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Unformatted text preview: ECON 101 – SOLUTIONS TO PS 1 GARTH BAUGHMAN (1) The firm Oz Productions makes boomerangs. Its cost function is given by C ( q ) = 1 3 q 3 , where q is the quantity of boomerangs produced. (a) Calculate the firm’s supply curve (i.e., what is the profit-maximizing choice of q as a function of the price of the good, q * ( p ))? Solution. We assume that the firm is a price taker and so the firm solves max q pq − C ( q ) . This gives the following first order condition: p = C ( q ). We know C ( q ) = d dq (1 / 3) q 3 = q 2 . So we solve p = q 2 for q and get q * ( p ) = √ p . (b) How does the profit-maximizing quantity change in response to price changes, i.e., what is the slope of the supply function calculated in part 1a? Solution. Whenever we see the word ‘slope’ we instantly think derivative. In this case, the derivative of q * ( p ) with respect to p . So, we calculate slope of supply = d dp q * ( p ) = d dp √ p = 1 2 p − 1 / 2 . (c) Directly verify that this slope is also given by the formula dq * ( p ) dp = 1 C ( q * ( p )) we derived in the first lecture. Solution. When we are asked to verify a formula like this, all we have to do is calculate each individual piece and then assemble them. After that, it’s just algebra to make the left hand side look like the right. We already know the left hand side from above: dq * ( p ) dp = 1 2 p − 1 2 Now, we need to find C ( ˙ ) and compose it with q * ( p ) C ( q ) = 1 3 q 3 = ⇒ C ( q ) = q 2 = ⇒ C ( q ) = 2 q and q * ( p ) = √ p from above. Putting it together, we get 1 C ( q * ( p )) = 1 2 q * ( p ) = 1 2[ √ p ] = 1 2 p − 1 2 which is what we wanted, so we’re done. 1 2 GARTH BAUGHMAN (2) Suppose boomerangs are produced using a single input, labour, according to the production function q = f ( ) = (3 ) 1 / 3 , i.e., a quantity of lobar will produce (3 ) 1 / 3 boomerangs. Suppose the price of lobar is w . (a) How much lobar is needed to produce q boomerangs? Solution. We know how many boomerangs are produced from a given amount of lobar, q = f ( ). So, we need only invert this function to arrive at the amount of lobar needed to produce a given amount of boomerangs. We need to solve q = (3 ) 1 / 3 for . To this end, q = (3 ) 1 / 3 = ⇒ q 3 = 3 = ⇒ = q 3 3 ≡ f − 1 ( q ) . (b) How much does it cost the firm to produce q boomerangs (i.e., what is the firm’s cost function)? Solution. We know it costs a firm w to use units of lobar. So, since f − 1 ( q ) units of lobar are used to produce q units, we can write C ( q ) = wf − 1 ( q ) = wq 3 / 3 for the cost of producing q units....
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This note was uploaded on 03/02/2011 for the course ECON 101 taught by Professor Dannicatambay during the Spring '08 term at UPenn.
- Spring '08