No there can be no fixed cost in long run because in long run all factors inputs are variable. Fixed cost only exists in short run.
The costs that a firm incurs to employ fixed inputs (overhead cost) is called total fixed cost (TFC). This cost is independent of the amount of output that the firm produces; the cost remains fixed for the firm.
The cost that a firm incurs to employ variable inputs like raw material, labour, etc. is called the total variable cost (TVC).
Total cost of a firm is summation of total fixed cost and total variable cost. (TC=TFC+TVC).
In short run, to increase the production of output, the firm needs to employ more of variable factors, which causes total variable cost to increase. And hence total cost also increases. Therefore,we can say that, as output increases, total variable cost and total cost increases whereas total fixed cost remains the same.
The following table highlights the relationship between TC, TFC and TVC.
Output 
Total Fixed cost (TFC) 
Total Variable Cost (TVC) 
Total Cost (TC= TFC +TVC) 
0 
20 
 
20 
1 
20 
50 
70 
2 
20 
80 
100 
3 
20 
100 
120 
4 
20 
130 
130 
5 
20 
180 
200 
6 
20 
260 
280 
As you can see,
A cost function shows the functional relationship between cost of production and output.
It is given as:
C = f(Q), ceteris paribus
Where, C = Cost
Q = Output
In order to produce output, the producers need to employ inputs. All inputs come at a price or cost. Every producer chooses the least cost input combination for different level of output.
Production function satisfy increasing returns to scale (IRS) when a proportional increase in all inputs results in an increase in output by more than the proportion.
That is, if all inputs are increased by 50% and the output is increased by more than 50%, say 80%, the production function holds increasing returns to scale.
The law of diminishing marginal product states that as we keep on employing more and more of a variable input to the production process, keeping other inputs constant, a point will be reached after which Marginal product (MP) from the variable inputs starts falling.
Marginal product of an input is defined as the change in the output per unit of change in the input when all other inputs are held constant.
MP_{1}= Change in output ∆Q /Change in input ∆x_{1}
MP_{1}=TP_{x1}TP_{x11}
Average product is defined as the output produced per unit of variable input, we calculate it as
AP_{1} = TP_{1}/x_{1}
For example: If total product with 10 units of a variable factor is 100 units, then, Average product AP will be equal to 100/10 i.e. 10 units.
Total product of an input or a variable input refers to the relationship between the input and output, keeping all other inputs constant.
Or
It refers to the total amount of goods produced by a firm with the available input during a specified period of time.
The SMC curve cuts the SAC curve at the minimum point of SAC because as long as SAC is falling the SMC is below SAC and when SAC rises SMC is above SAC. The point at which SMC is equal to SAC is the minimum point of SAC.
Given Q = 5L + 2K ……………………(equation 1)
L = 0 unit
K = 10 units
Putting the values of L and K in equation 1
Q = 5(0) + 2(10)
Q = 20 units
Thus, the maximum possible output for a firm with zero unit of L and 10 units of K with the given production function is 20 units.
Given,
Production function,Q = 2L^{2}K^{2}_{……………………………………}(equation 1)
(i) The maximum possible output that the firm can produce with 5 units of L and 2 units of K can be calculated as follows:
L = 5 units
K = 2 units
Putting the value of L and K in the above equation 1, we get
Q = 2 (5)^{2}(2)^{2}
Q = 2 × 25 x 4
Q = 2 × 25 x 4
Q = 200 units
Thus, the maximum possible output for a firm with 5 units of L and 2 units of K with the given production function is 200 units
(ii) The maximum possible output that the firm can produce with zero unit of L and 10 units of K can be calculated as follows
L = 0 unit
K = 0 units
Putting the new values of L and K in the equation 1 we get,
Q = 2 (0)^{2}(10)^{2}
Q = 0 units
Thus, with 0 units of L and 10 units of K and the given production function, the firm will produce 0 units of output.
Given,
Production function, Q = 5L^{1/2} K^{1/2}…..……(equation1)
L=100 units
K=100 units
Putting the values of L and K in the above equation 1, we get,
Q = 5 × (100)^{1/2} (100)^{1/2}
Q = 5√100√100
Q = 500 units
Thus, the maximum possible output that a firm can produce with 100 units of L and 100 units of K with the given production function is 500 units.
The long run marginal cost (LMC) and the long run average cost (LAC) curves look like ’U’. The reason behind their U shape is the operation of laws of returns to scale in long run.
According to the law, when the firm expands the size or production, they get some advantages of internal economies of scale due to which the perunit cost of production decreases. After a point it reaches an optimum level. When the scale of production is increased beyond the optimum level, the firm starts facing some disadvantages of internal economies due to which cost of production starts increasing.
The short run marginal cost curve is U shaped because of the operations of “Law of variable Proportion”. As output increases, MC curve slopes downward, reaches the minimum level and then starts rising upwards. The falling portion of the MC curve corresponds to the increasing returns to a factor, the minimum point on MC curve corresponds to the constant returns to a factor and the rising portion of the MC curve corresponds to the decreasing returns to a factor.
SMC curve cuts the AVC curve at the minimum point of the AVC Curve because to the left of minimum of AVC, SMC is below AVC, SMC and AVC both fall but SMC falls at a faster rate. At the minimum point E, SMC=AVC, beyond point E both SMC and AVC rise but SMC rises at a faster rate and lies above AVC.
Short run marginal cost (SMC), average variable cost (AVC) and short run average cost (SAC) curves all look like English alphabet ‘U’, i.e., they all are ‘U’ shaped. The reason behind the same is the law of variable proportion.
In the initial stages of production, all costs fall due to increasing returns to a factor or labour. In short run,marginal product of labour also increases, which implies that more and more output can be produced by employing additional unit of labour, leading all cost curves to fall. With the advent of constant returns to a factor or labour, the cost curve becomes constant and also reaches its minimum. Beyond this point,diminishing returns to a factor of labour set in, which implies additional unit of labour does not increase production levels.In fact,MP starts falling, hence all costs start increasing and all cost curves start rising. This gives SMC, SAC and SVC Curves their U shape.
Average fixed cost is obtained by dividing the total fixed cost by the quantity. Average fixed cost curve look like a rectangular hyperbola. As total fixed cost remains the same, an increase in output causes average fixed cost to diminish.However, it never becomes zero. At zero level of output, AFC is infinitely large. As output increases,it starts falling and when output is very large, AFC tends to zero but never becomes zero. This behavior gives AFC curve the shape of rectangular hyperbola as shown in below fig.
Production function satisfies decreasing returns to scale (DRS) when a proportional increase in all inputs results in an increase in output by less than the proportion.That is, if all inputs are increased by 50% and the output is increased by less than 50%, say 20%, the production function holds decreasing returns to scale.
Production function satisfies constant return to scale (CRS) when a proportional increase in all inputs results in an increase in output by the same proportion.That is, if all inputs are increased by 50% and the output is also increased by 50%, then the production function holds constant return to scale.
The law of variable proportion states that when more and more units of a variable factor are applied to increase the production, keeping quantities of other inputs fixed, the total product initially increases at an increasing rate, then increases at diminishing rate and finally starts falling. The law of variable proportion takes place in the shortrun.
The relationship between total product and Marginal product may be discussed as under –
The Production function represents the technological relationship between physical input and output of a product. In other words it shows that with a given state of technology and during a particular period of time,how much we can produce with the given inputs. Symbolically, production function can be written as follows
Q = f ( f_{1}, f_{2}, f_{3}.......f_{4} )
f_{1}f_{2} and f_{3} are factors of production.
Generally Land, Labour, Capital and Entrepreneurship are the factors of production used in the production process.

Short run 
Long run 
1.

It is the time period in which some factors are fixed and others are variable. 
It is the time period in which all factors of production can be changed. 
2. 
The production can be changed by changing only variable factors. 
The production can be changed by changing both the factors of Production. 
3. 
The factors of production can be grouped as (i) Fixed factors (ii) Variable factors 
The proportion of change in fixed factors and variable factors will be constant. 
4 
Market price exists in this period 
Normal price exists in this period. 
Q  TC 
1 2 3 4 5 6  50 65 75 95 130 185 
Q 
TC 
TFC 
TVC =TCTFC 
AVC = TVC/Q 
AFC=TFC/Q 
SAC=TC/Q 
SMC=TC_{n}TC_{n1} 
1 2 3 4 5 6 
50 65 75 95 130 185 
20 20 20 20 20 20 
30 45 55 75 110 165 
30 22.50 18.331 8.75 22 27.50 
20 10 6.66 5 4 3.33 
50 32.50 25 23.75 26 30.83 
30 15 10 20 35 55 
AFC at 4 unit = Rs 5
AFC = TFC/Q,
Putting the given value of AFC and Q in the formula we get,
TFC = AFC × Q
TFC = 5 × 4 = Rs. 20
L  MP_{L} 
1 2 3 4 5 6  3 5 7 5 3 1 
Total product of labour (TP_{L}) = ∑MP_{L}
Average Product of labour (AP_{L}) = Total product of labour/No. of Units of labour (TP_{L}/L)
L 
MP_{L} 
TP_{L} 
AP_{L} 
1 2 3 4 5 6 
3 5 7 5 3 1 
3 8 15 20 23 24 
3 4 5 5 4.6 4 
L  AP_{L} 
1 2 3 4 5 6  2 3 4 4.25 4 3.5 
Average Product of labour (AP_{L}) = Total product of labour/No. of Units of labour (TP_{L}/L)
Total product of labour (TP_{L})=Average Product of labour (AP_{L}) x No. of Units of labour (L)
Marginal Product of labour (MP_{L}) = TP_{L}TP_{L1}
L 
AP_{L} 
TP_{L} 
MP_{L} 
1 2 3 4 5 6 
2 3 4 4.25 4 3.5 
2 6 12 17 20 21 
2 4 6 5 3 1 
L  TP_{L} 
0 1 2 3 4 5  0 15 35 50 40 48 
Average Product of labour (AP_{L} )= Total product of labour/No. of Units of labour (TP_{L}/L)
Marginal Product of labour (MP_{L}) = TP_{L}TP_{L1}
L 
TP_{L} 
AP_{L} 
MP_{L} 
0 1 2 3 4 5 
0 15 35 50 40 48 
 15 17.50 16.67 10 9.60 
 15 20 15 10 8 
Average Fixed Cost (AFC):It is per unit fixed cost of producing a commodity. It is calculated by dividing the total fixed cost by the number of units produced.
Average Variable Cost (AVC): It is per unit variable cost of producing a commodity. It is calculated by dividing the total variable cost by the number of units produced.
Average Cost or Short Run Average Cost (SAC): It is the per unit total cost of production of a commodity. It is calculated by dividing the total cost by the number of units produced.
The relationship between AFC, AVC, AC can be studied with help of following table and Figure:
Output 
TFC 
TVC 
TC 
AFC 
AVC 
AC 
0 
20 
 
20 
 
 
0 
1 
20 
50 
70 
20 
50 
70 
2 
20 
80 
100 
10 
40 
50 
3 
20 
100 
120 
6.66 
33.33 
40 
4 
20 
130 
150 
5 
32.5 
37.5 
5 
20 
180 
200 
4 
36 
40 
6 
20 
260 
280 
3.33 
43.33 
46.66 
As you can see,
AC and AVC can never meet each other because AFC is a rectangular hyperbola which nereve reaches zero or cut xaxis.
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