# Die aggregierte Produktionsfunktion (BW)

Erik Hurst erläutert die Funktion:

"A production function is a mathematical relationship between the inputs a firm uses and the output the firm generates. For this class, we are going to assume that the firm only uses three broad inputs: labor (workers), capital (machines) and some technology (we are going to abstract from things like raw materials and land as of now - if it makes you more comfortable, we will include these in our measure of capital).

You should think of capital as ECONOMIC CAPITAL. Economic capital is simply the integral of all past investment - net of depreciation. That means if you bought a machine this year - it is investment this year. But, that machine also adds to the 'capital stock'. The capital stock is the value of all existing machines that your firm owns. Note: We measure labor and capital like we measure output - in terms of dollars. Our measure of labor, then, is the 'real' value of wages paid to workers and our measure of capital is the ‘real’ value of machines that we use.

Schema einer aggregierten Produktionsfunktion:

**Y = f(K, N, A)**

where **f(.)** is some mathematical relationship (a function), **K** is our symbol for the capital stock, **N** is our symbol for labor and **A** is our symbol for our technology measure (much more on this measure of technology in class)."

Oder auch:

**GDP = f(L, K, MFP)**(Labor, Capital, Multi-factor Productivity)

Welche Annahmen trifft man über MFP? Z.B.: kommt von außen, wird nicht von ökonomischen Faktoren (Preise) erfasst.

Es gibt 2 Gleichungen zur Erfassung einer Volkswirtschaft:

**Y = C(Konsum) + I(Investitionen) + G(Staatsausgaben) + NX(Nettoexport)**("Ausgaben", "expenditure").

Also, we have Y = f(K, N, A) (this is the supply side - what firms produce). As we learned in week 1, what is bought - by definition - must equal what is produced (again, once we account for inventory changes). These equations will become the foundation of our aggregate demand and aggregate supply equations later in the class! These equations will jointly determine both prices and output (and with some additional steps - interest rates and wages and other stuff). Prices will adjust in the economy to ensure that what is demanded is what is produced (store this away, this is one of the key insights from the class). We will have two equations (demand and supply) and two unknowns (Y and P) that we can solve.

**Constant Returns to Scale**. This simply means that if you double all non- technology inputs (labor and capital), you should be able to double output. Think about it. If you have a given level of technology and you have one plant that produces 10 units of output, shouldn't you be able to produce an identical plant (same amount of machines and workers) next to the original plant and produce an additional 10 units of output. Doubling the inputs (making another plant) should lead to a doubling of the output. We call this concept constant returns to scale. (This is distinct from concepts such as increasing returns to scale - ie, if you double all inputs, you more than double all output). Much existing research cannot reject that firms are close to constant returns to scale.

**Elasticities and Cost Shares**: Existing research has shown that the elasticity of output with respect to changes in labor tends to be about .7 and that the elasticity of output with respect to changes in capital tend to be about .3. This comes from empirical research. Elasticity is the percentage increase in Y (dependent variable) resulting from a 1% increase in X (independent variable). That means if labor (independent variable) increases by 1%, output (dependent variable) tends to increase by 0.7% - holding K constant.

Additionally, we find that the share of labor costs out of total GDP is about 70%. That means out of every $1 of GDP, 70% of that income flows to workers (as wages). Capital costs shares are the remaining 30%. These numbers are simply generated by taking labor income and dividing it by GDP or taking capital income and dividing it by GDP. The flows to capital owners include dividends, profits, interest payments, and rents.

**There are diminishing marginal returns to labor or capital individually.** Suppose you do not double both capital and labor, but instead only double labor. If there are diminishing returns to labor individually, you will less than double output. Furthermore, for each additional labor added to a fixed amount of capital, the extra amount of output that is generated gets smaller and smaller over time. This is just common sense (and a central tenant from micro). After some point, all production processes have diminishing returns to both capital and labor individually. Suppose you have a room with 3 computers and 2 employees. Adding a third employee gets you a lot of extra output. Adding the 4th worker will probably get you more total output, but likely not as much additional output as adding the 3rd worker. The 100th extra employee probably will not provide any additional value if there are only 3 computers. The more employees you add, the less additional (extra or marginal) amount of production that you would get holding the computers constant. Diminishing returns refers to the marginal product of labor (the extra output you get from adding additional workers). Marginal - in economics - refers to additional or extra. Product refers to output (ie, so putting them together, marginal product means additional output). The marginal product of labor is the additional output you get from adding one more unit of labor - holding the other inputs constant.

**Complementarity between A, K and **N. This just means that the extra amount of output that you get from the 4th worker (example above) will be higher if you have 4 computers than if you had 3 computers. The higher the fixed level of capital (or technology), the higher the marginal product of labor. Labor is more productive if you have more machines (and vice versa).

**zurück zu Ökonomische Ansätze (BW)**

Diese Seite entstand im Kontext von: **Besser Wissen (Vorlesung Hrachovec, 2006/07)**