Labour Supply Part 2: Constrained Optimisation

Labour Supply Part 2

• Constrained optimisation, Lagrangean method, primal and dual problem and estimation are covered.

Constrained Optimisation

• A new function, a Lagrangean function, is constructed to incorporate constraints.
• An extra parameter, the Lagrange multiplier (λ), is included.
• The derivatives of the function are taken and set equal to zero, proceeding as in an unconstrained problem.

Equivalence with Substitution

• Both methods give the same results, provided the constraint is satisfied with equality.
• Assumptions, such as non-satiation, regarding the utility function ensure this equivalence.

Lagrangean Method

• The Lagrangean is a linear combination of the objective function and constraint. (formula not provided)
• To find the maximum, derivatives are taken with respect to relevant variables (C, L, and λ).

First Order Conditions

• The three equations formed from the derivatives are solved simultaneously to find the optimal values for C, L, and λ.

Interpretation

• The slope of the budget constraint and the marginal rate of substitution (MRS) are equal.
• All income is spent, so the Lagrange multiplier (λ) must be non-zero.
• Kuhn-Tucker conditions are more general statements to check for slackness in the constraint.

Interpretation (continued)

• Equation (4) implicitly defines optimal demand for leisure (and supply of labour).
• With a known utility function, the optimal levels can be found analytically. Comparative static properties can be examined by analysing the effect of changing exogenous variables, as in earlier diagrams.

Too Much Maths

• Sign comparative static effects by examining signs of partial derivatives.
• Matrix techniques (e.g., Cramer's rule) can be utilized.
• Detailed mathematical methods are deferred for another time.

Worked Example

• A utility function is specified.
• A time constraint is given by 16 - L = h (where L is leisure time and h is hours of work). The hours of sleep is 8 hours.
• This leads to a budget constraint: wh + N = pC (where w is wage, N is non-labor income, p is price of consumption good, and C is consumption).

The Lagrangian Method (continued)

• The problem is to maximize utility subject to the time constraint w(16-L) + N = pC.
• The Lagrangia is composed.
• First order conditions will determine solutions.

Interpretation (continued)

• The solution yields the equation (2)÷(1).
• LHS is MRS, RHS is slope of budget constraint.

Worked Example (Continued)

• Sample numerical values are given (N=40, w=10, p=2). This information is used in diagrams.

Diagrammatically

• Diagram shows the budget constraint and indifference curves, providing a visual representation of the optimal consumption and leisure.

Now Let's Play

• Comparative statics involves examining how values change given a change in exogenous variables (w, N, and p).

Now Let's Play (Continued)

• Investigate the effects on labour supply when non-labor income (N) changes.
• Maintain the wages (w=10) and increase the non-labor income (N). This increase in non-labor income shifts the budget constraint outward.

Interpretation (Continued)

• As non-labor income (N) increases, leisure demand increases by ½ hour for every 10 increase in non-labor income(N).

Interpretation (Continued)

• As non-labor income (N) continues to increase, leisure time may exceed 16. Labour supply can end up as negative.

Investigating this Further

• The slope of indifference curves (IC) is given by MRS = C/L
• Working out the optimal values of C and L to calculate MRS at a given point.

Changing the Wage

• As the wage (w) increases, the optimal leisure time (L) decreases. This results in the labour supply curve.
• Using Calculus or numbers provides quantitative confirmation.

Calculus

• The slope of the relationship, calculated via calculus, shows a positive association between wage and labour.
• Results are unambiguous: An increase in wage will reduce demand for leisure and increase labour supply.

Using Some Numbers.

• Using numerical examples, hold N at 40, w at 10; then change the wage to investigate its effects.

Things to Note

• The slope of relationship is not constant: changes vary depending which side of 10 the wage change is.

As the value of w (Wage) goes to...

• As the wage (w) approaches zero, the term gets bigger and leisure demand approaches infinity.
• As the wage (w) approaches infinity, the term approaches zero and leisure demand approaches a set value.

This is the full curve...

• Graph the full labour supply curve with the wage varying from 0.1 to 100. This graph show the shape of the demand curve and the asymptotes.

In this graph we have removed ...

• Graph the labour supply curve, removing very small values of w to show clear asymptotes.

The next graph shows ...

• Show the labour supply function. Notice that it is monotonic (non-decreasing in positive direction).

Income and Substitution Effect

• The Slutsky decomposition technique can analyze the income and substitution effects.
• The substitution effect holds utility constant, is examined first.
• The income effect is examined by considering the effect of a change from one point A to point B on income.

Income and Substitution Effect (continued)

• An increase in income decreases labor supply (increases leisure time)
• A higher wage increases labor supply and decreases leisure demand.

The income effect is given by...

• The mathematical expression for the income effect.
• The total effect from a Slutsky equation.

Implies or from our results...

• A confirmation that the calculations are valid. The value found is greater than zero. A similar consideration will be made for leisure.

The Backward Bending Supply Curve

• There is a potential issue with the graphical labor supply model. Utility maximizing models will not result in a backward-bending supply curve.

Ad Hoc Specification

• Including additional terms in the model (i.e.: a linear and quadratic term for wage) could produce a backward-bending supply curve.

Effect of wages on hours

• A backward-bending curve is required for the wage to be positive over some range, and to be negative over other ranges.

We need and ...

• The issue is that utility maximizing models will not produce a backward bending supply curve.

Duality

• Microeconomic theory uses duality between maximization of one objective and minimization of another to explain economic concepts. Ex: utility maximization in the face of budget restrictions, also expenditure minimization to achieve a certain level of utility

Roy's Identity

• Roy's identity allows for a calculation of Marshallian demand from an indirect utility function. In the example, the indirect utility function is derived by substituting the optimal values of C and L into the utility function.

Example: Education

• More educated women are more likely to be employed.
• Education is related to wages; a higher level of education means higher wages on average

A Note on Estimation

• Estimating models involves maximum likelihood estimation.
• Least squares estimation fits a line to minimize the sum of squared deviations from the line.
• The likelihood function determines how likely the sample is given the parameters.

Gauss-Markov and Maximum Likelihood

• When Gauss-Markov assumptions hold, Least Squares and Maximum Likelihood produce the same results.
• The model of labour supply needs to incorporate both wage and sample probability.
• This involves employing maximum likelihood estimation.

Other Estimation Issues

• Mass points occur in distributions when significant observations coincide at certain points.
• Mass points at zero employment due to minimum employment hours. Labor supply might have mass points at established employment hours(e.g., 35 hrs/week).

Example: 

• Examples showing mass points and optimal choices that are negative in hours of labour supply.

Other Issues

• Mass points at kink points may confound analyses if the tax system is considered.

Kink Point

• At certain wage levels (h*), the amount of wage earned is subject to taxes, thus altering the slope of the budget constraint.
• This creates clustering of data points at the h* level, which creates problems in estimation.

Sample Selection Bias

• When the sample of individuals isn't a random sample of the target population, resulting bias might occur. A variable could influence labour supply/employment alongside the probability of being in the sample.

Sample Selection Bias (Continued)

• This leads to bias in estimated coefficients, as presented in a paper by Mroz.
• Examples of bias due to sampling biases include omitted variables resulting from a correlation between the omitted variable and selection, leading to biased estimates or other correlated effects.

Sample Selection Bias (Continued)

• A simple model explaining sample selection bias, showing how A or B being correlated with the variable Y affects the estimation of the model.

Sample Selection

• Sample selection bias arises when a sample isn't a random draw from the target population, and the selection process is correlated with the dependent variable and/or relevant explanatory factors.

The Backward Bending Supply Curve (Continued)

• This issue arises due to the required nonlinearity in the utility function for a backward-bending supply curve. This nonlinearity is rarely observed when modeling.

This quiz delves into the concepts of constrained optimisation, focusing on the Lagrangean method, primal and dual problems, and estimation techniques. It also discusses the equivalence between substitution methods and the conditions for their validity. Test your understanding of these advanced economic theories and applications.