 # Production Functions

Imagine you run a small factory that produces dog food. As the manager of the dog food factory, you are focused on being able to calculate your total output in order to make better business decisions.

The amount of dog food you can produce is dependent on the amount of labor, capital, and technology you have available to produce dog food at your dog food factory. Therefore, you’ll utilize the following function to determine your total output:

Yp = A * L½ * K½

The total output (Yp) is a function of the amount of Labor (L), Capital (K), and Technology (A) you have available.

Let's figure out how we can apply the total output equation above in mathematical terms!

## How to calculate total output

Consider the following situation in your dog food factory:

Question: You’ve determined the Labor Demand equation for your dog food factory is W = 100 – Ld, and the Labor Supply equation is W = 73 + 2Ls. What is the equilibrium value for L (Labor)? Additionally, Capital Stock from last year was 129, gross investments were 25, and depreciation was 10. What is the value for K (Capital)? Utilizing these variables, determine the total output (Yp), given that A is 15.

Utilizing the equation above…

Yp = A * L½ * K½

…let’s walk through this problem step-by-step and solve each part, starting with A!

### Understanding technology (A)

You’ll notice that in the problem, it defines A for us at the end.

Question: You’ve determined the Labor Demand equation for your dog food factory is W = 100 – Ld, and the Labor Supply equation is W = 73 + 2Ls. What is the equilibrium value for L (Labor)? Additionally, Capital Stock from last year was 129, gross investments were 25, and depreciation was 10. What is the value for K (Capital)? Utilizing these variables, determine the total output (Yp), given that A is 15.

Therefore, we can go ahead and plug it into our equation like so:

Yp = 15 * L½ * K½

For context, this value represents changes in technology in our dog factory. For example, if we upgraded our machine, our A value might be higher at 20. If we had worse machines, it might be lower at 10.

You'll never have to calculate A, it’ll always be given to you!

### Understanding labor (L)

To find L, we’re going to zone in on the following part of the problem:

Question: You’ve determined the Labor Demand equation for your dog food factory is W = 100 – Ld, and the Labor Supply equation is W = 73 + 2Ls. What is the equilibrium value for L (Labor)? Additionally, Capital Stock from last year was 129, gross investments were 25, and depreciation was 10. What is the value for K (Capital)? Utilizing these variables, determine the total output (Yp), given that A is 15.

Essentially, we need to know where Labor Demand and Labor Supply intersect on a supply and demand graph.  The point of intersection is Wage Equilibrium (because Labor Demand and Labor Supply are determined by wages).

Utilizing this point of intersection, we can find where L falls on the x-axis.

Visually, we’ve identified on the graph where the Wage Equilibrium point occurs (and therefore where our L value lies on the x-axis).

But how can we solve for this value mathematically?

By setting the Labor Supply and Labor Demand equations…

Labor Demand: W = 100 – Ld
Labor Supply: W = 73 + 2Ls

…equal to each other...

100 – Ld = 73 + 2Ls

...and solving for L!

100 - L = 73 + 2L
27 = 3L
L = 9

In simple terms: By setting the wage functions equal we are essentially asking… “at equilibrium, what amount of labor do we have available for production?”. And when we solve for that, we get an L value of 9!

Let’s go ahead and plug that into our overarching production function like so:

Yp = 15 * 9½ * K½

### Understanding capital (K)

To find K, we’re going to zone in on the following part of the problem:

Question: You’ve determined the Labor Demand equation for your dog food factory is W = 100 – Ld, and the Labor Supply equation is W = 73 + 2Ls. What is the equilibrium value for L (Labor)? Additionally, Capital Stock from last year was 129, gross investments were 25, and depreciation was 10. What is the value for K (Capital)? Utilizing these variables, determine the total output (Yp), given that A is 15.

To solve for K, we’re going to use the following formula:

K (this year) = K (last year) + Gross Investment - Depreciation

Considering this, all we need to do is plug in last year’s capital stock of 129…

Question: You’ve determined the Labor Demand equation for your dog food factory is W = 100 – Ld, and the Labor Supply equation is W = 73 + 2Ls. What is the equilibrium value for L (Labor)? Additionally, Capital Stock from last year was 129, gross investments were 25, and depreciation was 10. What is the value for K (Capital)? Utilizing these variables, determine the total output (Yp), given that A is 15.

K (this year) = 129 + Gross Investment - Depreciation

…the gross investments of 25…

Question: You’ve determined the Labor Demand equation for your dog food factory is W = 100 – Ld, and the Labor Supply equation is W = 73 + 2Ls. What is the equilibrium value for L (Labor)? Additionally, Capital Stock from last year was 129, gross investments were 25, and depreciation was 10. What is the value for K (Capital)? Utilizing these variables, determine the total output (Yp), given that A is 15.

K (this year) = 129 + 25 - Depreciation

…and the depreciation value of 10…

Question: You’ve determined the Labor Demand equation for your dog food factory is W = 100 – Ld, and the Labor Supply equation is W = 73 + 2Ls. What is the equilibrium value for L (Labor)? Additionally, Capital Stock from last year was 129, gross investments were 25, and depreciation was 10. What is the value for K (Capital)? Utilizing these variables, determine the total output (Yp), given that A is 15.

K (this year) = 129 + 25 - 10

…to result in a K value of 94!

K (this year) = 129 + 25 - 10 = 144

Let’s go ahead and plug that into our total output equation like so:

Yp = 15 * 9½ * 144½

### Calculating total output (Yp)

Based on the work we did above, we have the following production function, which we can easily solve to find our final total output of 540!

Yp = 15 * 9½ * 144½
Yp = 15 * 3 * 12
Yp = 540

This means that based on our amount of labor, capital, and technology available, we can product 540 units of dog food.

This concept does not only apply to small-scale production, it can also be applied to much larger scales—such as nations and countries—which is how this concept applies to Macroeconomics. Using this same framework, in theory, we can determine the potential output of a large economy by utilizing a production function.

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