Short problems

1. “When a good is inferior, the demand for the good increases when its price increases.” Is this statement correct? Explain precisely.2. “In the short-run, the marginal rate of technical substitution(RTS) will not necessarily be equal to the ratio of input prices.” Is thisstatement correct? Draw a graph and explain.3. What is the short-run supply for a price-taking firm? Il-lustrate your answer with short-run marginal cost, short-run averagecost, and short-run average variable cost curves and clearly mark theprofit level at the intersections of these curves.

4. A firm produces outputqusing a production functionf(x1,x2) = (min{αx1,βx2})1γ. At the input prices(w1,w2), using Sh-phard’s lemma, calculate the contingent demand forx1andx2.

5. Labor is used in everything from the harvesting of avocadosto the manufacture of zippers, but not all processes are equally laborintensive. Assume that the industries A and Z both use inputs oflabor and capital, but that the production of A’s product (avocados) isinherently more labor-intensive than Z’s product (zippers).

1. (a) Depict their respective positions on an isoquant/isocost diagram,assuming that both face the same price for labor and capital andthat both are operating at the same level of cost.
2. (b) Assume that the price of capital now falls. Which industry willend up expanding the most? Defend your conclusion by referringto changes in the diagram.

Long problems

6. On mount Olympus everyone lives forever and everygood is available in unlimited amounts. But, alas, even the blessedOlympians have only 24 hours in each day.

(a)Explain how scarcity persists even in Olympus. Is it fair to saythat even in Olympus, “There ain’t no such thing as a free lunch”?

(b)If it takes 2 hours to eat a kilo of ambrosia and 1 hour to drink aliter of nectar, diagram the daily budget constraint of an Olympianwho does nothing but eat and drink. What is the real price ofambrosia in terms of nectar? What is an Olympian’s real dailyincome in terms of nectar?

(c)Draw the indifference curves and optima (1) of an ambrosia loverand (2) of a nectar lover.

(d)If you observe Zeus both eating ambrosia and drinking nectarwhat must be his marginal rate of substitution (MRS) at his pointof consumption? What can you say about his MRS if he eats butdoesn’t drink? Illustrate your answers.

7. A firm produces outputqusing a production functionf(L,K) = (18)L12K12whereLis labor andKis capital. CapitalKis fixedat a levelK=16, and its price isr=10. Denote the price of labor asw, and the price of output asp.

(a)Forw=2, calculate the firm’s short-run total cost functionSTC(q), short-run average cost functionSAC(q), and short-runmarginal cost functionSMC(q)?

(b)If this a price-taking firm and the market price isp, calculate thefirm’s profit functionπ= (q,p)? What is the optimal level ofoutputq∗(p)? ifp=160what are the optimal levels of outputq∗and profitπ∗? Assuming there is free market entry and exit,what will happen in the long-run?

(c)If this firm faces the following downward-sloping inverse de-mand functionp=a−bq(aandbare both positive real num-bers), calculate the firm’s profit function? What is the optimallevel of outputq∗? What would happen asbdecreases (b→0)?

(d) At the input prices(r,w), calculate the firm’s long-run total costfunctionLTC(q), long-run average cost functionLAC(q), andLong-run marginal cost functionLMC(q)?