Given a short run cost function as T
2 60 100
3
1 3 2 C Q Q Q
, find the minimum value
of AVC and MC.
To find the minimum value of AVC, we need to calculate the average variable cost (AVC) at different levels of output (Q) and identify the lowest value.
AVC = TC/Q
To find the minimum value of MC, we need to calculate the marginal cost (MC) at different levels of output (Q) and identify the lowest value.
MC = ΔTC/ΔQ
First, let's calculate the values of AVC and MC for different output levels using the given short-run cost function T(Q).
Q = 1:
T(1) = 2(1)^2 + 60(1)^1 + 100 = 2 + 60 + 100 = 162
T(0) = 2(0)^2 + 60(0)^1 + 100 = 0 + 0 + 100 = 100
TC(1) = T(1) - T(0) = 162 - 100 = 62
AVC(1) = TC(1)/Q = 62/1 = 62
MC(1) = ΔTC/ΔQ = (T(1) - T(0))/(1 - 0) = (162 - 100)/(1 - 0) = 62
Q = 2:
T(2) = 2(2)^2 + 60(2)^1 + 100 = 8 + 120 + 100 = 228
T(1) = 2(1)^2 + 60(1)^1 + 100 = 2 + 60 + 100 = 162
TC(2) = T(2) - T(1) = 228 - 162 = 66
AVC(2) = TC(2)/Q = 66/2 = 33
MC(2) = ΔTC/ΔQ = (T(2) - T(1))/(2 - 1) = (228 - 162)/(2 - 1) = 66
Q = 3:
T(3) = 2(3)^2 + 60(3)^1 + 100 = 18 + 180 + 100 = 298
T(2) = 2(2)^2 + 60(2)^1 + 100 = 8 + 120 + 100 = 228
TC(3) = T(3) - T(2) = 298 - 228 = 70
AVC(3) = TC(3)/Q = 70/3 ≈ 23.33
MC(3) = ΔTC/ΔQ = (T(3) - T(2))/(3 - 2) = (298 - 228)/(3 - 2) = 70
Therefore, the minimum values of AVC and MC can be observed at Q = 2, with a value of 33 for both.