August 22, 2017

Short run cost

  • Total Cost (\(TC\)) - the cost of all the factors of production used by the firm.
  • We can divide it by two components - the total fixed cost (\(TFC\)) and the total variable cost (\(TVC\)).
  • \[ TC=TFC+TVC \]

Short run cost cont.

Marginal Cost

  • Marginal Cost (\(MC\)) - the change in total cost that results from a one-unit increase in output.
  • Marginal cost shows the slope of tangent line to the total cost curve.
  • \[ MC=\frac{\Delta TC}{\Delta Q}=\frac{\Delta TVC}{\Delta Q} \]

Average cost

  • Average total cost (\(ATC\)) characterize total cost per unit output.
  • \[ ATC=\frac{TC}{Q} \]
  • From the total cost formula, we can derive the values for average variable cost (\(AVC\)) and average fixed cost.
  • \[ \frac{TC}{Q}=\frac{TFC}{Q}+\frac{TVC}{Q}=AVC+AFC \]

Average Cost cont.

Properties of cost curves

  • \(AFC\) is a convex decreasing function because \(FC\) does not change as \(Q\) goes up.
  • Normally, \(MC\) is U-shaped because of the optimality of the short run factor of production. When production is low (\(Q=0\)), the factor is highly productive (efficient). As we add more of a factor, its productivity diminishes. At some point, any additional unit of a factor start contributing more to the cost, thus \(MC\) cost start to increase.
  • More often however, \(MC\) is assumed to have U-shape, because that give us nice way to finding a unique equilibrium.
  • \(ATC\) is U-shaped due to the influence of two opposite forces: spread of \(TFC\) over the output and decreasing marginal returns.

Properties of cost curves cont.

  • When production is low \(Q=0\), \(TFC\) constitutes a large amount of \(TC\). As the output goes up, the contribution slowly diminishes as \(TVC\) takes over. This force drives \(ATC\) down.
  • Decreasing marginal returns imply that adding more of a factor decreases its productivity (similar to \(MC\)). As a result, as an output goes up, it drives \(ATC\) up.
  • At the start, the first force is dominant due to high \(AFC\), thus \(ATC\) decreases. As the output become large, the second force takes over, increasing \(ATC\).

Example

  • Quantity Total Cost Quantity Cost
    0 200 360 600
    100 300 400 700
    220 400 420 800
    300 500 430 900
  • Find Average Cost (\(ATC\)), Marginal Cost (\(MC\)) for each quantity of labor.
  • Draw the average cost, marginal cost curves.

Quiz

  • Quantity Total Cost Quantity Cost
    0 100 16 400
    5 200 18 500
    12 300 19 600
  • What is Total Fixed Cost (\(TFC\)) for each quantity?
  • Find Average Cost (\(ATC\)), Marginal Cost (\(MC\)) for each quantity of product.

Another representation of total cost

  • So far we considered \(TC\) to be a function of an output we want to produce.
  • In other words, we can write \(TC\) as \[ TC=f(Q) \]
  • However, we can also think of \(TC\) as a function of resources needed to produce \(Q\) (usually capital and labor): \[ TC=f(K,L)=w_L L+ w_K K \]

Another representation of total cost cont.

  • In the short run, we assume that the only factor we can change is labor.
  • In other words, if labor can produce output in some way \(L=\mathcal{L}(Q)=Q^2\), then the total cost: \[ TC=f(K,L)=w_L L+ w_K K\\ TC=f(K,L)=w_L \mathcal{L}(Q)+ w_K K\\ TC=f(K,L)=w_L Q^2+ w_K K=w_L Q^2 + const= f(Q) \]

Going from short run to long run

  • Unlike the short run, in the long run all resources are variable.
  • The total cost now depends on the several choice variables, namely labor \(L\) and capital \(K\).
  • How to minimize cost in the long run?
  • Think of a collection of the short run cost functions, each given different value for \(K\).
  • On each of them, choose the region that minimal amoung different short-run costs. Then combining them together will give us the long run total cost curve.

Economies of scale

  • Economies of scale are features of a firm's technology that make average total cost fall as output increases.
  • In other words, economies of scale is a part of the \(LATC\) curve that is decreasing.
  • The reason for it is in a specialization of labor and capital, making them more efficient as the production increases.
  • Diseconomies of scale are features of a firm's technology that make average total cost rise as output increases.
  • At diseconomies of scale, the efficiency of resources diminishes.
  • The point in between the scales is called constant returns to scale.

Long run average cost curve

Example

  • Consider following total cost for three values of \(K\).
  • Quantity Total Cost (\(K=1\)) Total Cost (\(K=2\)) Total Cost (\(K=3\))
    1 10 40 80
    2 5 20 60
    3 10 9 40
    4 20 7 20
    5 40 10 10
  • Find and draw LRAC curve.
  • Identify regions with economies of scale, diseconomies of scale, and constant returns to scale.

Quiz

  • A company has one factory producing integrated circuits. CEO of this company considers buying another factory to scale the production. The total cost of producing circuits is given in the table.

  • Find Long Run Average Cost (LATC) for each quantity and draw its curve.

  • Suppose market conditions are such that the company have to produce 1 unit of \(Q\). Should the company buy a second factory?

  • Quantity Total Cost (1 factory) Total Cost (2 factories)
    0 2 N/A
    1 1 5
    2 2 2
    3 5 1
    4 N/A 2

Wrap up

  • We considered the cost analysis of the firm and showed the relationship between different types of short run costs.
  • We calculated the short run cost for a given level of long run resource and extrapolated it to find long run cost.
  • We introduced three possible scenarios for long run average cost - economies of scale, constant return to scale, diseconomies of scale.