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Profit Maximization. Beattie, Taylor, and Watts Sections: 3.1b-c, 3.2c, 4.2-4.3, 5.2a-d. Agenda. Generalized Profit Maximization Profit Maximization with One Input and One Output Profit Maximization with Two Inputs and One Output Profit Maximization with One Input and Two Outputs.

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### Profit Maximization

Beattie, Taylor, and Watts

Sections: 3.1b-c, 3.2c, 4.2-4.3, 5.2a-d

Agenda
• Generalized Profit Maximization
• Profit Maximization with One Input and One Output
• Profit Maximization with Two Inputs and One Output
• Profit Maximization with One Input and Two Outputs
Defining Profit
• Profit can be generally defined as total revenue minus total cost.
• Total revenue is the summation of the revenue from each enterprise.
• The revenue from one enterprise is defined as price multiplied by quantity.
• Total cost is the summation of all fixed and variable cost.
Defining Profit Cont.
• Short-run profit () can be defined mathematically as the following:
Revenue
• In a perfectly competitive market revenue from a particular enterprise can be defined as p*y.
• When the producer can have an effect on price, then price becomes a function of output, which can be represented as p(y)*y.
Marginal Revenue
• Marginal Revenue (MR) is defined as the change in revenue due to a change in output.
• In a perfectly competitive world, marginal revenue equals average revenue which equals price.
Marginal Revenue Cont.
• When the market is not perfectly competitive, then MR can be represented as the following:
Marginal Value Of Product
• Marginal Value of Product (MVP) is defined as the change in revenue due to a change in the input.
• To find MVP, you need to substitute the production function y=f(x) into the TR function.
Cost Side of Profit Maximization
• Marginal Cost (MC) and Marginal Input Cost (MIC) can be derived from the cost side of the profit function.
• Marginal cost is defined as the change in cost due to a change in output.
• From the cost minimization problem, it was shown the different forms that marginal cost could take.
• Marginal Input Cost is the change in cost due to a change in the input.
• MIC is equal to the price of the input.
Profit Maximization with One Input and One Output
• Assume that we have one variable input (x) which costs w.
• Assume that the general production function can be represented as y = f(x).
Notes on Profit Maximization
• By solving the profit maximization problem, we get the optimum decision rule where MVP=MIC.
• With minor manipulation we can transform the result from the previous slide using the production function into the other form of the optimum decision MR = MC.
Notes on Profit Maximization Cont.
• There are two primary ways to solve the profit maximization problem.
• Solve the constrained profit max problem w.r.t. x and y.
• Transform the constrained profit max problem into an unconstrained problem by substituting the production function or its inverse into the profit max problem and solve w.r.t. to the appropriate variable.
Solving the Profit Maximization Problem W.R.T. Inputs
• Assume that we have one variable input (x) which costs w.
• Assume that the general production function can be represented as y = f(x).
Solving the Profit Maximization Problem W.R.T. Outputs
• Assume that we have one variable input (x) which costs w.
• Assume that the general production function can be represented as y = f(x) with an output price of p.
Profit Max Example 1
• Suppose that you would like to maximize profits given the following information:
• Output Price = 10
• Input Price = 200
• TFC = 100
• y=f(x)=50x-x2

### Question: How would you find the loss in profit (π) if you were a revenue maximizer instead a profit maximizer?

Loss = ππ-Max- πRevenue-Max

Profit Max Example 2
• Suppose that you would like to maximize profits given the following information:
• Output Price = 20
• Input Price = 200
• TFC=100
• y=f(x)=50x-x2
Profit Maximization with Two Inputs and One Output
• Assume that we have two variable inputs (x1 and x2) which cost respectively w1 and w2. Also, let TFC represent the total fixed costs.
• Assume that the general production function can be represented as y = f(x1,x2), where y sells at a price of p.
First Order Conditions for the Constrained Profit Maximization Problem with Two Inputs
First Order Conditions for the Unconstrained Profit Maximization Problem with Two Inputs
Summary of Profit Max Results
• At the optimum, each input selected will cause the MPP with respect to that input to equal the ratio of input price to output price.
• For example:
• MPPx1= w1/p
• MPPx2= w2/p
Summary of Profit Max Results Cont.
• From the profit max problem you will get a relationship between the two inputs.
• This relationship is called the expansion path.
• Once you selected a certain output, your revenue becomes trivially given to you when output price is fixed.
• Hence, you are just minimizing cost.
Example 1 of Profit Maximization with Two Variable Inputs
• Suppose you have the following production function:
• y = f(x1,x2) = 40x1½ x2½
• Suppose the price of input 1 is \$1 and the price of input 2 is \$16. Let the total fixed cost equal \$100.
• What is the optimal amount of input 1 and 2 if you have a price of 20 for the output and you want to produce y units?
• What is the profit?
Example 1 of Profit Max with Two Variable Inputs Cont.
• Summary of what is known:
• w1 = 1, w2 = 16
• y = 40x1½ x2½
• p = 20
Example 2 of Profit Max with Two Variable Inputs Cont.
• Summary of what is known:
• w1 = 1, w2 = 16
• y = 40x11/4 x21/4
• p = 20
Example 2: Finding the Profit Max Inputs Using the Production Function and MPPxi=wi/p
Profit Maximization with Two Outputs and One Input
• Assume that we have two production functions (y1 and y2) which have a price of p1 and p2 respectively.
• Assume that you have one input X that can be divided between production function 1 (y1=f1(x1)) and production function 2 (y2=f2(x2)).
Profit Maximization with Two Outputs and One Input Cont.
• The amount of input allocated to y1 is defined as x1 and the amount of input allocated to y2 is x2.
• The summation of x1 and x2 have to sum to X, i.e., x1+x2=X.
• The price of the input is w.
First Order Conditions for the Constrained Profit Maximization Problem with Two Outputs
Summary of Profit Max Results
• At the optimum, the marginal value of product of the first production function with respect to input 1 (MVPy1) is equal to the marginal value of product of the second production function (MVPy2).
• This gives you the optimal allocation of inputs.
• For example:
• MVPy1= MVPy2
Summary of Profit Max Results Cont.
• With some manipulation of the previous fact, the optimum rule for output selection occurs where the slope of the PPF, i.e., MRPT, is equal to the negative of the output price ratio.
• This gives you the optimal allocation of outputs.
• MRPT=-p1/p2
• Suppose you have the following production functions:
• y1 = f1(x1) = 300x11/3
• y2 = f2(x2) = 300x21/3
• Suppose the price of output 1 is \$4 and the price of output 2 is \$1.
• The price of the input w is 1 and the total fixed cost is 1000.
• What is the optimal amount of output 1 and 2 if you have 9000 units of input X to allocate to both productions?
• What is the profit?
Example 1 of Profit Max with Two Outputs and One Input Cont.
• Summary of what is known:
• w=1, p1=4, p2=1, X=9000, TFC=1000
• y1 = 300x11/3
• y2 = 300x21/3