Intermediate Microeconomics Spring 2008

ANSWERS TO MIDTERM 1 1. A firm produces game consoles using equipment and labor. When it uses E machine-hours of equipment and hires L person-hours of labor, it can manufacture up to Q game consoles. The relationship between Q, E, and L is as follows: Q = E 1/3 L2/3 . The firm must pay PE for each machine-hour of equipment it uses and PL for each personhour of labor it hires. Suppose the production manager is told to produce Q = 500 game consoles per month and that she wants to choose E and L to minimize costs while achieving that production target. (a) What is the objective function for this problem? (b) What is the constraint? (c) Which of the variables (Q, E, L, PE , and PL ) are exogenous? Which are endogenous? Explain. (d) Write a statement of the constrained optimization problem. ANSWER (This is end of chapter problem 1.3 slightly modified) a) The production manager wants to minimize total costs T C = PE E + PL L. b) The constraint is to produce Q = 500 units, so the manager must choose E and L so that Q = 500. c) The endogenous variables are E and L, because those are the variables over which the production manager has control. By contrast, the exogenous variables are Q, PE, and PL because the production manager has no control over their values and must take them as given. d) min T C = PE E + PL L such that Q = 500. 2. Suppose that guacamole and salsa are substitutes. Suppose also that in January you observe an increase in the price of salsa, while good weather increases the output of avocados. What will be the effect of these simultaneous changes on the equilibrium price and quantity of guacamole? Explain your answer graphically. ANSWER A rise in the price of salsa increases demand for guacamole, shifting its demand curve to the right (increasing QD and putting upward pressure on its price). The rise in the output of avocados will reduce their price. Since avocados are an input to guacamole, a drop in their price will shift the supply of guacomole to the right as well (increasing QS and putting downward pressure on its price). The net effect of these simultaneous 1

changes is a rise in equilibrium quantity. However, the change in the equilibrium price is ambiguous. 3. Suppose the demand curve for chips is given by Qd = 10−P +I where I is income. You observe that when I = 20 the market for chips clears at P = 15 and Q = 15. You also observe that when I = 30 the market for chips clears at P = 20 and Q = 20. (a) Estimate (i.e. write the equation for) the supply curve, ∆y x .) (b) What is the price elasticity of supply when I = 20?(Recall εy,x = ∆x y (c) What is the price elasticity of demand when I = 30? (d) What is the income elasticity of demand when I = 20? ANSWER s 20−15 = 20−15 = 1. Plug in a (a) Slope of the supply curve Qs = c + dP is d = ∆Q ∆P S pair {P, Q } into the supply equation to get c : 15 = c + 15 this yields c = 0. So the supply curve is QS = P. Note that income affects demand but not supply (Principles of Microeconomics). S P 20 = 1. = 1 20 (b) The formula is (see above) εQS ,P = ∆Q ∆P QS ∆QD P = −1 20 ∆P QD 20 ∆QD I εQD ,I = ∆I QD = 1 20 15

(c) The formula is (see above) εQD ,P =

= −1.

(d) The formula is (see above) = 43 . Note that we use the quantities demanded and not the equilibrium quantities above, as we are trying to find the income elasticity of demand. 4. Suppose the demand curve for SUV’s is given by Q = 10 − P. At what price will demand be unitary elastic? ANSWER D D P P P The formula we need to use is: εQD ,P = ∆Q = ∆Q = 1 10−P = −1. This ∆P QD ∆P 10−P yields P = −(10 − P ) which, as any 5-year old knows, yields P = 5. 5. For the following goods, explain whether you would expect the elasticity (of demand or supply as specified) to be greater in the long run or the short run. (a) supply of gas (b) demand for gas. ANSWER In both cases the long-run price elasticity is higher than the short-run, as agents have time to adjust over time. Thus, consumers can switch to more fuel-efficient vehicles, while suppliers can, for instance, build more capacity.

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