If a firm's total cost function Tc= 100-12Q+0.6Q2 where TC is total cost in birr and Q is units of output then find the following
MC,TFC,TVC,AVC
To find the marginal cost (MC), we need to find the derivative of the total cost function (TC) with respect to the quantity of output (Q).
TC = 100 - 12Q + 0.6Q^2
To find the marginal cost, we take the derivative of TC with respect to Q:
MC = dTC/dQ = -12 + 1.2Q
To find the total fixed cost (TFC), we need to identify the portion of total cost that does not change with the level of output. In this case, the fixed cost is the constant term of the total cost function, which is 100.
TFC = 100
To find the total variable cost (TVC), we need to identify the portion of total cost that changes with the level of output. In this case, the variable cost is the portion of the total cost function that depends on the level of output, which is -12Q + 0.6Q^2.
TVC = -12Q + 0.6Q^2
To find the average variable cost (AVC), we need to divide the total variable cost by the quantity of output (Q):
AVC = TVC/Q = (-12Q + 0.6Q^2)/Q = -12 + 0.6Q
Assume that a firms production function is given as Q= L2/3 k1/3. Determine MPL,MPK,MRTSLK and whether production function exhibits increasing, decreasing or constant returns to scale
To find the marginal product of labor (MPL), we take the derivative of the production function with respect to labor (L):
Q = L^(2/3) * K^(1/3)
Taking the derivative with respect to L:
MPL = (2/3) * L^(-1/3) * K^(1/3)
To find the marginal product of capital (MPK), we take the derivative of the production function with respect to capital (K):
MPK = (1/3) * L^(2/3) * K^(-2/3)
To find the marginal rate of technical substitution of labor for capital (MRTSLK), we take the derivative of L with respect to K:
MRTSLK = -∂Q/∂K / ∂Q/∂L
MRTSLK = -[(1/3) * L^(2/3) * K^(-2/3)] / [(2/3) * L^(-1/3) * K^(1/3)]
Simplifying,
MRTSLK = -K / 2L
The production function exhibits increasing returns to scale if doubling the inputs (L and K) more than doubles the output (Q).
To check for returns to scale, we can examine the relationship between the total product and total input. Let's consider doubling the inputs:
Q(2L, 2K) = (2L)^(2/3) * (2K)^(1/3) = 2^(2/3) * L^(2/3) * 2^(1/3) * K^(1/3)
Q(2L, 2K) = 2 * (L^(2/3) * K^(1/3)) = 2 * Q(L, K)
Since doubling the inputs leads to doubling the output, the production function exhibits constant returns to scale.
To find the marginal cost (MC), we need to take the derivative of the total cost function (TC) with respect to the quantity of output (Q).
TC = 100 - 12Q + 0.6Q^2
MC = dTC/dQ
Taking the derivative of TC with respect to Q:
MC = -12 + 1.2Q
Therefore, the marginal cost (MC) is -12 + 1.2Q.
To find the total fixed cost (TFC), we can observe that the fixed cost is the intercept when output (Q) is zero. In this case, it is equal to 100 birr.
Therefore, the total fixed cost (TFC) is 100 birr.
To find the total variable cost (TVC), we can subtract the total fixed cost (TFC) from the total cost (TC).
TVC = TC - TFC = 100 - 100 = 0 birr.
Therefore, the total variable cost (TVC) is 0 birr.
To find the average variable cost (AVC), we need to divide the total variable cost (TVC) by the quantity of output (Q).
AVC = TVC / Q
When TVC is 0 birr, the average variable cost is also 0 birr.
Therefore, the average variable cost (AVC) is 0 birr.
To find the marginal cost (MC), we need to take the derivative of the total cost function with respect to the quantity of output (Q).
Given the total cost function: TC = 100 - 12Q + 0.6Q^2
1. Marginal Cost (MC):
First, differentiate the total cost function with respect to Q:
dTC/dQ = -12 + 1.2Q
The derivative of the total cost function with respect to Q gives us the marginal cost function. Thus, MC = -12 + 1.2Q.
2. Total Fixed Cost (TFC):
Total fixed cost (TFC) is the cost that does not vary with the quantity of output (Q). In this case, since there is no explicit fixed cost term, TFC is zero.
3. Total Variable Cost (TVC):
To find the total variable cost (TVC), subtract the total fixed cost (TFC) from the total cost (TC):
TVC = TC - TFC = TC - 0 = TC
In this case, TVC is equal to the total cost function itself: TVC = 100 - 12Q + 0.6Q^2.
4. Average Variable Cost (AVC):
To calculate the average variable cost (AVC), divide the total variable cost (TVC) by the quantity of output (Q):
AVC = TVC / Q = (100 - 12Q + 0.6Q^2) / Q
Simplifying further, we get:
AVC = 100/Q - 12 + 0.6Q
Now you have the equations for MC, TFC, TVC, and AVC based on the given total cost function.