Maximizing profit and utility functions

In summary: Using Lagrange multipliers, we obtain the system:y=\lambda(5)x=\lambda(10)We find this implies:x=2ySubstituting for $x$ into the constraint, we have:5(2y)+10y-100=0\implies (x,y)=(10,5)We find:U(10,5)=50Picking another point on the constraint, such as $(12,4)$, we the find:U(12,4)=48<50And so we may conclude that:U_{\max}=50We also find:\lambda=-
  • #1
mrjericho1991
5
0
Dear friends, I am new to this forum and I need urgent help in solving these two questions as I am due to submit them tomorrow early morning. Please help me in solving these two questions. Waiting for urgent response.

1: A farm’s profit is given by π = 100x + 80y + 2xy− x(square) − 2y(square) − 5000, where x is the number of turkeys produced and ( is the number of beef cattle produced.
How many of each should be produced to maximise the profit?
Prove that profit is indeed maximised at this level of production?
What is the maximum profit?

2: A consumer’s utility function is given by U(x,y) = xy ( with the budgetary constraint 5x + 10y = 100.

Find the values of x and y ( that maximise the utility function, subject to the budgetary constraint.
What is the value of the Lagrange multiplier λ?
 
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  • #2
Hello, and welcome to MHB! (Wave)

I changed the thread title to reflect the nature of the questions being asked. A thread title simply stating help is urgently requested is not useful to the community, nor is it good for SEO. :)

We ask that our users show their progress (work thus far or thoughts on how to begin) when posting questions. This way our helpers can see where you are stuck or may be going astray and will be able to post the best help possible without potentially making a suggestion which you have already tried, which would waste your time and that of the helper.

Can you post what you have done so far?
 
  • #3
I have just entered into Economics course and I am going nowhere, I am unable to understand it. I need your complete help step by step in solving these questions.
 
  • #4
mrjericho1991 said:
I have just entered into Economics course and I am going nowhere, I am unable to understand it. I need your complete help step by step in solving these questions.

Okay, let's look at the first problem. Let's write the profit function as:

\(\displaystyle P(x,y)=100x+80y+2xy-x^2-2y^2-5000\)

We need to set the partial derivatives $P_x(x,y),\,P_y(x,y)$ equal to zero...can you do that?
 
  • #5
Not really Sir. I don't want to lie. I want you to solve the question and mention your steps side by side in English. Please assume, I know nothing. Its my first class.
 
  • #6
mrjericho1991 said:
Not really Sir. I don't want to lie. I want you to solve the question and mention your steps side by side in English. Please assume, I know nothing. Its my first class.

What methods are you given in your lecture notes and/or textbook for maximizing multi-variable functions?
 
  • #7
MarkFL said:
What methods are you given in your lecture notes and/or textbook for maximizing multi-variable functions?

Maxima and Minima of Functions of Two Variables

- - - Updated - - -

Something like critical point, second order derivatives test is also mentioned in here.
 
  • #8
mrjericho1991 said:
Maxima and Minima of Functions of Two Variables
- - - Updated - - -
Something like critical point, second order derivatives test is also mentioned in here.

Yes, how have you been instructed to optimize such functions (with and without constraint)?

In order to find the critical point(s) without constraint, you need to equate the first partials to zero...do you know how to find partial derivatives?
 
  • #9
without constraint. Yes I know but tell me what values do I need to input for partial derivatives?
 
  • #10
You don't 'input' values- you set the formulas equal to 0 and solve the equations for x and y.
 
  • #11
mrjericho1991 said:
without constraint. Yes I know but tell me what values do I need to input for partial derivatives?

You need to equate the first partials to zero, and then solve the resulting system. We have:

\(\displaystyle P(x,y)=100x+80y+2xy-x^2-2y^2-5000\)

And so we compute the first partials, and equation them to zero to obtain the system:

\(\displaystyle 100+2y-2x=0\)

\(\displaystyle 80+2x-4y=0\)

So, what critical point do you get from solving the above linear system?
 
  • #12
As a follow-up, I would begin by adding the two equations to eliminate $x$ and we obtain:

\(\displaystyle 180-2y=0\implies y=90\)

Now, substituting for $y$ into the first equation, there results:

\(\displaystyle 100+2(90)-2x=0\implies x=140\)

And so our critical point is:

\(\displaystyle (x,y)=(140,90)\)

Now, in order to use the second partials test for relative extrema, we need to compute:

\(\displaystyle P_{xx}(x,y)=-2\)

\(\displaystyle P_{yy}(x,y)=-4\)

\(\displaystyle P_{xy}(x,y)=2\)

And we find:

\(\displaystyle D(x,y)=P_{xx}(x,y)P_{yy}(x,y)-\left(P_{xy}(x,y)\right)^2=8-4=4>0\)

And so we conclude that the critical point is at the global maximum, which is:

\(\displaystyle P(140,90)=5600\)

For the second problem, we have the objective function:

\(\displaystyle U(x,y)=xy\)

Subject to the constraint:

\(\displaystyle g(x,y)=5x+10y-100=0\)

Using Lagrange multipliers, we obtain the system:

\(\displaystyle y=\lambda(5)\)

\(\displaystyle x=\lambda(10)\)

We find this implies:

\(\displaystyle x=2y\)

Substituting for $x$ into the constraint, we have:

\(\displaystyle 5(2y)+10y-100=0\implies (x,y)=(10,5)\)

We find:

\(\displaystyle U(10,5)=50\)

Picking another point on the constraint, such as $(12,4)$, we the find:

\(\displaystyle U(12,4)=48<50\)

And so we may conclude that:

\(\displaystyle U_{\max}=50\)

We also find that:

\(\displaystyle \lambda=1\)
 

Related to Maximizing profit and utility functions

1. How do profit and utility functions differ?

Profit functions are used in economics to determine the maximum amount of profit that can be obtained from a certain level of production. Utility functions, on the other hand, are used in consumer theory to determine the maximum level of satisfaction that a consumer can obtain from a given set of goods or services.

2. What is the goal of maximizing profit and utility functions?

The goal of maximizing profit and utility functions is to find the optimal level of production or consumption that will result in the highest possible profit or satisfaction. This involves finding the point where the derivative of the function is equal to zero, known as the maximum point.

3. How are profit and utility functions optimized?

Profit and utility functions are optimized by taking the derivative of the function and setting it equal to zero. This will give the critical point or the maximum point of the function. The critical point can then be plugged back into the original function to find the maximum profit or utility.

4. What are some factors that can affect profit and utility functions?

Profit and utility functions can be affected by a variety of factors such as changes in market conditions, input costs, consumer preferences, and technological advancements. These factors can shift the function and change the optimal level of production or consumption.

5. Are profit and utility functions always accurate in predicting outcomes?

No, profit and utility functions are based on assumptions and may not always accurately predict outcomes. These functions are useful for making decisions and understanding trends, but they may not account for all variables and unexpected events that could affect profits and utility levels.

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