7. Shaki makes and sells backpack danglies. The total cost in dollars for Shaki to make q danglies is given by C(q) = 10+.02q³. Find the quantity that minimizes Shaki's average cost for making danglies.
8. Note You may recall that elasticity of demand at price p with quantity offer q is defined as E = that is the ratio between the relative (percent) change of da da 2/p the demand and the relative (percent) change of price. The "Leibniz notation" for derivatives (writing, say, instead of f'(x)) is so popular because it allows us to treat that "fraction" as if it actually was one (it isn't), so that we can also write E = or E= |d, and there is a rigorous way to prove that these are indeed equivalent expressions, even if we obtained them by "cheating", as if we were dealing with actual fractions. It's common to ignore the sign of dade (which is usually negative, since p >,q> 0, but, practically always, d<0), hence the formulas with the absolute value. How does this affect revenue? Note that looking at elasticity does not address the global maximum of revenue, but shows how sensitive revenue is for small changes in pricing a local, not a global feature. Now, revenue is price time demand (that is, sales), or pq. With a small change in p and q, call them dp and dq, we can check the corresponding small change in revenue d(pq). Up to first order (we ignore products of two small changes) that gives d(pq) = pdq+qdp (rules of differentiation: dp dq is considered negligible compared to pdq+gdp). Now, let dp < 0 (lower the price, and since <0, it is dq > 0), so, since both [pdg p and q are positive, qdp < 0. But if E> 1 (elastic demand). > 1, or odp pdq> qdp, so that pdq+qdp> 0 (note that the second sum term is negative). That is revenue marginally increases, for a marginal drop in price. The demand function for Shaki's danglies is given by q = 12 - 2p (q is the number of danglies, p is the price in dollars per dangly). Find the elasticity of demand when p = $3.5. 1. Should Shaki raise or lower his price to increase revenue? 2. We know Shaki's cost from question 7. What price will maximize profit (profit is revenue minus cost, and we know how to write revenue as a function of price) (note that you will have to find cost as a function of the price p, from the info in question 7 (cost as a function of quantity produced/sold), and in this question (demand function). Note Make sure you take into account the domains for p and q due to the constraints p≥ 0,42 0, when deciding whether your answers to these question are acceptable or spurious.