# How can a producer maximize profit

## Short and long term profit maximization

Profit maximization is the central goal of companies in economics. A distinction is made between short-term and long-term profit maximization.

You know the terms short-term and long-term profit maximization from microeconomics, but are no longer sure what they mean? Check out our Video at! Here we explain the two forms of profit maximization to you and show you an example of how to calculate them.

• Short-term profit maximization: definition
in the text
• Long-term profit maximization: definition
in the text

### Short-term profit maximization: definition

With short-term profit maximization, one input factor is always kept constant. A good example here would be renting a production hall. In the short term, the rent must always be paid, regardless of whether the hall is used or not. Long term you can then decide whether you want to continue renting the hall or cancel the lease.

The second factor is variable in the short-term production function. An example of this would be the heating for the production hall. Of course, the hall only needs to be heated when it is in use. The input factor therefore only arises during production.

### Calculate profit maximization

So let's get down to the mathematical solution of the optimization problem. For this we first need a production function. It could look like this:

As a fixed factor, we take the capital employed in our company, that is , as the price of capital , as a wage rate and as a price for our product . The employee factor is variable.

We have to calculate the profit-maximizing production volume. To do this, we first have to determine the number of employees that we need for our company. To do this, we first set up the maximization approach and insert the given numbers:

Now we have to observe the first order condition. It demands that the amount of the variable factor be expanded until the marginal value product and the associated factor price match. The marginal value product tells you what additional revenue you can achieve if you employ an additional unit, in this case an additional employee. Simply put: the marginal product multiplied by the price.

### Profit maximization example

In our example, this means that we have to hire more employees until the added value that they create, i.e. the marginal value product, corresponds to their wage income. First we follow the maximization rate from:

Then we rearrange the equation so that the marginal value product is on the left and the associated factor price is on the right:

The 2nd order condition requires that the variable factor has a decreasing marginal productivity. Therefore, by expanding production, the marginal value product falls and thus achieves a balance with costs. This means that the more employees you have already hired, the less you gain from another. This means that the first employee you hire brings you the most, the second a little less, the third even less and the fourth nothing but trouble. In other words, you hire new employees until the added value you receive from the additional employee corresponds to the wage rate. That is then our optimum. In mathematical terms, it looks like this:

With the right graphics, this might be a little clearer:

Here you can see the falling power function of the marginal product and the linear function of the costs. First of all, the marginal product is greater than the costs, so you hire more employees. Up to the point where both functions intersect. That is your optimum. So you have hired exactly the right number of employees to maximize your profit in the short term.

At this point the marginal product is less than the cost, so you shouldn't hire more people.

### Result of short-term profit maximization

Our function from earlier let's just solve it now , i.e. the number of employees we need for our company. is therefore 256. That means we need 256 units of our variable factor. So we need 256 employees.

Now we only calculate our profit maximum out. Thatâ€™s what we do in our production function:

So in the short-term maximum profit we produce 128 units of our product. Now you can put everything into your profit equation to calculate the maximum profit in the short term! Our maximum profit in the short-term profit maximum is thus 320.

### Summary

Let's summarize everything again briefly: So first you set up your maximization rate, then you derive it according to the variable factor and rearrange the whole thing so that on the one hand there is the marginal value product and on the other hand the factor price. Then you dissolve according to your variable factor, put everything into the production function and already have the profit-maximizing output volume. If you now use that in the profit function, this results in your profit maximum.

It continues with long-term profit maximization:

### Long-term profit maximization: definition

Hopefully you have understood short-term profit maximization. In the further part of the article, we will explain briefly and concisely the long-term profit maximization and also show you here, using an example, how you can easily calculate the long-term profit maximization.

In contrast to short-term profit maximization, both factors are variable here. So you have to change both factors in order to get to your profit maximum.

The long-term profit maximization is also the actual goal of the company, because a company should of course achieve the highest possible profits in the long term. In the short term, it just has to aim for higher sales.

### Calculate long-term profit maximization

As with short-term profit maximization, you also have a production function here. In our case it looks like this:

Let's say your start-up produces lemonade. Your factors of production could then be liters of water and lemons be. The lemonade gets a bit sour without any sugar, but it definitely tastes good anyway!

Let's set the factor costs and the lemonade price: , and p = 15. As with short-term profit maximization, one must again set up the maximization rate and use all available information.

Since both factors are variable here, we have to derive the maximization rate according to both factors. So we calculate the marginal product both for the liters of water and for the lemons and apply the optimization condition. This reads and is also simply called the input rule. It will look like that:

Then we bring the factor prices back to the other side and multiply the respective marginal product by the appropriate production factor. This enables us to use the original production function and thus obtain the factor input quantity that is dependent on y. It looks like this:

The calculation for the lemon input factor looks like this:

Now you rearrange your equations according to the production factors and get the factor input quantities depending on the output:

### Result of long-term profit maximization

So that you get the profit-maximizing output, you put the whole thing into your production function: