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A production function for a perfectly competitive firm is given as Q=AX1^(1/2)X2^(1/2) where Q is the output (in tons) and Xi are quantities of inputs used in the production of Q (in tons). The cost of producing Q is given as C=summation of WiXi where i=1,2, where Wi are per unit input prices of input 1 and 2 respectively. 1. Show that the cost function in it's general form is constant with concavity, homogeneity and shephard's lemma properties of cost function?
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A production function for a perfectly competitive firm is given as Q=A...

Properties of the Cost Function

Cost function in its general form is constant with concavity, homogeneity, and Shephard's lemma properties, as explained below:

Constant Cost
- The cost function is constant because the cost of producing the output Q is given by the sum of the prices of inputs used, which do not vary with the level of output. This implies that the cost function remains constant regardless of the quantity produced.

Concavity
- The cost function exhibits concavity because the production function given is concave with respect to inputs. As the inputs increase, the marginal product of each input diminishes, leading to increasing marginal costs. This concavity property of the production function translates into a concave cost function.

Homogeneity
- The cost function demonstrates homogeneity because it is a linear function of input prices and quantities. This means that if we were to double all input prices and quantities, the cost function would also double. This property of homogeneity allows for easy analysis of cost behavior with changes in input levels.

Shephard's Lemma
- Shephard's lemma states that the cost function can be derived from the production function by taking partial derivatives with respect to input prices. In this case, the cost function C can be obtained by summing the product of input prices and quantities, which aligns with Shephard's lemma.

In conclusion, the cost function for the perfectly competitive firm exhibits constant, concavity, homogeneity, and Shephard's lemma properties, making it a valuable tool for analyzing cost behavior in production.
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A production function for a perfectly competitive firm is given as Q=AX1^(1/2)X2^(1/2) where Q is the output (in tons) and Xi are quantities of inputs used in the production of Q (in tons). The cost of producing Q is given as C=summation of WiXi where i=1,2, where Wi are per unit input prices of input 1 and 2 respectively. 1. Show that the cost function in it's general form is constant with concavity, homogeneity and shephard's lemma properties of cost function?
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