Microeconomics and Profit Model

by Max Yi Ren

Disclaimer

This lecture note is created by Dr. Namwoo Kang and Dr. Panos Papalambros. The spreadsheet for demonstration is from Dr. Alex Burnap and Dr. Panos Paplambros. Others from the ODE lab also contributed to these materials throughout years.

The following activities are based on the course spreadsheet.

Activity 1: Market simulation with pricing

Consider you are competing companies selling exactly the same product, say a teddy bear. The consumer’s part-worth on the price is -1. Discuss among yourselves to come up with a MSRP.

Note: The market share of your product depends on not only the price you set, but also those of your competitors. A common model for predicting the market share is the following:

where $s_i$ is the $i$th companies market share, $\theta$ is the part-worth of the price (here we assume a homogeneous preference model for all consumers), $p_i$ is the price set by company $i$.

In this case, the profit of a company is simply $s_ip_i$.

• Task 1: Compete in a simulated market and see who has the highest profit.
• Task 2: Learn from your previous lesson and compete again. How will you change your price?
• Task 3: If all competitors are driven by profit, does there exists an equilibrium of prices, i.e., a set of prices that no one is motivated to change?

Activity 2: Market simulation with pricing and product positioning

Now include design in your decision making: Let the design variables be the size of the bear, and the fluffiness. The artificial part-worths are -0.1, 0.0008, and 1.5 for price, size and fluffiness. The unit cost is 0.001 (dollar per inch cubed) and 5 (dollar per fluffiness). The range of size is 1000 to 100000, and 1 to 10 for fluffiness. Make a decision on your MSRP, size and fluffiness.

Note: The profit in this case will be $s_i(p_i-c_i)$, where $c_i$ is the unit cost.

Activity 3: Optimal decision making

From Activity 2, you might find it hard to make a decision on what design attribute levels and prices to set, in order to maximize your profit given all your competitors. To this end, develop an optimization routine in MATLAB to automatically adjust your design decisions according to the market.

A template of the code can be found here.

Once done, run the competition again, with your algorithm rather than gut feeling.

Brief intro of fmincon: The optimizer for nonlinear constrained problems

Nonlinear constrained optimization problems have the following general form:

Here $f(x)$, $g(x)$ and $h(x)$ are the objective, the inequality constraints, and the equality constraints, respectively. $x$ are the variables. In our case, the objective is the profit, and constraints are related to bounds of design variables and other engineering considerations.

The usage of fmincon can be found by typing help fmincon in the MATLAB terminal. The most common call has the following form

[x,f] = fmincon(@(x)objective(x,arg1,arg2,...),x0,A,b,Aeq,beq,lb,ub,@(x)nonlinear)

Here arg1, arg2 are inputs to the objective function. In our case, these could be part-worths and product costs that influences the profit, yet are not part of the variables. Ax <= b are the linear inequality constraints; Aeq x = beq are the equality constraints. lb and ub are the lower and upper bounds on the variables. @(x)nonlinear is the function handle for nonlinear constraints.

Activity 4: The equilibrium

You may noticed from Activity 3 that there exists an equilibrium of the market. Conduct analysis to find out what that equilibrium is.

A template of the code can be found at the bottom of “optimal_profit_template.m”. Need to download files from Activity 3.

Activity 5: Formulate your own decision problem

If your company plans to upgrade manufacturing capabilities with a max budget of $500,000, how will you rigorously plan the upgrade? Consider the following upgrade costs:

• Upper bound on teddy bear size: $100 for each additional unit improvement
• Upper bound on teddy bear fluffiness: $100,000 for each additional unit improvement

Does your decision depend on your competitors’ product positioning?

For the project you are currently working on, define your design variables, associate them to manufacturing cost, engineering feasibility, and eventually to the profit. Study your competitors. Derive a design decision for this market.

Activity 6: A more complicated game

If you want to form an organization with companies to maximize your profit, how many companies will you include in the organization? Consider a penalty of $1,000,000 for betrayal.