Finance 00 Practice Midterm # Soutions ) Consider the foowing production function. Suppose that capita is fied at. Q K. L.05L For what vaues of Q is margina cost increasing? For what vaues of Q is margina cost decreasing? The margina product of abor for K = is MP.8 L.5L If the margina product of abor is increasing, the first derivative wi be positive. MP L.8.0L So, the margina product of abor is increasing for L between 0 and.67 and decreasing for L greater than.67. With K = and L =.67, Q =.9 So, Margina cost is increasing when Q is greater than.9 and decreasing for Q<.9 ) Suppose that ou have the foowing production process: k..8 a) Given the above production function what wi margina cost and average cost functions ook ike in the short run? Epain. The margina product for the above production function is MP.8k.. The margina product of abor is decining in abor. Margina costs are equa to w MC MP
As quantit increases in the short run, abor increases. As abor increases, the margina product of abor decines, so margina costs increase. If margina costs are increasing, then average costs are increasing. b) Given the above production function, what wi margina cost and average cost ook ike in the ong run? Epain. This function has increasing returns to scae..8...8 k k So average costs are decining in the ong run as Q increases. If average costs are decining, margina costs are decining. c) What percentage of this firm s ong run costs be capita costs? Labor costs? In the ong run, capita s share of epenses are./(.+.8) =. (.%) and abor s share of epenses are.8/(.) =.667 (66.7%) d) Suppose that the government offers this firm a subsid that owers its capita costs b 0% per unit. What wi happen to this firm s demand for abor? What about the demand for capita? B how much wi margina costs decrease in the short run? How about the ong run? The easticit of abor demand with respect to the price of capita is. p k.. The easticit of capita demand with respect to the price of capita is.8 p k.67. So, when the price of capita fas b 0%, abor demand fas b (- 0)(.) = 6.6% and capita demand rises b (-0)(-.67) =.%. The easticit of margina costs with respect to the price of capita is. p k.. So, margina costs fa b (0)(.) = 6.6%
) Suppose that ou have the foowing technoog for producing output. k The price of abor is $7 per hour and capita costs $50 per unit. You are current using 6 units of capita. You need to produce 00 units of output. a) Cacuate our optima choice for capita and abor. We need to set up the maimization probem min $7L k, $50K Subject to the constraint that output is at east 00 units. ( k, 00 ) Setting up the probem: k, $7L $50K k 00 Take the derivatives with respect to K and L and set them equa to zero L K k, $7 k 0 k, $50 k 0 Now, sove each for ambda. $7 k $50 k Rearranging, we get
k k $50 $7 This simpifies to L K The cost minimizing combination of capita and abor is 5 hours of abor for ever unit of capita. Now, use the output constraint 00 k k k Sove for k and 00 k k 6 b) Cacuate our tota cost of production. TC $ 7L $50K $7(6) $50() $6,5 c) Cacuate our average (unit) cost. $6,5 AC 00 $65. d) Cacuate our ependiture shares for capita and abor. That is, what percentage of our costs are abor costs, what percentage are capita costs)? TC $ 7L $50K $7(6) $50() $6,5 Labor Costs = $7(6) = $, Capita Costs = $50() = $,00 w $,.50 TC $6,5 pk k $,00.50 TC $6,5
) Suppose that we have the foowing observations of consumer behavior: There are two products avaiabe to purchase: Cheeseburgers and Mikshakes Observation #: When the price of cheeseburgers was $ and the price of mikshakes was $, an individua purchased 5 cheeseburgers and mikshakes. Observation #: When the price of cheeseburgers was $ and the price of mikshakes was $, an individua purchased cheeseburgers and 5 mikshakes. Are these two observations consistent with a rationa choice? What we are ooking for here is something that indicates the persona preferences of this consumer. Specifica, we know the foowing: If a consumer chooses option A over option B when A is more epensive than B, then the consumer is reveaing that he/she strict prefers option A to option B. Take the first observation: With the prices of cheeseburgers and mikshakes given b $ and $ respective, the cost of 5 cheeseburgers and mikshakes is $(5) + $() = $8. Meanwhie, the cost of the aternative choice ( cheeseburgers and 5 mikshakes) is $() + $(5) = $6. Therefore, we can definite sa that the combination (5 CB, MK) is preferred to ( CB, 5MK). Now, take the second observation: With the prices of cheeseburgers and mikshakes given b $ and $ respective, the cost of 5 cheeseburgers and mikshakes is $(5) + $() = $7 whie, the cost of the aternative choice ( cheeseburgers and 5 mikshakes) is $() + $(5) = $7. Therefore, we can definite sa that the combination ( CB, 5 MK) is preferred to (5 CB, MK). Observation #: (5 CB, MK) is preferred to ( CB, 5MK) Observation #: ( CB, 5 MK) is preferred to (5 CB, MK) No. This is not consistent with rationa behavior! 5) Suppose that the price of good X is $6 and the price of good Y is $. You have $0 to spend and our preferences over X and Y are defined as, U a) Cacuate the margina utiit of X (remember, this is the change in utiit resuting from a sight increase in consumption of X)
The margina utiit of X is the derivative with respect to X U, b) Cacuate the margina utiit of Y. The margina utiit of Y is the derivative with respect to Y U, c) Cacuate the margina rate of substitution. What does the margina rate of substitution measure? The MRS is the ratio of the above two epressions U, MRS U, The margina rate of substitution measures the rate at which the consumer is wiing to trade Y for X. d) Suppose that ou chose to consume 0 units of X, 0 units of Y. Is this an optima choice at the current prices? Epain. The rue for maimizing utiit is U MRS U, P, P Here, the ratio of the price of X to the price of Y is, but when X = 0, Y = 0 0 MRS 0 This choice can t be optima! e) If the answer to (d) is no, cacuate our optima choice of X and Y. I am going to answer this the ong wa.fee free to use shortcuts!
We have the utiit function, U Reca the genera method for maimization probems: Where, f and g 6 0,. Therefore, we can set up the agrangian as 6 0, Taking the first order conditions 0, 0 6, Soving for ambda 8 8 Setting the above two equa to each other 8 8 If we simpif this down a bit: 0,, ma, g to subject f
8 8 (Mutip both sides b 8) (Divide both sides b ) (Divide both sides b ) Now use the constraint to sove the rest of the probem. 6( ) 0 7.5 6 0 Y = 7.5, X = 7.5 Note that a income is spent. ($6)(7. 5) + ($)(7.5) = $0. Further, note that ou are spending (/) of our income on X, (/) of our income on Y. 6) Microeconomics focuses on two primar paers: consumers and firms. Both of these paers are soving optimization probems. a) Brief, epain the probem that each economic paers faces. Consumers are endowed with a set of preferences. The have a imited amount of income to spend on various goods and services. The face a set of market prices and must seect a choice of consumption to maimize their we being. The resut of this process is a demand curve. Firms actua have a coupe optimization probems: First, the face a production decision. The are endowed with a technoog to convert inputs into output. The face a set of input prices (these are determined in factor markets). The must choose a production process that wi generate enough output to match their epected saes and minimize the costs of production. The second stage is deciding the price to charge. Taking costs as given, the firm must seect a price to maimize profits given the properties of the demand the face.
b) Brief describe the ogic behind the soution process. Both the consumer probem and the producer are soved in virtua identica was. Step #: Given a set of prices, seect the proper mi of goods: U MRS U, P, P (Consumer) TRS Fk F k, Pk k, P (Firm) Step #: Choose an appropriate scae to satisf our constraint: p p I (Consumer) k (Firm) c) In what was are these probems simiar? Are there an important differences? The on rea notabe difference is that the firms decisions are restricted in the short run (capita can t be adjusted). Therefore, whie the consumer is awas acting efficient, the firm isn t. 7) Suppose that ou have estimated the foowing demand curve: Q 5.5P. 0I Where I represents income and P is price. Suppose that average income is equa to $5,000. Cacuate the price easticit of demand at P=$65. If ou were a revenue maimizing firm, woud it be optima to charge a price of $65? At a price of $65, demand is equa to Q 5.5(65).0(5,000) 8.5
dq dp P Q 65.5.5 8.5 Demand is ver eastic at this price (specifica, bigger than ). Therefore, this firm shoud consider owering its price to attract more saes. 8) Suppose that ou have estimated the foowing regression (standard errors associated with each are beow in parentheses). Your sampe size is 50. Q d 00 P (6.5) (.) (60.5) a) Cacuate our forecast at the sampe average of $50. Q d 0 00 00 50 b) Cacuate the 95% confidence interva for our forecast. A 95% confidence is equa to our estimate +/- standard errors where the error is cacuated as: Var Q P SD Q d d / $50 60.5, 68.65 50 / P $50, 68.65 60.70 c) Cacuate our estimated demand easticit at the sampe average of $50. % Q % P Q P 50 P Q 00 d) Wh might ou be worried about cacuating an estimate of demand at a price of $70? You are starting to stra quite a was from the sampe mean. The estimates get worse!! e) Suppose that ou knew that Income is high correated with quantit, but that Income was negative correated with price. Shoud income be incuded in this regression? Epain.
If I eave income out of the regression, I have mode specification error. When price is high, income is ow and when income is ow, quantit is ow, so m estimation of the coefficient on price is too negative (it is capturing some of the effect of income on quantit). However, if I incude income, I have a muticoinearit probem which wi screw up m standard errors. Idea, I shoud add a variabe to this regression that is high correated with income, but not correated with price. 9) Suppose that ou did a time series anasis on E-Commerce retai saes. Your data is quarter from 999 Q unti 06 Q. Saes are in miions of doars. Your regression equation incudes a time trend and quarter dummies. E t D D D SUMMARY OUTPUT R Squared 0.97 Standard Error 665.6 Observations 67 Coefficient Standard error t-stat Intercept 66.0 00.69.55 Time 7.55.05 0.9 Q -966.9 8.8 -.06 Q -96.6 8.98 -.06 Q -096. 7.9 -.9 a) Is there a significant seasona component in the data? Epain. Yes, the data suggests that e-saes are around 9-0B arger in the th quarter than the are in the other three quarters. These coefficient are significant at the 95% eve. b) B how much do retai saes grow per ear? e-saes grow b.7 biion per quarter, so $5B per ear. c) The Durbin Watson Statistic for this regression is.7. What does that mean?
It sas that there is positive autocorreation in the residuas. This means that whie m estimated coefficients are unbiased, the standard errors wi not be correct. Suppose that ou run the regression again, but this time, ou take the natura og of e- saes. n E t D D D SUMMARY OUTPUT R Squared 0.98 Standard Error.58 Observations 67 Coefficient Standard error t-stat Intercept 9..05 7.8 Time 0.00.00 8.8 Q -.0.055 -.65 Q -.9.055 -.98 Q -..055 -.8 d) How do our answers to (a) and (b) change? Now, e-saes grow b % per quarter (6% per ear) and that th quarter saes are around 0% higher than the rest of the ear. 0) Suppose that ou have the foowing data on heating oi usage: Heating Oi Usage (in Thousands of Barres) 995Q,500 997Q 5,00 995Q,750 997Q,75 995Q,000 997Q,500 995Q 5,00 997Q 0,750 996Q,00 998Q,50 996Q,50 998Q,5 996Q,50 998Q 6.00 996Q 0,680 998Q 5,5 a) Cacuate a forecast for usage in the first quarter of 999 using a moving average with a ength of. Your forecast woud be equa to the average of the previous quarters:
,50,5 6,00 5,5 MA ( ),5 b) Repeat (a) using an eponentia smoothing mode with a smoothing parameter of. (assume that our forecast for 998Q was,500). 5,5.6,500, 790 999Q. c) How woud ou compare the performance of the methods in (a) and (b)? For various methods, ou coud compare root mean squared errors. The smaer the RMSE, the better the forecast. d) Wh shoud ou be carefu to check for the presence of a trend or seasonait before using the methods in (a) and (b)? If there is seasonait or a trend in the data, there are better methods for forecasting.