Lagrangian Analysis

One of the prerequisites for this course is a solid background in mathematics and in particular the Lagrange multiplier technique.

This finds the maximum or minimum of a multivariable function f(x,y...) where there is a constraint on the input values you are allowed to use, in the form of g(x, y, ...) = c, where g represents another multivariate function with the same input space as f and c is some constant.

The objective is to find points where f and g are tangent to each other which, in economics, often is a Pareto value.

The process sets the gradient of a certain function, called the Lagrangian, equal to the zero vector.

When you want to maximize or minimize a multivariable function f(x,y, ..) subject to the constraint that another multivariable function equals a constant, g(x,y..) = c do the following.

1. Introduce a new variable, lamda (the Lagrange multiplier), and define a new function LG (the Lagrangian).

LG(x, y, .. lamda) = f(x,y ..) - lamda(g(x, y ..) - c)

2. Set the gradient of LG equal to the zero vector.

∇LG = (x, y, ... lamda) = 0

3. Consider each solution, which will look something like (x_0, y_0, ... lamda_0) then plug it into f, since f does not have lamda as an input. Whichever one gives the greatest (or smallest) value is the maximum (or minimum) point.

Consider the following example from Jeffrey M. Perloff, Microeconomics (6th edition), Addison-Wesley, 2012, p A6-A9

Lisa’s objective is to maximize her utility, U(B, Z), subject to (s.t.) a budget constraint:

max(bz) U(B, Z)

s.t. Y = p_B B + p_Z Z, (4B.1)

where B is the number of burritos she buys at price p_B, Z is the number of pizzas she buys at price p_Z , Y is her income, and Y = p_B B + p_Z Z is her budget constraint (her spending on burritos and pizza can’t exceed her income). The mathematical statement of her problem shows that her control variables (what she chooses) are B and Z, which appear under the "max" term in the equation. We assume that Lisa has no control over the prices she faces or her budget.

To solve this type of constrained maximization problem, we use the Lagrangian method:

max(B,Z,λ)λ = U(B, Z) - λ(p_B B + p_Z Z - Y) (4B.2)

where λ is called the Lagrange multiplier. With normal-shaped utility functions, the values of B, Z, and λ determined by the first-order conditions of this Lagrangian problem are the same as the values that maximize the original constrained problem. The first-order conditions of Equation 4B.2 with respect to the three control variables, B, Z, and λ are:

∂LG/∂B = MU_B(B,Z) - λp_B = 0 (4B.3)
∂LG/∂Z = MU_Z(B,Z) - λp_Z = 0 (4B.4)
∂LG/∂λ = Y - p_B B - p_Z Z = 0 (4B.4)

[Where ∂ is the partial derivative,'tho', ≡ is 'identical to' ]

where MU_B (B, Z) ≡ U(B, Z)/∂B is the partial derivative of utility with respect to B (the marginal utility of B) and MU_Z (B, Z) is the marginal utility of Z. Equation 4B.5 is the budget constraint. Equations 4B.3 and 4B.4 say that the marginal utility of each good equals its price times λ.

What is λ? If we equate Equations 4B.3 and 4B.4 and rearrange terms, we find that

λ = MU_B/p_B = MU_Z/p_Z

Because the Lagrangian multiplier, λ, equals the marginal utility of each good divided by its price, λ equals the extra pleasure one gets from one’s last dollar of expenditures. Equivalently, λ is the value of loosening the budget constraint by one dollar. 5 Equation 4B.6 tells us that, to maximize her utility, Lisa should pick a B and Z so that, if she got one more dollar, spending that dollar on B or on Z would give her the same extra utility.

There is an alternative interpretation of this condition for maximizing utility. Taking the ratio of Equations 4B.3 and 4B.4 (or rearranging 4B.6), we find that

MU_Z/MU_B = p_Z/p_B (4B.6)

The left side of Equation 4B.7 is the absolute value of the marginal rate of substitution, MRS = -MU_Z /MU_B, and the right side is the absolute value of the marginal rate of transformation, MRT = -p_Z/p_B. Thus, the calculus approach gives us the same condition for an optimum that we derived using graphs. The indifference curve should be tangent to the budget constraint: The slope of the indifference curve, MRS, should equal the slope of the budget constraint, MRT.

For example, suppose that the utility is Cobb-Douglas, as in Equation 4A.3: U = AB^α Z^β . The first-order condition, Equation 4B.5, the budget constraint, stays the same, and Equations 4B.3 and 4B.4 become

∂LG/∂B = α( U(B, Z)/B) = λp_B = 0 (4B.8)
∂LG/∂Z = β( U(B, Z)/B) = λp_Z = 0 (4B.9)

Using Equations 4B.8 and 4B.9, we can write Equation 4B.6 as

λ = α ( U(B, Z)/p_B B) = β ( U(B, Z)/p_Z Z)

Taking the ratio of Equations 4B.8 and 4B.9 and rearranging terms, we find that

β p_B B = α p_Z Z (4B.10)

Substituting Y - p_B B for p_Z Z, using Equation 4B.5, into Equation 4B.10 and rearranging terms, we get

B = (α/(α+β)) * (Y /p_B) (4B.11)

Similarly, by substituting Equation 4B.11 into Equation 4B.10, we find that

Z = (β/(α+β)) * (Y / p_Z) 4B.12)

Thus, knowing the utility function, we can solve the expression for the B and Z that maximize utility in terms of income and prices.

Equations 4B.11 and 4B.12 are the consumer’s demand curves for B and Z, respectively.

If α = β = 1/2 , A = 20, Y = 80, and p_Z = p_B = 10, then B = Z = 4 and the value of loosening the budget constraint is λ = MU_B /p_B = MU_Z /p_Z = 10/10 = 1. If p_B rises to 40, then Z = 4, B = 1, and λ = 20/40 = 5/10 = 12 .

Reading: Nicholson, W., Intermediate Microeconomics and its application (sixth edition), Dryden Press, 1994

Chapter One: Economic Models

"The mostly widely quoted definition describes economics as the 'study of the allocation of scarce resources among alternative end uses.' This definition introduces two important aspects of society that concern economists: *scarce resources* and *alternative end uses*."
p3

"Microeconomics [is] The study of economic choices individuals and firms make and how those choices create markets"
p3

"Models [provide] Simple theoretical descriptions that capture the essentials of how the economy works"
"Supply and demand [is] A model describing how a good's price is determined by the behaviour of the individuals who buyt the good and the firms that sell it"
p5

"..Smith believed that most differences in relative prices could ultimately be tracedback to differences in underlying labor costs.." p6-7

Ricardo generally agreed with Smith but added diminishing returns. "Law of diminishing returns [is the] Hypothesis that for the cost associated with producing one more unit of a good rises as more of that good is produced" p8

Marshall added the principle by focussing on the last, or maginal, unit bought. Principles of Economics (1890) showed "how the forces of demand and supply *simultaneously* determine price" p9

"Equibrium price [is] The price at which the quantity demanded by buyers of a good is equal to the quantity of the good supplied by sellers" p10

"Partial equilibrium model [is] An economic model of a single market" p13

"General equilibrium model [is] An economic model of a complete system of markets" p15

"Production possibility frontier [is] a graph showing all possible combinations of goods that can be produced with a fixed amount of resources" p15

"Opportunity cost [is] The cost of a good or service as measured by the alternative uses that are forgone by producing the good or service" p16

"Direct approach [is] A method of verifying economic models that examines the validity of the assumption which the model is based"
"Indirect approach [is] A method of verifying economic models that asks if the model can accurately predict real-world events" p18

"Positive economic analysis [are] Theories that explain how resources actually are used in an economy"
"Normative analysis [are] Theories that make judgments about how the economy's resouces should be used"
p19

Chapter Two: Mathematical Tools

Functional relationships, Y=f(X), value of Y depends on the value of X. Value of Y is dependent on X, value of X is the independent variable.

Y as a linear function, Y = a + bX, Y as a quadratic function, Y = a + bX + cX^2.

Linear functions are represented by straight-line graph.

The intercept of Y when X equal 0.

The slope is the direction of the graph. Shows the change in Y that results from a change in X; delta Y / delta X = slope

Multi-variable functions are expressed, Y = f(X, Z)

"Simultaneous equations [are] A set of equations with more than one vairable that must be solved together for a particular solution" p45

"Simultaneous equations determine solutions for two (or more) variables that satisfy all the equations. An important use of such equations is to show how supply and demand cruves determine equilibrium prices. For that reasons, such equations are widely encountered in economics." p50

Almost all econometric models deal with positive values. Also, economists tend to put the dependent variable on the horizontal access.