Reviewing Some Tools Using Production as a Context
1. Some basic mathematical manipulations
(1.1)
Y  AL
Question 1.1: What is the marginal product of labor? (Definition of Marginal Product, partial
Derivative)
Question 1.2: What is the average product of labor? (Definition of average product)
Question 1.3: How does the growth rate of output relate to the growth rate of labor? (Natural log, time
derivative, differential)
Question 1.4: What is the elasticity of output with respect to labor input? (Definition of elasticity,
Natural log, time derivative, differential)
Question 1.5: Find the comparative static multipliers for this model and interpret them. (Definition of
comparative static multiplier, partial derivative)
2. Constructing and plotting a function to examine something of interest
(2.1)
  PY  WL
Question 2.1: How would you define the variables in equation (2.1) in order for equation (2.1) be a
profit function for a firm? (Defining variables)
Question 2.2: Use equation (1.1) to eliminate the variable Y from equation (2.1). Plot the resulting
profit function as it depends upon L . You should see that the plot you obtain depends upon the values
of the variables P , W , and A . Plot three different curves for all three cases. (Plotting a relationship)
3. Constructing an optimization problem
Question 3.1: Use equations (1.1) and (2.1) to write down a profit maximization problem for a firm,
and present the problem as a one variable mathematical programming problem.
Question 3.2: What is the “first order condition” or “necessary condition” for profit maximization for
this problem? (Finding an extremum, first order condition, necessary condition)
Question 3.3: What are the two cases in which the first order condition is not satisfied? For each case,
determine what the producer should do. (What to do when the choice variable does not appear in the
first order condition)
Question 3.4: Suppose the first order condition is satisfied.
does in this situation? (An example of indeterminacy)
Why does it not matter what the producer
Question 3.5: What are the Kuhn Tucker Conditions for profit maximization for this problem? What
case does the Kuhn Tucker conditions rule out? (Kuhn-Tucker Conditions).
Question 3.6: Do you see that constructing an optimization problem that satisfies the Kuhn Tucker
conditions ensures that the problem will have a solution? What restriction would you need to put on the
parameters P , W , and A in order for the Kuhn-Tucker conditions to be satisfied? What restriction
would you need to put on the parameters P , W , and A in order for the first order condition to be
satisfied? (Placing restrictions on parameters so that your model is functional)
Question 3.7: What is the “second order condition” or “sufficient condition” for profit maximization for
this problem? Show that it is not satisfied. (Second order condition)
4. Constructing an optimization problem that has a unique solution
(4.1)
Y  AL ,
0  1
Question 4.1: What is the marginal product of labor? (Definition of Marginal Product, partial
Derivative)
Question 4.2: What is the average product of labor? (Definition of average product)
Question 4.3: Plot the production function (4.1) as it depends upon the labor level L . In the diagram,
find a way to represent the marginal product of labor for a given labor level, and do the same for the
average product of labor.
Question 4.4: How does the growth rate of output relate to the growth rate of labor? (Natural log, time
derivative, differential)
Question 4.5: What is the elasticity of output with respect to labor input? (Definition of elasticity,
Natural log, time derivative, differential)
Question 4.6: Why is the production function (4.1) with the restriction 0    1 probably more
representative of real world production process than the production function (1.1), where   1 ?
(Usefulness of restrictions)
Question 4.7: Use equations (4.1) and (2.1) to write down a profit maximization problem for a firm,
and present the problem as a one variable mathematical programming problem. (Writing down an
optimization problem)
Question 4.8: Obtain the first and second order conditions for profit maximization for this problem.
(First and second order conditions)
Question 4.9: Explain why the second order condition always holds (Importance of parameter
restrictions)
Question 4.10: Use the first order condition to obtain the optimal amount of labor the employer should
hire. (Deriving a labor demand curve)
Question 4.11: Use the first order condition to show that labor is paid a real wage that is equal to the
marginal product of labor (Definition of real wage)
Question 4.12: Use the first order condition to show that the parameter  can be interpreted as the
share of output being paid to labor. (Definition of labor share of output)
5. Comparative Statics
Question 5.1: Use the first order condition obtained in answering question 4.7 to obtain the equation
PY  WL . (Obtaining an equation for a model from an optimization problem)
Question 5.2: Use the following equations to create a model of production and employment. Define
and classify each variable, assuming the model determines the economy’s output level and employment
level. (Endogenous variable, exogenous variable, parameter)
(5.1)
Y  AL ,
(5.2)
PY  WL
0  1
(Answer, except for definitions)
Classification of variables:
Variables (3): Y , L , A , P , W
Endogenous (2): Y , L
Exogenous (3): A , P , W
Parameters (1): 
Question 5.3: Totally differentiate equations (5.1) and (5.2). (Use of natural log, total differentiation)
Question 5.4: Convert the total differentials you obtained for question 5.3 into partial differentiation,
by assuming the parameter  is constant. (partial differentiation)
Question 5.5: Treat the two partial differentials you obtained from Question 5.4 as a system of two
equations with the two unknowns dY and dL . Using substitution, solve the two equations for dY and
dL . (Using substitution to solve as system of equations)
Question 5.6: Prove that the two equations of question 5.5 are linearly independent, and then use
Cramer’s rule to solve the system. (Cramer’s rule, checking a system for linear dependence and nonsingularity)
Questions 5.7: Use the solutions you obtained from questions 5.5 or 5.6 to find the comparative static
multipliers for the model. (Total derivative, independence of exogenous variables, comparative static
multipliers
Questions 5.8: Determine the sign of each comparative static multiplier, and interpret each multiplier
(i.e., explain what you have learned). In particular, discuss what happens to the magnitude of each
multiplier as the parameter  increases toward one. (Interpreting comparative static multipliers)
Question 5.9: The model using equations (5.1) and (5.2) is said to be “in the levels.” Convert this
model into a model that is “in the growth rates.” Perform a comparative static analysis of this model in
the growth rates.