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A price-taking and profit-maximizing firm produces one output at the rate y> 0 using one input r>0 by way of the production function () , where f(x)竺2x2 . The firms output sells at the price p >0 while the input is purchased at the price wo (a) (b) (c) Determine the lalue of the input that solves the FONC, and denote it by x (p,w). Is Set up the profit maximization problem. Derive the FONC and SOSC. x(p,w) unique? Explain. Does x(p,w) yield a local or a global solution to the profit maximization problem? Explain What is x(p,w), ie., is it a demand function, supply function, neither, or something else? Explain. (d) (e) Substitute x-x(p,w) in the FONC and confirm that an identity in (p,w) results (f) Show that the law of demand holds for x(p,w) (g) Show that the firm uses more of the input when the output prices increases. (h) Show that x* (67,Bw)E x*(p.w) for all θ > 0 , and interpret the result. (i) Find the profit maximizing value of output, say y (p,w) ) Show that the law of supply holds for y(p,w) (k) Show that the output of the firm decreases as the input price increases. (1) Show that y, (θρ.0w) y*(PM (m) Find the maximum value of profit, say T (p,w) (n) Show that on (p,w)/ap y(p,w). Is the firm better off if the price at which it sells its ) for all >0, and interpret the result. output is higher? (o) Show that on (p,w)aw- (p,w). Is the firm better off if the price at which it buys its input is higher? Show that π*(9p,9w) Ξθπ(p, w) for all θ > 0 , and interpret the result. (p)

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