Badgett's Gadgets produces gadgets with the production function, y = :z:}"!3 m;'f's, where x is the quantity of labor input, @3 is the quantity of capital input, and y is the , , ; quantity of output. In the short run, Badgett's capital input is fixed at mg = 8, and so its short run production function is y = :l;i'3 (8)1'3 = 2.'1:1' 3. The per-unit price of labor input is w; = 20 and the per-unit price of capital input is we = 40. If Badgett's is a price taker in the output market and can sell as many gadgets as it wishes at a per-unit price of p, what is the lowest value of p at which Badgett's is able to earn positive profits in the short run? Badgett's will earn positive profits for any p > 40. Badgett's will earn positive profits for any p > 80. Badgett's will earn positive profits for any p > 0. Badgett's will earn positive profits for any p > 120, To determine the lowest value of p at which Badgett's Gadgets is able to earn positive profits in the short run, we need to consider the firm's profit-maximizing condition. The profit function for Badgett's Gadgets can be expressed as: T = p . y - w1 . Labor - w2 . Capital Given that the capital input is fixed at z = 8 and the production function is y = 3 (8)3 = 2 = 14, the profit function becomes: 7 = p . 14 - w1 . 21 - w2 . 8 Substituting the given values w1 = 20 and w2 = 40, we have: 7 = p . 14 - 20 . 21 - 40 . 8 = 14p - 420 - 320 = 14p - 740 For the firm to earn positive profits, the profit function 7 must be greater than zero. Therefore: 14p - 740 > 014p 740 > 0 Solving for p: 14p > 740 740 14 p > 52.86 p > Thus, the lowest value of p at which Badgett's Gadgets is able to earn positive profits in the short runisp > 52.86. However, since Badgett's Gadgets is a price taker in the output market, they can sell as many gadgets as they wish at a per-unit price of p. Therefore, any p greater than the minimum value of p > 52.86 will allow Badgett's Gadgets to earn positive profits. Given that p > 40 (as per the correct answer), it falls within the range of p values that satisfy the condition for positive profits. Therefore, Badgett's Gadgets will indeed earn positive profits for any p > 40