specific topics in optimisation

Lecturer: Farzad Javidanrad

Specific Topics in Optimisation(for MSc & PhD Business, Management & Finance Students)

(Autumn 2014-2015)

Envelope Theorem & Optimisation Under Intertemporal Choice

• Parameter: The variable that helps in defining a mathematical/statistical model but its values are given or determined outside of the model.

In the linear model 𝑦 = 𝑚𝑥 + 𝑐,

𝑦 and 𝑥 variables (their values determined inside the model)

𝑐 and 𝑚 parameters (y-intercept and slope parameter respectively)

Basic Terminologies

Adopted and altered from http://www.met.reading.ac.uk/pplato2/h-flap/math2_2.html

• Objective Function: A function which is going to be optimized.

• Decision Variables: The independent (or explanatory) variables in a model.

• Optimal Values: Maximum or minimum values of variables in a model..

• The envelope theorem focus on the relation between the optimal value of the dependent variable in a particular function and the parameter(s) of that function. How the optimal value does change when a parameter (or one of parameters of the model) changes?

Consider the following quadratic function which contains on parameter

𝑦 = 𝑥2 − 𝑏𝑥

The optimal value of 𝑥 = 𝑥∗ can be obtained by the first order condition 𝑓′ 𝑥 = 0:

2𝑥 − 𝑏 = 0 → 𝑥∗ =𝑏

2and 𝑦∗ = 𝑥∗2 − 𝑏𝑥∗ ⟹ 𝑦∗ = −

𝑏2

4

Obviously, 𝑥∗ and consequently, 𝑦∗ are functions of 𝑏, i.e. 𝑥∗ = 𝑥 𝑏 , 𝑦∗ = 𝑦(𝑏).

Basic Terminologies

• The following table shows how the optimal values of 𝑥 and 𝑦 change when 𝑏 changes.

Investigate how the optimal value of the dependent variable in the quadratic function 𝑦 = 𝑎𝑥2 + 𝑏𝑥 + 𝑐 changes when each parameter 𝑎, 𝑏 and 𝑐 change.

Basic Example

𝒃𝒙∗ =

𝒃

𝟐𝒚∗ = −

𝒃𝟐

𝟒

-2 -1 -1

-1 -0.5 -0.25

0 0 0

1 0.5 -0.25

2 1 -1 -1.2

-1

-0.8

-0.6

-0.4

-0.2

0

-3 -2 -1 0 1 2 3

b

𝒚

• Case 1 (unconstraint model): Consider a general optimization model with two independent (decision) variables 𝑥 and 𝑦 and one parameter 𝛼:

𝑧 = 𝑓(𝑥, 𝑦, 𝛼)

The first-order necessary conditions are: 𝜕𝑧

𝜕𝑥= 𝑓𝑥 = 0 𝑎𝑛𝑑

𝜕𝑧

𝜕𝑦= 𝑓𝑦 = 0

Assume the second-order conditions hold, the optimal values of 𝑥 and 𝑦 for any given value of 𝛼 are the function of 𝛼, i.e. 𝑥∗ = 𝑥∗ (𝛼) and 𝑦∗ = 𝑦∗ 𝛼 . By substituting these optimal values into the objective function we obtain the indirect objective function, which is an indirect function of the parameter 𝛼 :

𝜑 𝛼 = 𝑓(𝑥∗ 𝛼 , 𝑦∗ 𝛼 , 𝛼)

The Envelope Theorem (Unconstraint Model)

Objective function with two decision variables and one parameter

𝜑 𝛼 gives the optimal value of 𝑧, for any given value of 𝛼

• Some of the properties of the indirect objective functions are:

A. For any given set of parameters, the objective function is optimized (maximized or minimized)

B. When the parameters vary, the indirect objective function outlines the optimum values of the direct objective function.

C. The graph of the indirect objective function is an envelope of the graphs of optimized objective functions when parameters vary.

• In order to find the variation of indirect objective function 𝜑 𝛼 in terms of 𝛼, we need to differentiate it:


𝜕𝜑

𝜕𝛼= 𝑓𝑥

𝜕𝑥∗

𝜕𝛼+ 𝑓𝑦

𝜕𝑦∗

𝜕𝛼+ 𝑓𝛼

However, we know that at the optimal values 𝑓𝑥 = 𝑓𝑦 = 0, so:

𝜕𝜑

𝜕𝛼= 𝑓𝛼

This equation says that the rate of change of the optimal value of the dependent variable 𝑧 in terms of 𝛼, where the independent variables 𝑥 and 𝑦 are optimally obtained (for any given 𝛼), can be calculated directly by differentiating the indirect objective function 𝜑 in terms of 𝛼.

• Geometric Interpretation: If we plot 𝜑 𝛼 for various 𝛼′𝑠,

the plot will be the envelope of the different curves obtained

from the family function 𝑧 = 𝑓(𝑥, 𝑦, 𝛼), when 𝛼 changes.

• Here we have a function with one variable and one parameter.


𝑧 = 𝑓(𝑥, 𝛼)

𝛼 𝑥∗(𝛼)

𝜑 𝛼

Adopted and altered from http://exactitude.tistory.com/229.

Economic Example• Here we have the relation between short-run (SR) and long-run (LR) average cost curves when the

size of production (parameter)

changes.

• The long-run average cost (LRAC) is the envelope

curve for different short-run average cost (SRAC)

curves when the size of the company changes in

the long-run.

• The theorem can be generalised to include more than one parameter. For example, in microeconomics, a price-taker producer face with the following profit function:

𝜋 = 𝑝. 𝑓 𝐿, 𝐾 − 𝑤𝐿 − 𝑟𝐾

Where 𝑝 is the output price, 𝑤 and 𝑟 are factor production prices or simply the cost of employing labour and capital in the production process. They are parameters of the model.

Size of the company (production) increasing

Ad

op

ted

and

altere

d fro

m h

ttp://m

aaw.in

fo/Eco

no

mie

sOfScale

No

te.h

tm

= 𝜑 𝛼

• To know how profit 𝜋 changes when one of these parameters change, say 𝑤, we can simply differentiate the objective function with respect to that parameter, assuming other parameters and variables constant, i.e.:

𝜕𝜋

𝜕𝑤= −𝐿

• Using the envelope theorem leads us to the same result but 𝐿 should be evaluated at its optimal value, 𝐿∗ (the value that maximise 𝜋 at any given values of 𝑝, 𝑤 and 𝑟 ), because:

Through profit maximising we reach to the optimal level of demand for labour and capital as:𝐿∗ = 𝐿∗ 𝑝,𝑤, 𝑟 𝑎𝑛𝑑 𝐾∗ = 𝐾∗(𝑝, 𝑤, 𝑟)

So, the indirect profit function will be:𝜋∗ 𝑝,𝑤, 𝑟 = 𝑝. 𝑓 𝐿∗, 𝐾∗ − 𝑤𝐿∗ − 𝑟𝐾∗

Now,𝜕𝜋∗

𝜕𝑤= 𝑝 𝑓𝐿

𝜕𝐿∗

𝜕𝑤+ 𝑓𝐾

𝜕𝐾∗

𝜕𝑤− 𝑤

𝜕𝐿∗

𝜕𝑤− 𝐿∗ − 𝑟

𝜕𝐾∗

𝜕𝑤𝜕𝜋∗

𝜕𝑤= 𝑝𝑓𝐿 − 𝑤

𝜕𝐿∗

𝜕𝑤+ 𝑝𝑓𝐾 − 𝑟

𝜕𝐾∗

𝜕𝑤− 𝐿∗

Knowing that at the optimal values of 𝐿 and 𝐾 the terms in parentheses are zero, therefore, we will reach to: 𝜕𝜋∗

𝜕𝑤= −𝐿∗


This means if this particular employer faces with the increase in wage rate by £1, in case he/she has employed 1000 workers and this is the optimal level of labour which makes the maximum profit, he/she will lose -£1000 by £1 increase in the wage.

A profit-maximising firm, in this situation would reduce the number of workers up to a point that the increase in cost is compensated by reduce in payment in total.

• Case 2 (constraint model): It rarely happens to optimise a function without any constraint. In the real world any optimisation is subject to one or more constraints. Again, imagine a simple example of optimisation with two independent variables and one parameter 𝛼:

𝑍 = 𝑓(𝑥, 𝑦, 𝛼)Subject to the constraint

𝑔 𝑥, 𝑦, 𝛼 = 𝑐The Lagrangian Function can be written as:

𝐿(𝑥, 𝑦, 𝜆, 𝛼) = 𝑓 𝑥, 𝑦, 𝛼 + 𝜆[𝑐 − 𝑔 𝑥, 𝑦, 𝛼 ]

Setting the first partial derivatives of 𝐿 equal to zero,

𝐿𝑥 = 𝑓𝑥 − 𝜆𝑔𝑥 = 0𝐿𝑦 = 𝑓𝑦 − 𝜆𝑔𝑦 = 0

𝐿𝜆 = 𝑔(𝑥, 𝑦, 𝛼) = 𝑐

Assuming the second-order conditions are satisfied, solving these equations gives the optimal values of 𝑥, 𝑦 and 𝜆, in terms of the parameter 𝛼.

The Envelope Theorem (Constraint Model)

A

𝑐 is a constant

𝜆 is the Lagrange Multiplier

𝑥∗ = 𝑥∗(𝛼)𝑦∗ = 𝑦∗(𝛼)𝜆∗ = 𝜆∗(𝛼)

And the indirect objective function will be:

𝑍∗ = 𝑓 𝑥∗ 𝛼 , 𝑦∗ 𝛼 , 𝛼 = 𝜑(𝛼)

𝜑(𝛼) is the maximum value of 𝑍 for all values of 𝑥 and 𝑦 that satisfy the constraint and also it shows the maximum value of 𝑍 for any value of 𝛼.

How does 𝜑(𝛼) change when 𝛼 changes? To answer this, we need somehow to consider the constraint function 𝑔 𝑥, 𝑦, 𝛼 = 𝑐 in our analysis. To do that, we make indirect Lagrangian function 𝐿∗, using optimal values 𝑥∗ and 𝑦∗ and then differentiate the new indirect function with respect to 𝛼.

𝐿∗ 𝑥∗ 𝛼 , 𝑦∗ 𝛼 , 𝜆, 𝛼 = 𝜑 𝛼 + 𝜆[𝑐 − 𝑔 𝑥∗ 𝛼 , 𝑦∗ 𝛼 , 𝛼 ]

By maximising 𝐿∗ we have, in fact, maximised 𝜑 𝛼 , why?


Indirect objective function

= 0

• By differentiating 𝐿∗ with respect to 𝛼 we have:

𝜕𝐿∗

𝜕𝛼= 𝑓𝑥 − 𝜆𝑔𝑥

𝜕𝑥∗

𝜕𝛼+ 𝑓𝑦 − 𝜆𝑔𝑦

𝜕𝑦∗

𝜕𝛼+ 𝑓𝛼 − 𝜆𝑔𝛼

Or

𝜕𝐿∗

𝜕𝛼= 𝐿𝑥

𝜕𝑥∗

𝜕𝛼+ 𝐿𝑦

𝜕𝑦∗

𝜕𝛼+ 𝐿𝛼

Where 𝐿𝑖 is the partial derivative of the Lagrangian function with respect to element 𝑖.

Base on the first order conditions , 𝐿𝑥 and 𝐿𝑦 are zero, then:

𝜕𝐿∗

𝜕𝛼= 𝐿𝛼 = 𝑓𝛼 − 𝜆𝑔𝛼

• Using the Lagrange method allow us to change the optimisation of a constraint model to the optimisation of an unconstraint model.

• Note: If the parameter enters just in the objective function the result for constraint and unconstraint models are similar.


A

• Solving the first two equations of , simultaneously (by ignoring 𝛼, which means considering it as a fixed value without any variation), gives the value of 𝜆 as:

𝜆 =𝑓𝑥𝑔𝑥

=𝑓𝑦

𝑔𝑦

By re-arranging the terms, we will have:

𝜆 =𝑓𝑥𝑓𝑦

=𝑔𝑥𝑔𝑦

This means 𝜆 is the slope of the level curve of the objective function 𝑓(𝑥, 𝑦) at the optimal points

𝑥∗and 𝑦∗, which should be equal to the slope of the level curve of the constraint function 𝑔(𝑥, 𝑦) at those points.

Interpretation of the Lagrange Multiplier A

Both pictures adopted from http://en.wikipedia.org/wiki/Lagrange_multiplier

• But more information about 𝜆 can be obtained through the envelope theorem:

By solving all equations in simultaneously (again consider 𝛼 as a constant), the optimal values for variables and 𝜆 will be:

𝑥∗ = 𝑥∗ 𝑐 , 𝑦∗ = 𝑦∗ 𝑐 , 𝜆∗ = 𝜆∗ 𝑐

Substituting these results into the Lagrangian function 𝐿(𝑥, 𝑦, 𝜆), we have:

𝐿∗ 𝑥∗ 𝑐 , 𝑦∗ 𝑐 , 𝜆∗ 𝑐 = 𝑓 𝑥∗ 𝑐 , 𝑦∗ 𝑐 , 𝜆∗ 𝑐 + 𝜆∗ 𝑐 [𝑐 − 𝑔 𝑥∗ 𝑐 , 𝑦∗ 𝑐 ]

Differentiating with respect to 𝑐, we have:

𝜕𝐿∗

𝜕𝑐= 𝑓𝑥

𝜕𝑥∗

𝜕𝑐+ 𝑓𝑦

𝜕𝑦∗

𝜕𝑐+ 𝑐 − 𝑔 𝑥∗, 𝑦∗

𝜕𝜆∗

𝜕𝑐+ 𝜆∗ [1 − 𝑔𝑥

𝜕𝑥∗

𝜕𝑐− 𝑔𝑦

𝜕𝑦∗

𝜕𝑐]

By rearranging, we get:

𝜕𝐿∗

𝜕𝑐= 𝑓𝑥 − 𝜆∗𝑔𝑥

𝜕𝑥∗

𝜕𝑐+ 𝑓𝑦 − 𝜆∗𝑔𝑦

𝜕𝑦∗

𝜕𝑐+ 𝑐 − 𝑔 𝑥∗, 𝑦∗

𝜕𝜆∗

𝜕𝑐+ 𝜆∗

Interpretation of the Lagrange Multiplier

A

• The terms in the brackets are zero (why?), so;

𝜕𝐿∗

𝜕𝑐= 𝜆∗(𝑐)

• Considering the fact that 𝑐 comes through the constraint (and not through the objective function), the change of the maximum (or minimum) value of the objective function 𝑓 with respect to change of 𝑐 (which is the 𝜆∗), can be called as the marginal value of the 𝑐, and in some cases related to the production function, it can be called as the shadow price of the resources.

• If the objective function is the utility function 𝑈 = 𝑈 𝑥, 𝑦 and the constraint is the budget line 𝑥. 𝑃𝑥 + 𝑦. 𝑃𝑦 = 𝑚, then 𝜆∗ can be interpreted as the marginal utility of money spent on 𝑥 and 𝑦 .

• Note: In the dual analysis of the above maximisation, when we try to minimise the expenditure 𝑥. 𝑃𝑥 + 𝑦. 𝑃𝑦 subject to maintain the utility at a specific level 𝑈 𝑥, 𝑦 = 𝑈∗, the new Lagrange

multiplier 𝜆 is the inverse of the Lagrange multiplier in the primal analysis 𝜆∗, i.e.: 𝜆 = 1 𝜆∗

Interpretation of the Lagrange Multiplier

• One of the important optimisation cases in microeconomics is when consumers try to maximise their utilities subject to their wealth constraints. This case has many implications in finance theory when an investor with a multi-period planning horizon is going to split his/her wealth between present consumption and investment on different assets (future consumption).

• Here we focus on two period maximization but it can be easily extended to n-period case. So, we can define the followings:

Two periods: Period 1 (the present) and period 2 (the future, e.g. next year)

𝑦1=Income of period 1 and 𝑦2= Income of period 2

𝑐1=Value of the consumption in period 1 and 𝑐2= Value of the consumption in period 2

Saving can happen in period 1 but not in period 2 as it is the final period, so the only saving we have is 𝑠 =𝑦1 − 𝑐1.

The investor can borrow from or lend to the capital market at the interest rate 𝑟. He/she is able to have extreme choices: sacrifice the current consumption and invest all his/her current and future income (wealth) in the financial market (so, he/she is a lender in period 1) or sacrifice the future consumption and use all the life-time money on the present consumption (so, he/she is a borrower in period 1)

Intertemporal Choice

Deriving the Intertemporal Budget Constraint

• Period 2 budget constraint is:

𝑐2 = 𝑦2 + 1 + 𝑟 𝑠

= 𝑦2 + 1 + 𝑟 (𝑦1 − 𝑐1)

• By rearrange the terms, we have:

1 + 𝑟 𝑐1 + 𝑐2 = 𝑦2 + 1 + 𝑟 𝑦1

By divide through by 1 + 𝑟 , we have:

𝑐1 +𝑐2

1 + 𝑟= 𝑦1 +

𝑦21 + 𝑟

= 𝑊

present value of lifetime consumption

present value of lifetime income

Total wealth of the investor

The Intertemporal Budget Constraint

• The budget constraint shows all combinations of 𝑐1and 𝑐2that just exhaust the consumer’s resources.

𝑐1

𝑐2

𝑦1

𝑦2

Borrowing

Saving

Consumption = income (in both periods)

The point (𝑦1, 𝑦2) is always on the

budget line because 𝑐1=𝑦1, 𝑐2=𝑦2• The Budget Constraint:

𝑐1 +𝑐2

1 + 𝑟= 𝑦1 +

𝑦21 + 𝑟

Or

(𝑦1−𝑐1) +𝑦2 − 𝑐21 + 𝑟

= 0

𝑦2 + 1 + 𝑟 𝑦1

𝑦1 +𝑦2

1 + 𝑟Adopted and altered from Chapter 17 Mankiw produced by Ron Cronovich © 2010 Worth Publishers

The Intertemporal Budget Constraint

The slope of the budget line

equals 𝑑𝑐2

𝑑𝑐1= −(1 + 𝑟)

𝑐1

𝑐2

𝑦1

𝑦2

1

𝑐1 +𝑐2

1 + 𝑟= 𝑦1 +

𝑦21 + 𝑟

Considering no change in income(𝑑𝑦1 = 𝑑𝑦2 = 0):

𝑑𝑐1 +𝑑𝑐21 + 𝑟

= 0

𝑑𝑐2𝑑𝑐1

= −(1 + 𝑟)

1 + 𝑟

Adopted and altered from Chapter 17 Mankiw produced by Ron Cronovich © 2010 Worth Publishers

Consumer Preferences

• The utility function 𝑈 = 𝑈(𝑐1, 𝑐2) shows the utility associated to each value of 𝑐1and 𝑐2.

• An indifference curve shows all combinations of 𝑐1 and 𝑐2 that make the consumer equally happy, i.e. equal level of utility.

𝑐1

𝑐2

IU1

IU2

Higher indifference

curves represent higher

levels of happiness.

Adopted from http://www.cengage.com/economics/book_content/032427470X_nechyba/chapters/partA.html


Without any money constraint, a rational consumer try to reach to the highest possible utility level.

Utility Surface

Indifference Utility Curves

• Marginal Rate of Substitution (MRS ): the amount of 𝑐2 (in terms of value) that the consumer would be willing to substitute (sacrifice) for getting one unit of 𝑐1.

𝑈∗ = 𝑈 𝑐1, 𝑐2

𝑑𝑈∗ =𝜕𝑈

𝜕𝑐1𝑑𝑐1 +

𝜕𝑈

𝜕𝑐2𝑑𝑐2

As 𝑈∗ is constant (for indifference curve) so, 𝑑𝑈∗ = 0, and we have:

𝜕𝑈

𝜕𝑐1𝑑𝑐1 +

𝜕𝑈

𝜕𝑐2𝑑𝑐2 = 0

And

𝑀𝑅𝑆 =𝑑𝑐2𝑑𝑐1

= −

𝜕𝑈𝜕𝑐1𝜕𝑈𝜕𝑐2

= −𝑀𝑈𝑐1𝑀𝑈𝑐2

IC1

The slope of an

indifference curve at

any point is the MRS

at that point.1

MRS

Consumer Preferences


𝑐1

𝑐2

Optimization

• The optimal O(𝑐1,𝑐2) is where the budget line just touches the highest indifference curve.

• At the optimal point the slope of the indifference curve is equal to the slope of the budget line.

𝑐1

𝑐2

O

At the optimal point, MRS = −(1 + 𝑟)

Adopted from http://www2.hawaii.edu/~fuleky/anatomy/anatomy.html


Maximisation of the Intertemporal Utility• Maximise: 𝑈 = 𝑈(𝑐1, 𝑐2) , subject to:(𝑦1−𝑐1) +

𝑦2−𝑐2

1+𝑟= 0

• The Lagrange function for this problem is:

𝐿 𝑐1, 𝑐2, λ = 𝑈 𝑐1, 𝑐2 + λ[(𝑦1−𝑐1) +𝑦2 − 𝑐21 + 𝑟

]

Producing the first-order conditions:

𝐿1 =𝜕𝑈

𝜕𝑐1− 𝜆 = 0

𝐿2 =𝜕𝑈

𝜕𝑐2−

𝜆

1 + 𝑟= 0

𝐿𝜆 = (𝑦1−𝑐1) +𝑦2 − 𝑐21 + 𝑟

= 0

Solving for the first two equations:

𝜕𝑈

𝜕𝑐1𝜕𝑈

𝜕𝑐2

= 1 + 𝑟 ⟹𝑀𝑈𝑐1𝑀𝑈𝑐2

= 1 + 𝑟In the optimal point the slope of the indifference curve should be equal to the slope of the constraint.

• Assuming the second-order conditions hold, from the first-order equations we can reach to the consumption (Marshallian demand) functions:

𝑐1 = 𝑐1 𝑟, 𝑦1, 𝑦2𝑐2 = 𝑐2 𝑟, 𝑦1, 𝑦2

• In these equations, consumption in each period relates inversely to the interest rate 𝑟 and directly to the levels of present and future incomes. So, the current consumption 𝑐1is not confined by the current income 𝑦1, because individuals can borrow or lend between two periods and they choose their consumption strategy based on the present value of their lifetime income.

• If 𝑦1 or 𝑦2 increases for any reason, the budget constraint

shifts to the right and allows more consumption in both

periods as the investor can be on the higher level of utility.

Deriving the Intertemporal Consumption Functions

𝑐1

𝑐2

A

The Impact of the Change in Interest Rate

𝑐1

𝑐2

𝑦1

𝑦2

A

B

• But the change of interest rate is different.

• An increase in 𝑟 rotates the budget line

around the point (𝑦1, 𝑦2 ).

• This increase encourages the investor to reduce the present consumption and invest the money in the capital market for a higher consumption in the future. This means that the investor is on the new budget line on point B, where 𝑐2 is higher than 𝑐1.


The new budget line (the red line)is the rotation of the previous line around the

point (𝑦1, 𝑦2 ), such that, lesser 𝑐1but

greater 𝑐2 available.

Budget Line with a Constraint on Borrowing

𝑐1

𝑐2

𝑦1

𝑦2

• The borrowing constraint does not allow the investor to increase his/her consumption beyond the income limit:

𝑐1 ≤ 𝑦1

• So, the extension of the budget line below the point 𝑦1, 𝑦2 does not exist. But it does not make any difference for an investor who is more saver than consumer in the first period as it is already 𝑐1 ≤ 𝑦1.

c1

c2𝐴


Budget Line with a Constraint on Borrowing

• If our investor prefer the present consumption to the future consumption he/she wish to be at point D but the borrowing constraint does not allow that therefore, he/she must choose to reduce his/her utility level to point E.

• In this case we have a corner solution, 𝑐1 = 𝑦1 and 𝑐2 = 𝑦2.

• Point E is the best possible solution for our investor as he/she cannot choose any point below or above E. Why?

𝑐1

𝑐2

𝑦1

DE

𝑦2


specific topics in optimisation

Education