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##### Find an expression for the price elasticity of demand for this equipment

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solution

**Question**

1.;A policy adviser suggests that in order to improve its balance of trade with China;Canada should lower the price of some heavy machinery equipment. Suppose the;demand function for this equipment is;Q;5,000;(3 p 1) 2;where Q is quantity per year and p is price (measured in hundreds of thousands of;dollars).;a) Find an expression for the price elasticity of demand for this equipment.;Suppose the current p is equal to 3, at this price, is the demand elastic or;inelastic?;b) Continuing from a), explain (in words) whether you think Canada will;succeed in raising its revenue if the policy advisers suggestion is;implemented.;*;As.sign.ment (-sin-mnt) n. A thing or task that is assigned to a person (Oxford American Dictionary, 1980).;(15);2.;A reasonably realistic model of a firms costs is given by the short-run CobbDouglas cost curve;C Tq;1;F;where C is total cost, q is output, is a positive parametric constant, F is the fixed;cost, and T measures the technology available to the firm (also a parameter).;Assume T > 0.;a) Comment on whether this total cost function is (strictly) concave or;(strictly) convex if < 1.;b) Assuming that < 0.5, use calculus of optimization to find the value;of q that minimizes the average cost. (Note: Your answer for optimal;q will be in terms of the parameters of the model.) Be sure to check;your second-order condition.;(15);3.;You run a small furniture business. You sign a deal with a customer to deliver up;to 400 chairs, the exact number to be determined by the customer later. The price;will be $90 per chair up to and including 300 chairs, and above 300, the price will;be reduced by $0.25 per chair (on the whole order) for every additional chair over;300 ordered. What are the largest and smallest revenues your company can make;under this deal? [Be sure to use calculus of optimization to work out your answers;and check your second-order condition.];(10);4.;A landscape architect plans to enclose a 3000 square-foot rectangular region in a;botanical garden. She will use shrubs costing $45 per foot along three sides and;fencing costing $20 per foot along the fourth side. By using calculus and checking;your second order condition, find the minimum total cost.;(10);5.;At a price of $8 per ticket, a musical theatre group can fill every seat in the;theatre, which has a capacity of 1500. For every additional dollar charged, the;number of people buying the ticket decreases by 75. Use calculus to find what;ticket price maximizes revenue. Be sure to check your second order condition.;(25);6.;The attendance (denoted by the variable F, measured in thousands of fans) at a;Blue Jays home game is approximated by;F = 150WP-;where W is the fraction of the games they have won so far (0 W 1) and P;is the price of a general admission ticket (suppose 7 P 10 dollars).;i);Find the partial derivatives F / W;and F / P.;ii);Suppose currently W = 0.512 and P = $8. Find the numerical value of;the partial derivatives above and tell me in words what they mean.;iii);Find the partial elasticity of F with respect to P.;iv);Now suppose attendance was determined by W alone in the form of F =;150W. Is this function (strictly) concave or convex? Explain. Why do;you think your answer makes sense? (Be brief.);v);Consider the equation for F given in part iv) above. Suppose W depends;on the team coachs ability A in the following manner;W = 2A1/5.;Use the Chain rule to find an expression for;dF. (Note: Your final;dA;answer must be in terms of A alone.);(15);7.;The supply and demand equations of a good are given by;Qs = 8 + P;Qd = (80/3) ()P;respectively. P is measured in dollars. Suppose the government decides to;impose a constant per unit tax of $t on the supplier.;i) Find the equilibrium quantity in terms of t.;ii) Using the expression in part i) find the value of t that maximizes the;governments total tax revenue. (Make sure you verify that your secondorder condition is satisfied.)

Paper#15530 | Written in 18-Jul-2015

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