TO SUE:

The production function shows the relationship between the production of a good and the inputs (factors of production) required to produce it.

**It usually takes the following general form:**

Q = f (K, L, t usw.)

TO SUE:

Where Q is production, K is capital input, L is labor input, t is “technology or art of manufacture”, and the term “etc.” indicates that other inputs may also be relevant (e.g. land or raw materials). The production function shows how changes in output are related to changes in inputs or factors of production. It is also an efficiency metric that indicates the maximum amount of production that can be obtained from a fixed amount of resources.

Production functions can take different forms. Economists generally work with homogeneous production functions. An example of such a function is the famous Cobb-Douglas production function.

**Production isoquants:**

The long run production function involving the use of two factors (e.g. capital and labor) is represented by isoquants or equal product curves (or production indifference curves).

TO SUE:

**Definition:**

An isoquant is a curve or locus of points showing all possible combinations of inputs that are physically capable of producing a given fixed level of output. An isoquant sitting above another indicates a higher level of production.

Cowardly. 1 shows two typical isoquants - capital input is measured on the vertical axis and labor input on the horizontal axis. Isoquant Q_{1}shows the locus of the combinations of capital and labor that produce 100 units of production. The producer can produce 100 units of output with 10 units of capital and 75 units of labor, or 50 units of capital and 15 units of labor, or with any other combination of inputs at Q_{1}= 100. Likewise the isoquant Q_{2}shows the various combinations of capital and labor that can produce 200 units of production.

We can use any number of isoquants in Fig. 1 because there are infinitely many possible production levels between 100 and 200 pieces (as well as under 100 pieces or over 200 pieces).

TO SUE:

**Characteristics:**

Isoquants have four important properties.

**These are the following:**

1. First, an isoquant that is above and to the right of another indicates a higher level of production. Therefore, any point on a higher isoquant is always better than any point on a lower isoquant.

2. Second, isoquants cannot meet or intersect. If this were the case, a combination of K and L would produce two different levels of output. Vendor technology is inconsistent. We exclude such events.

3. Third, as in Fig. 1, the isoquants fall off along the relevant production area. This negative slope indicates that as the producer decreases the amount of capital employed, more labor must be added to keep the rate of production constant. Or, if labor input decreases, capital input must increase to keep output constant. Thus, the two inputs can be swapped with each other to maintain a constant output level.

**The marginal rate of technical substitution (TMST):**

**The rate at which one input can be substituted for another versus an isoquant is called the marginal rate of technical substitution (MRT) and is defined as:**

TO SUE:

MRTS_{EU}_{Gabel}= – ∆K/∆L

where K is capital, L is labor and ∆ denotes each change. The minus sign is added to make MRTS a positive number since ∆K/∆L, the isoquant slope, is negative.

For any motion along an isoquant, the MRTS is the ratio of the marginal products of the two inputs.

**To prove this, assume that the consumption of L increases by 3 units and K by 5. If in this step the MP _{EU}4 units of Q per unit of L and that of K is 2 units of Q per unit of K, the resulting change in output (Q) is:**

TO SUE:

∆Q = (4 x 3) + (2 x 5) = 22

This means that if L and K are allowed to vary slightly, the change in Q resulting from the change in the two inputs is the marginal product of L times the amount of change in L plus the marginal product of K times its change.

**Generally:**

∆Q = MP_{EU}. ∆L + MP_{k}. ∆K.

TO SUE:

**Q is constant along an isoquant; therefore ∆Q is equal to zero. Setting ∆Q equal to zero in the above equation and solving for the isoquant slope, ∆K/∆L, we get:**

∆K/∆L = MP_{EU}/MP_{k}= MRT_{EU} _{Gabel}

Since K and L must vary inversely along an isoquant, ∆K/∆L is negative.

4. Fourth, the MRTS decreases throughout the relevant phase. This means that the isoquants are convex with respect to the origin. This point is in Fig. 1. If capital is reduced by 10 units from 50 to 40, labor need only be increased by 5 units from 15 to 20 to keep the production level at 100 units. If capital falls by 10 units from 20 to 10, labor must increase by 35 units from 40 to 75 to keep production at 100 units.

**Optimal combination of properties:**

A rational producer aims to either maximize production at a fixed budget (e.g. Rs. 300 per day) or to minimize costs at an output required for production (e.g. 150 units). Each goal is a constrained optimization problem.

TO SUE:

Our task here is to determine the specific combinations of inputs that a company should select when constrained. Here we will observe that a firm achieves the highest possible level of output for any level of cost, or the lowest possible cost of production for any level of output, when the MRMS for any two inputs is equal to the ratio of their prices.

**Entrance fees and iso cost lines:**

The total cost C of using any value of K and L is C = rk + w L, the sum of the cost of K units of capital at a price of r per unit and L of units of labor at a price of w per unit.

Suppose the cost of capital is Rs. 30 per unit (r = Rs. 30) and the labor receives a wage of Rs. 15 per day (w = Rs. 15). If only capital is used then: C = rK + 0 and the maximum amount of capital that can be purchased is K = C/r = Rs. 300/Rs. 30 = 10 units. Similarly, if only labor is hired, C = 0 + w L and the maximum number of workers that can be hired (per day) is L = C/w = Rs. 300/Rs. 15 = 20. We can also think of various other combinations of capital and labor that can be bought (rented) with the same budget.

The budget equation is in Fig. 2. The AB line is called the iso-cost line or equal-cost line. It is indeed the producer's budget line.

**Definition and scope:**

TO SUE:

The isocost line is a point location showing alternative combinations of K and L that can be purchased with a fixed amount of money at prevailing market prices.

**Its slope is:**

This is called the factor price ratio or actual factor substitution rate. where w is the price of labor (P_{EU}) and r the capital price (P_{k}).

**Production maximization under cost pressure:**

TO SUE:

A rational producer whose goal is to maximize output under cost constraints will always seek to achieve the highest achievable isoquant permitted by the isocost line. This point is in Fig. 3. Here the producer reaches Q_{2}with its Isocost AB line and producing 150 units at a price of Rs 300. Hence the cost per unit is Rs. 2. Can the total cost be further reduced?

NO. If the producer moves to point F or G along the same isoquant by accident or miscalculation, his total cost (expense) remains the same, but his production drops to 100 units. So your unit cost increases to 150 units. So only point E can be an optimal point. And the combination of K and L corresponding to the point (namely K_{1}and me_{1}) is referred to as the most cost-effective combination. Thus, a rational producer maximizes output by choosing the least expensive combination of inputs whose prices are given (i.e., determined by market forces).

**At point E, the slope of the isoquant or MRTS equals the slope of the isocost line:**

** **

TO SUE:

**Production of a fixed production at the lowest cost:**

Now suppose that the producer's goal is to produce exactly 150 production units, no more, no less. This objective can also be achieved by choosing the most cost-effective combination of inputs or by meeting the condition above. In Fig. 4, the single isoquant indicating production of 150 units just touches the isocost line A_{2}B_{2}at point E

This means that the minimum production cost of a given production of 150 units is Rs. 2. If the producer moves to the right or left of point E along the same isoquant, the cost increases. Thus, E is the optimal point, indicating a combination of lower cost inputs. For example at the exit point it is 30 units but the total cost is Rs. 100 which means the cost per unit is Rs. 3.

**Diploma:**

Thus, the two alternative strategies presented here deliver the same results. To maximize output for a given cost, or to minimize cost for a given output, the manufacturer must use inputs in amounts equal to the marginal rate of technical substitution and the factor price ratio.