Labour Supply Part 2: Constrained Optimisation
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What is the function of the Lagrangian method in this context?

  • To find the slope of the indifference curve
  • To determine the total time available for work
  • To maximize utility subject to constraints (correct)
  • To solve for the non-labour income N
  • What does an increase in non-labour income (N) by 10 result in?

  • An increase in labour supply by 1/2 hour
  • An increase in leisure demand by 1/2 hour (correct)
  • A decrease in leisure demand by 1/2 hour
  • No change in labour supply or leisure demand
  • What happens to the slope of the budget constraint when non-labour income increases?

  • It shifts leftward
  • It becomes less steep (correct)
  • It becomes steeper
  • It remains constant
  • What will occur if leisure demand exceeds 16 due to increasing N?

    <p>Labour supply becomes negative</p> Signup and view all the answers

    What does MRS represent in this context?

    <p>Marginal Rate of Substitution between consumption and labour</p> Signup and view all the answers

    At N=50 and holding L=10, how is the MRS calculated?

    <p>52.5/10</p> Signup and view all the answers

    What does the equality of MRS and slope of the budget constraint signify?

    <p>That consumption and leisure are optimally balanced</p> Signup and view all the answers

    How is the budget constraint expressed in terms of labour and consumption?

    <p>w(16 - L) + N = pC</p> Signup and view all the answers

    What does an increase in non-labour income do to labour supply?

    <p>It decreases labour supply.</p> Signup and view all the answers

    What is essential for a backward bending supply curve to exist?

    <p>Wage effects must be positive over some range and then negative.</p> Signup and view all the answers

    Which equation helps in separating the income and substitution effects?

    <p>The Slutsky equation.</p> Signup and view all the answers

    In the context of labour supply, what happens when wages increase?

    <p>Labour supply increases while leisure demand decreases.</p> Signup and view all the answers

    What role does the quadratic term play in the labour supply model?

    <p>It allows for the incorporation of backward bending supply.</p> Signup and view all the answers

    Why is the existence of a backward bending supply curve problematic in utility maximisation theory?

    <p>It cannot be derived from linear utility functions.</p> Signup and view all the answers

    Which term is NOT needed to establish a backward bending supply of labour?

    <p>Negative income effects.</p> Signup and view all the answers

    The substitution effect holds under what condition?

    <p>When utility remains constant.</p> Signup and view all the answers

    What does the concept of duality in microeconomic theory imply?

    <p>Maximising one objective can align with minimising another.</p> Signup and view all the answers

    Which of the following is a result of duality that aids in econometric modelling?

    <p>Shephard’s Lemma</p> Signup and view all the answers

    What is Roy’s identity used for in microeconomic theory?

    <p>To derive Marshallian demand from an indirect utility function.</p> Signup and view all the answers

    What does the term 'mass points' refer to in labor supply data?

    <p>Points in the distribution with a significant number of observations</p> Signup and view all the answers

    What is the purpose of the indirect utility function?

    <p>To evaluate the maximum utility achieved with given constraints.</p> Signup and view all the answers

    What does the TOBIT model address in labor supply analysis?

    <p>It accounts for situations where the dependent variable cannot take negative values.</p> Signup and view all the answers

    In the context of the indirect utility function, what parameters can affect its calculation?

    <p>Wage, prices, and available income.</p> Signup and view all the answers

    What is a potential outcome when running an OLS regression on labor supply data that includes negative hours?

    <p>The slope coefficient would be biased downwards.</p> Signup and view all the answers

    Why is understanding Roy’s identity important in economics?

    <p>It helps derive demand functions without starting from utility maximisation.</p> Signup and view all the answers

    In a typical labor supply model, what does the variable Y represent?

    <p>The intended number of work hours supplied per week</p> Signup and view all the answers

    What is the purpose of incorporating the Lagrange multiplier, λ, in the Lagrangean method?

    <p>To handle constraints in the optimization problem.</p> Signup and view all the answers

    What is the Marshallian demand function based on?

    <p>The relationship between money income and the price of good.</p> Signup and view all the answers

    What condition must be met for the equivalence of the Lagrangean method and substitution method to hold?

    <p>The constraint must be satisfied with equality.</p> Signup and view all the answers

    Which of these statements accurately describes the indirect utility function?

    <p>It can be generalized using properties derived from earlier methods.</p> Signup and view all the answers

    How do mass points at certain hours, such as 0 or 35 hours per week, affect labor supply observations?

    <p>They indicate that optimal choices cannot be observed.</p> Signup and view all the answers

    Which of the following statements correctly describes the first-order conditions (FOCs) in the context of the Lagrangean method?

    <p>They involve taking derivatives of the Lagrangean with respect to all parameters.</p> Signup and view all the answers

    What is indicated by a negative Y* in a labor supply model?

    <p>The individual does not work at all.</p> Signup and view all the answers

    What could be a problem of having ad hoc models in labor supply analysis?

    <p>They complicate the understanding of underlying behaviors.</p> Signup and view all the answers

    What do the Kuhn Tucker conditions refer to in constrained optimization?

    <p>Criteria to check for optimality in the presence of slackness.</p> Signup and view all the answers

    In the interpretation of the first-order conditions, what does the Left-Hand Side (LHS) represent?

    <p>The marginal rate of substitution (MRS).</p> Signup and view all the answers

    Why is it important to treat observations with mass points appropriately in labor supply analysis?

    <p>To avoid misleading conclusions from the data.</p> Signup and view all the answers

    What does the statement 'all income is spent' imply about the Lagrange multiplier, λ?

    <p>λ must be non-zero.</p> Signup and view all the answers

    What is one potential method to analyze comparative static effects in constrained optimization problems?

    <p>Employing Cramer's rule for matrix techniques.</p> Signup and view all the answers

    Which factor does NOT influence the optimal demand for leisure or supply of labor according to the content?

    <p>The length of working hours.</p> Signup and view all the answers

    What effect does OLS regression have on the slope and intercept estimates in the context of Tobit analysis?

    <p>It results in a downwards-biased slope and upwards-biased intercept.</p> Signup and view all the answers

    What complication arises in the analysis when including a tax system in the Tobit model?

    <p>It complicates the estimation due to potential clustering at kink points.</p> Signup and view all the answers

    How does sample selection bias affect the independence assumption of OLS?

    <p>It violates the independence assumption by selecting non-random samples.</p> Signup and view all the answers

    What is a potential problem when reaching a tax threshold according to the Tobit analysis?

    <p>There is a consideration on whether to work more hours or remain at the kink point.</p> Signup and view all the answers

    What methodological challenge is associated with estimating optimal work hours in the presence of a kink point?

    <p>Estimating these models significantly increases in difficulty.</p> Signup and view all the answers

    What does the Tobit model analysis imply about individuals' choices related to work?

    <p>Work decisions are influenced by deterministic factors leading to non-random selection.</p> Signup and view all the answers

    What happens to individuals at the kink points within the context of the Tobit model?

    <p>Individuals experience a change in their wage rate due to taxation.</p> Signup and view all the answers

    Which statement about Tobit analysis is most accurate?

    <p>It can be used to correct bias from non-random sample selection.</p> Signup and view all the answers

    Study Notes

    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.

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    Description

    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.

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