Question

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.

Answer 1

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.