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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. 3. Given your understanding of Hotelling's Lemma explain why the supply function is positively sloped?
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Hotelling's Lemma and the Positively Sloped Supply Function

Understanding Hotelling's Lemma is crucial in explaining why the supply function of a perfectly competitive firm is positively sloped. Hotelling's Lemma states that the cost-minimizing input demand for a firm is equal to the marginal cost of production.

Cost Minimization and Hotelling's Lemma

In the production function Q=AX1^(1/2)X2^(1/2), the firm minimizes its cost by choosing the optimal combination of inputs X1 and X2 to produce output Q. The cost of producing Q is given as C=summation of WiXi, where Wi are the per unit input prices.

Supply Function and Hotelling's Lemma

Hotelling's Lemma implies that the marginal cost of production is equal to the cost-minimizing input demand for the firm. As the firm increases its output Q, it needs to increase the quantities of inputs X1 and X2 to maintain the optimal input combination. This leads to an increase in the cost of production, resulting in a positively sloped supply function.

Positive Slope of the Supply Function

Therefore, the positively sloped supply function of a perfectly competitive firm can be explained by Hotelling's Lemma. As the firm increases its output, it needs to increase its input quantities, leading to higher production costs and a higher supply price. This relationship between output, input quantities, and production costs results in a positively sloped supply curve.
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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. 3. Given your understanding of Hotelling's Lemma explain why the supply function is positively sloped?
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