{K}>0
  • More inputs yield more output.

3 The Marginal Product of Labour decreases as Quantity of Labour increases

  • The more labour input added, the less marginal product can be obtained.
  • Slope (concavity) decreases in a production function.

4 The Marginal Product of Capital decreases as Quantity of Capital increases

  • The more capital input added, the less marginal product can be obtained.

5 The Marginal Product of Labour increases as the Quantity of Capital Input Increases

  • Adding more capital increases the Marginal Product of Labour for each quantity of labour – if there is a worker to utilize the capital, then MPN increases.

The Effect of a Change in Total Factor Productivity on the Production Function

  • An increase in total factor productivity zz has 2 effects.
  • There is an upward shift in the production function as more output can be produced given capital and labour inputs
  • Marginal product of labour shifts to the right, as it increases when zz increases.
  • Caused by -
  • Anything that permits more output to be produced for given inputs, such as technological innovation (i.e. assembly line) increases zz.
  • Good weather (for agricultural and construction) increases z.z.
  • Government regulations may decrease zz
  • Increase in relative price of energy decreases zz

Image result for ford model t

Profit Maximization Problem

  • Representative firm behaves competitively – It takes the real wage as given. (Price at which labour trades for consumption goods)
  • It wants to maximize its profits, given YwNdY – wN^{d}
  • YY – Total revenue received from selling output, wNdwN^{d} – total real cost of the labour input (total real variable costs).
  • Subbing in the production function Y=zF(K,Nd)Y = zF(K,N^{d}), the firm must choose NdN^{d} to maximize π=zF(K,Nd)wNd\pi = zF(K,N^{d}) - wN^{d}
  • zF(K,Nd)zF(K,Nd) – Firm’s revenue,
  • wNdwNd – Variable cost
  • KK is fixed, and π\pi is real profit.
  • To maximize profits, the firm will choose Nd=NNd = N^{\ast} to get maximized profits π\pi^{\ast} (which is distance ABAB in this graph)
  • Located at Marginal Revenue = Marginal Cost, or MPN=wMPN=w.

A firm’s marginal product of labour schedule is its demand curve for labour.


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