EconS 301 ‐ REVIEW EXERCISES FOR MIDTERM #1 Chapter 2 Review Questions 2.12. Consider a linear demand curve, Q = 350 – 7P a) Derive the inverse demand curve corresponding to this demand curve. Q = 350 − 7 P 7 P = 350 − Q (Remember, the inverse demand curve is the demand curve solved for P) P = 50 − 17 Q b) What is the choke price? The choke price occurs at the point where Q = 0 . Setting Q = 0 in the inverse demand equation above yields P = 50 . That is, the choke price shows us the price at which consumers will not demand any quantity of the good. c) What is the price elasticity of demand at P = 50? At P = 50 , the choke price, the elasticity will approach negative infinity. Since elasticity equals the percentage change in Quantity over the percentage change in price, at Q=0, the elasticity will approach negative infinity. 2.16. Consider the following demand and supply relationships in the market for golf balls: Qd = 90 – 2P – 2T and QS = ‐9 + 5P – 2.5R, where T is the price of titanium, a metal used to make golf clubs, and R is the price of rubber. a) If R = 2 and T = 10, calculate the equilibrium price and quantity of golf balls. Substituting the values of R and T, we get Demand : Q d = 70 − 2 P Supply : Q s = −14 + 5P In equilibrium, 70 – 2P = –14 + 5P, which implies that P = 12. Substituting this value back, Q = 46. b) At the equilibrium values, calculate the price elasticity of demand and the price elasticity of supply. Elasticity of Demand = –2(12/46), or –0.52. Elasticity of Supply = 5(12/46) = 1.30. Notice that -2 is simply the (∆Q/∆P), or derivative with respect to P of the demand function. 1 c) At the equilibrium values, calculate the cross‐price elasticity of demand for golf balls with respect to the price of titanium. What does the sign of this elasticity tell you about whether golf balls and titanium are substitutes or complements? 10 ) = −0.43 . The negative sign indicates that titanium and golf balls are 46 complements, i.e., when the price of titanium goes up the demand for golf balls decreases. Remember that people like to consumer complements together so the increase in the price of one is essentially like an increase in the price of the other, and therefore, the demand will decrease. ε golf ,ti tan ium = −2( 2.20. Suppose that the market for air travel between Chicago and Dallas is served by just two airlinesm United and American. An economist has studied this market and has estimated that the demand curves for round‐trip tickets for each airline are as follows: = 10,000 – 100PU + 99PA (United’s demand) = 10,000 – 100PA + 99PA (American’s demand) Where PU is the price charged by United, and PA is the price charged by American. a) Suppose that both American and United charge a price of $300 each for a round‐trip ticket between Chicago and Dallas. What is the price elasticity of demand for United flights between Chicago and Dallas? QUd = 10000 − 100(300) + 99(300) QUd = 9700 Using PU = 300 and QUd = 9700 gives ⎛ 300 ⎞ ⎟ = −3.09 (Simply plug numbers into the price elasticity of demand ⎝ 9700 ⎠ ε Q , P = −100 ⎜ equation) b) What is the market‐level price elasticity of demand for air travel between Chicago and Dallas when both airlines charge a price of $300? (Hint: Because United and American are the only two airlines serving the Chicago‐Dallas market, what is the equation for the total demand for air travel between Chicago and Dallas, assuming that the airlines charge the same price?) Market demand is given by Q d = QUd + QAd . Assuming the airlines charge the same price we have… 2 as x increases (holding y constant). Consider the utility function U(x. This implies an elasticity equal to ⎛ 300 ⎞ ⎟ = −. b) Do the consumer’s preferences exhibit a diminishing marginal utility of x? Is the marginal utility of y diminishing? b) Since MU x = y 2 x . marginal utility of y. In the following pictures. more of each good is better. U2 > U1. MU y = x . a) I like both peanut butter and jelly.0309 ⎝ 19400 ⎠ ε Q . and always get the same additional satisfaction from an ounce of peanut butter as I do from 2 ounces of jelly. a) Does the consumer believe that more is better for each good? Since U increases whenever x or y increases.4. P = −2 ⎜ Chapter 3 Review Questions 3. However. Therefore the marginal utility of x is diminishing.Q d = 10000 − 100 PU + 99 PA + 10000 − 100 PA + 99 PU Q d = 20000 − 100 P + 99 P − 100 P + 99 P (PA and PU simply become P) Q = 20000 − 2 P d When P = 300 . MU x falls. Therefore the preferences exhibit a constant. This is also confirmed by noting that MUx and MUy are both positive for any positive values of x and y . Draw indifference curves to represent the following types of consumer preferences. 3.y) = y√ with the marginal utilities MUx = y/(2√ ) and MUy = √ . As y increases.13. MUy does not change. 3 . not diminishing. Q d = 19400 . but dislike jelly. 4 . Jelly U1 U2 Peanut Butter d) I like peanut butter and jelly. but neither like nor dislike jelly. Jelly U1 U2 Peanut Butter c) I like peanut butter.Jelly 4 2 U1 U2 1 2 Peanut Butter b) I like peanut butter. but I only want 2 ounces of peanut butter for every ounce of jelly. draw a typical indifference curve (it need not be exactly to scale.y) = 3x + y. The MUx equation is simply a constant of 3.y? b) MRS x . y = 3 (because MUx/MUy=3/1=3) d) Is the MRS diminishing. but it needs to reflect accurately whether there is a diminishing MRSx.Jelly U1 U2 2 1 2 4 Peanut Butter 3. remain constant. the total utility increases. with MUx = 3 amd MUy = 1. Yes. e) On a graph with x on the horizontal axis and y on the vertical axis. constant. Consider the utility function U(x. so the change in x or y doesn’t change MUx. The marginal utility of x remains constant at 3 for all values of x. c) What is MRSx.y). 5 .15. Label the curve U1. As you add more of either x or y. or increase as the consumer buys more of x? Explain. Also indicate on your graph whether the indifference curve will intersect either or both axis. a) Is the assumption that more is better satisfied for both goods? 2. y remains constant moving along the indifference curve (look at above equation). the “more is better” assumption is satisfied for both goods since both marginal utilities are always positive. or increasing as the consumer substitutes x for y along an indifference curve? The MRS x . b) Does the marginal utility of x diminish. the MRS x . and β are positive constants. 6 . Regardless. When α = 1 . the indifference curves will never intersect either axis. the “more is better” assumption is satisfied for both goods since both marginal utilities are always positive. e & f) The graph below depicts indifference curves for the case where A = 1 and α = β = 0. y will diminish because x is in the denominator and drives down the entire fraction. the marginal utility of x diminishes as x increases. y ) = x 0. we need to specify three possible cases: When α < 1 . The marginal utilities are MUx = αAxα – 1yβ and MUy = βAxαyβ – 1.Y U1 U2 X f) On the same graph draw a second indifference curve U2. where A. the marginal utility of x increases as x increases.15 for this utility function.21. Thus U ( x. a) Yes.5. Suppose a consumer’s preferences for two goods can be represented by the Cobb‐Douglas utility function U = Axαyβ. Answer all parts of Problem 3.5 y 0. When α > 1 . with U2>U1. b) Since we do not know the value of α . the marginal utility of x remains constant as x increases. (See above graph) 3. only that it is positive. c) MRS x . y = αAxα −1 y β αy = βAxα y β −1 βx d) As the consumer substitutes x for y .5 . α. y) = xαy1‐α.y of 4 when x = 4 and y = 8.24. the expression for MRSx. y = MU x MU y αxα −1 y1−α = (plugging in the MU’s and then rearranging) (1 − α )xα y −α α y = 1−α x c) Since we know that MRSx. One type of Cobb‐Douglas utility function is given by U(x. where MUx = αxα‐1y1‐α and MUy = (1‐α)xαy‐α.y is MRS x . α 8 1−α 4 α 2= 1 − α (we set MRS equal to 4 and then plug in x. Suppose that you are told that a person with Cobb‐Douglas preferences has an MRSx.450 U2 400 350 Y 300 250 200 150 U1 100 50 0 0 5 10 15 20 25 30 35 X 3.y values and solve for 2 − 2α = α 4= d) α= 2 3 α) 7 .y = 4 when x = 4 and y = 8. What is the numerical value of α? First. Are the indifference curves bowed in toward the origin? Yes. Food costs $1 a unit. 12 Clothing 10 8 6 4 2 0 0 5 10 15 20 25 30 35 Food c) Draw the indifference curve associated with a utility level of 36 and the indifference curve associated with a utility of 72. Remember: Use the Budget Line and set Py to zero to find horizontal intercept and Px equal to zero to find the vertical intercept. The marginal utilities of food and clothing are MUF = C + 1 and MUC = F. Place the number of units of clothing on the vertical axis and the number of units of food on the horizontal axis.C) = FC + F.Chapter 4 Review Questions 4. She is buying 8 units of food. note that they intersect the F-axis.e. The utility function that Ann receives by consuming food F and clothing C is given by U(F. a) Ann is currently spending all of her income. and clothing costs $2 a unit. Plot her current consumption basket. the indifference curves are convex. If Ann is spending all of her income then… F + 2C = 22 8 + 2C = 22 (Simply plugging in what we know to the Budget Line) 2C = 14 C=7 b) Graph her budget line. How many units of clothing is she consuming? 3.. Also. Ann’s income is $22.2. 8 . i. bowed in toward the origin. find the utility maximizing choice of food and clothing.80 70 U=72 Clothing 60 50 40 U=36 30 20 10 0 0 5 10 15 20 25 30 35 Food d) Using a graph (and no algebra). 80 70 U=72 Clothing 60 50 40 U=36 30 Optimum at F=12. find the utility‐maximizing choice of food and clothing. That is. The tangency condition requires that MU F PF (where slope of BL equals slope of IC) = MU C PC Plugging in the known information yields 9 . find the intersection of the budget line and the indifference curves. C=5 20 10 0 0 5 10 15 20 25 30 35 Food e) Using algebra. Consider a consumer with the utility function U(x. 10 .5y). However. Algebraically.C +1 1 = F 2 2C + 2 = F Substituting this result into the budget line. Combining these two conditions.C = C +1 5 +1 1 = = The marginal rate of substitution is equal to the price ratio. 12 2 F g) Does Ann have a diminishing marginal rate of substitution of food for clothing? Show this graphically and algebraically. plugging y into the budget line. f) What is the marginal rate of substitution of food for clothing when utility is maximized? Show this graphically and algebraically. MRS F . The equation of all these “elbow” points is 3x = 5y. At the optimum the consumer chooses 5 units of clothing and 12 units of food. he could always move to such a point. Yes. y) = (20.6x.C diminishes. 12). 5 x + 10 y = 220 . The prices of the two goods are Px = $5 and Py = $10. such as (5. As F increases and C decreases along an isoquant. or y = 0.C . We can see this in the graph in part c) because the indifference curves are bowed in toward the origin. 6) etc. MRS F . This question cannot be solved using the usual tangency condition. Determine the optimum consumption basket. you can see from the graph below that the optimum basket will necessarily lie on the “elbow” of some indifference curve. MRS F . we get (x. That is. (10. that is. and the consumer’s income is $220.C = C +1 F . the indifference curves do exhibit diminishing MRS F . The usual budget constraint must hold of course. Therefore the optimum point must be such that 3x = 5y. F + 2C = 22 results in (2C + 2) + 2C = 22 4C = 20 C =5 Finally. keeping utility constant and decreasing his expenditure. If the consumer were at some other point.3.y) = min(3x. the two goods are perfect complements in the ratio 3:5. 3). 4. that is. plugging this result back into the tangency condition implies that F = 2(5) + 2 = 12 . even when C = 0. she cannot give up any more CDs. Thus. since she is already at a corner point with C = 0. the marginal utility per dollar spent on CDs is lower than on sandwiches. Instead. S) = (0. Therefore the best Helen can do is to spend all her income on sandwiches: (C.6) (5. Thus the S = 0 corner cannot be an optimum. the optimum will consist of C = 0 and Helen spending all her income on sandwiches: S = 10. the corner optimum is reflected in the fact that the slope of the budget line is steeper than that of the indifference curve. which is of course entirely feasible at this corner point. PC / PS = 3 > MRSC. and Helen can spend a combined total of $30 each day on these goods. [Note: At the other corner with S = 0 and C = 3.S = 0.S = 2. even at the corner point. 10) we have PC / PS = 3 > MRSC. 10). 9C + 3S = 30. S) = (0. Combining this with the budget constraint. or S = 3C + 20.3) x 4. Helen’s preferences over CDs (C) and sandwiches (S) are given by U(S. Thus. Specifically. our assumption that there was an interior solution must be false. Graphically.3. we find that the optimal number of CDs would be given by 18C = −30 which implies a negative number of CDs. notice if there was an interior solution. Since it’s impossible to purchase a negative amount of something.] 11 . note that at (C. with MUC = S + 10 and MUS = C + 10. However.C) = SC + 10(S + C).7. The fact that Helen’s indifference curves touch the axes should immediately make you want to check for a corner point solution.75. the tangency condition implies (S + 10)/(C +10) = 3. If the price of a CD is $9 and the price of a sandwich is $3. See the graph below.12) (10. find Helen’s optimal consumption basket. Helen would prefer to buy more sandwiches and less CDs.y (20. To see the corner point optimum algebraically. Paul’s optimal basket contains all hamburgers and no pizza. allowing him to purchase at most 50 hamburgers or at most 100 pizzas. 50) = 200. Paul’s optimal basket is at point B. Though Paul is happy to get this gift. because MUH /PH = 4/6 > MUP / PP = 1/3. The new budget constraint is ABD and he can now buy a maximum of 120 pizzas.15. H) = (0. How much would she have needed to give him in cash to make him just as well off as with the gift certificate? Paul’s initial budget constraint is the line AC. Knowing that he likes pizza. The $60 cash certificate shifts out his budget constraint without changing the maximum number of hamburgers that he can buy. Paul’s grandmother gives him a birthday gift certificate of $60 redeemable only at Pizza Hut. When he gets the gift certificate. his grandmother did not realize that she could have made him exactly as happy by spending far less than she did. spending 12 . 50). Hamburgers 60 55 50 B A 20 C 100 D 120 Pizza Initially. and Paul’s monthly income is $300. at point A where (P. The price of pizza is $3 and the price of hamburgers is $6.H) = P + 4H. as shown by his utility function: U(P. 4. Paul consumes only two goods. pizza (P) and hamburgers (H) and considers them to be perfect substitutes. His utility level at point A is U(0. So point B is where (P. 50) = 220. that would have given him more minutes. and y denote spending on other goods. So. 50) with a utility of U(20. Plan B: Pay a $20 monthly fee and make calls for $0. He mobile phone company offers him two plans: • • Plan A: pay no monthly fee and make calls for $0.22. If Plan A is better for him. what is the set of baskets he may purchase if his behavior is consistent with utility maximization? What baskets might he purchase if Plan B is better for him? Let x denote the number of phone calls. He spends this money making telephone calls home (measured in minutes of calls) and no other goods. Under Plan A. However. Darrell’s budget line is JLM. These budget lines intersect at point L. Similarly. Thus. including either of these points. Under Plan B. No point between L and M would be optimal under this plan because then Darrell could have chosen a point under Plan B. Graph Darrell’s budget constraint under each of the two plans. y 60 40 J K L M 67 N 120 200 x If we know that Darrell chooses Plan A.50 per minute. Any point between L and K (but not including point L) would be dominated by a point under 13 . we cannot exclude point L itself (Darrell could.20 per minute. Paul could also achieve a utility of 220 by consuming 220/4 = 55 hamburgers. have perfect complements preferences with an “elbow” at point L). To buy the extra 5 hamburgers he would require 5*6 = $30. H) = (20. 4. his optimal bundle must lie on the line segment JL. Darrel has a monthly income of $60. or about x = 67. it is JKLN. between L and N.all of his regular income on hamburgers and the $60 gift certificate on pizza. if Darrell chooses Plan A his optimal basket could be anywhere between J and L. if he had received a cash gift of $30 it would have made Paul exactly as well off as the $60 gift certificate for pizzas. if he chose Plan B then his optimal basket must lie between L and N. However. for instance. and left him with more money to buy other goods. with associated marginal and MU y = 1. when the price of x goes up. Using the tangency condition. Is y a normal good? What happens to the demand for y as Px increases? From the budget constraint. 14 . Chapter 5 Review Questions 5. The demand for x does not tangency condition is = 2 x py 4 px2 1 depend on the level of income. and the budget constraint. a) Calculate his optimal basket when Px=4 and Py=1. a) Derive David’s demand curve for x as a function of the prices. including either of these points. Denoting the level of income by I. y = py I − px x I = − .Plan A between L and J.19 Lou’s preferences over pizza (x) and other goods (y) are given by U(x. 60) with a utility of 900. the demand curve for y is. x y = 4 x Insert int o budget constraint 4 x + 4 x = 120 8x = 120 4(15) + y = 120 y = 60 x = 15 Lou’s initial optimum is the basket (x. Moreover. y) = utility functions MU x = 1 2 x x + y .7 David has a quasi‐linear utility function of the form U(x. b) Derive David’s demand curve for y. 4 x + y = 120 . 5. Thus. with associated marginal utilities MUx=y and MUy=x. the budget constraint implies that p x x + p y y = I and the py2 px . indicating that y is a normal good. Verify that the demand for x is independent of the level of income at an interior optimum. y) = (15. if Darrell chooses Plan B his optimal basket could be anywhere between L and N. which means that x = . His income is $120. the demand for y increases as well. Px and Py. y = 4 .y)=xy. py p y 4 px You can see that the demand for y increases with an increase in the level of income. First we need the decomposition basket. xy = 900 . x This implies that ( x. i. This latter income equals: 3*17. y ) = (10 3 .9 = 103. and the income needed to buy the decomposition basket at the new prices.3.3) + 69. c) Calculate the compensating variation of the price change. That is he 15 . 60). y ) = (10 3 .3 = 2. d) Calculate the equivalent variation of the price change. This would satisfy the new tangency condition.30 3 ) or approximately (17.4.3. The compensating variation is the amount of income Lou would be willing to give up after the price change to maintain the level of utility he had before the price change.8. y = 4 .51.9). The equivalent variation is the amount of income that Lou would need to be given before the price change in order to leave him as well off as he would be after the price change. How much income would Lou need to purchase this bundle under the original prices? He would need 4(17. y =3 x y = 3x 3x + 3x = 120 6x = 120 x = 20 3(20) + y = 120 y = 60 Together.8 = $16. these conditions imply that (x. y = 3 and x y = 3x x(3x) = 900 x 2 = 300 x = 17. and the income effect is 20 – 17.9 This gives ( x.3.40 3 ) or approximately (17. The substitution effect is therefore 17. and also such that xy = 1200. This equals the difference between the consumer’s actual income.3 + 1*51. After the price change his utility level is 20(60)=1200. Therefore the additional income should be such that it allows Lou to purchase a bundle (x.e. y) = (20.2 = 138. which satisfies the same tangency condition as the decomposition basket and also the new budget constraint: 3x + y = 120. y) satisfying the initial tangency condition.3 – 15 = 2.2). The compensating variation thus equals 120 – 103. $120. would give him as much utility as before.2. Now we need the final basket. 69.32y = 900 y = 51.32 17.b) Calculate his income and substitution effects of a decrease in the price of food to $3.7. MUF = C + 1 MUC = F Tangency: MUF/MUC = PF / PC. and draw a graph illustrating these effects. (C + 1)2 = 36. (Eq 3) Eq 5 can be written as F(C + 1) = 36. Her marginal utility of food is MUf= C+1 and her marginal utility of clothing is MUc=F.) Decomposition Basket: Must be on initial indifference curve. Also. the initial level of utility is U = FC + F = 12(2) + 12 = 36. She has an income of $20. so the demand for F is F = 12/PF 6 Demand for food PF5 4 3 2 1 2 3 4 6 F 12 b) Calculate the income and substitution effects on Carina’s consumption of food when the price rises from 1 to 4. and C = 2. PFF + 4C = 20.4. (C + 1)/ F = PF/4 => 4C + 4 = PFF.4 – 120) dollars in order to be as well off as if the price of pizza were to decrease instead. Thus C = 2. and draw this demand curve for prices of food between 1 and 6. (Also. we know that F = 12/4 = 3. and thus. So the decomposition basket is F = 6. Initial Basket: From the demand for food in (a). 5. F = 6. (Eq 1) Budget Line: PFF + PCC = I . F = 12/1 = 12.would need to increase his income by (138. 16 . with U = FC + F = 36 (Eq 5) Tangency condition satisfied with final price: MUF/MUC = PF / PC. food F and clothing C. with the Utility function U=FC+F. U = 3(2) + 3 = 9. (Eq 2) Substituting (Eq 1) into (Eq 2): 4C + 4 + 4C = 20. but it should be consistent with the data. C = 5. Income effect on F: Ffinal basket – Fdecomposition basket = 3 – 6 = ‐3. Using Eq 3. Final Basket: From the demand for food in (a). Also. (C + 1)/ F = 4/4 => C + 1 = F. C = 5. From the budget line. independent of PF. and C = 2. by Eq 3. Substitution effect on F: F decomposition basket – Finitial basket = 6 – 12 = ‐6. we see that PFF + 4(2) = 20. Therefore his equivalent variation is $18. Your graph does not need to be to scale.20 Carina buys two goods. The price of clothing is $4. a) Derive the equation representing Carina’s demand for food. Donna’s utility function is U(x. Hence.c) Determine the numerical size of the compensating variation (in monetary terms) associated with the increase in the price of the good from 1 to 4.y)=x2y. Jim’s optimal basket is a solution to equations MUx / MUy = P / Py and P x + Py y = IJ. we have 2xy / x2 = P and P x + y = 100 with solution x = 200 / (3P) and y = 100 / 3. with associated marginal utility functions MUx=2xy and MUy=x2. So she would need an additional income of 24 (plus her actual income of 20). The compensating variation associated with the increase in the price of food is ‐24. Analogous system of equations for Donna is y / x = P and P x + y = 150 with solution x = 75 / P and y = 75. PFF + PCC = 4(6) + 4(5) = 44. 17 .22 There are two consumers on the market: Jim and Donna. a) Find optimal baskets of Jim and Donna when the price of y is Py=1 and price of p is x.y)=xy. Clothing 11 10 9 Initial U 8 7 Final U 6 Decomp Basket 5 4 Final BL 3 Final 2 Initial Basket Decomp BL 1 0 Initial BL 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 Food Inc Effect Subst Effect = -6 5. Approximate shape of the demand curve for Jim and Donna is depicted below. with associated marginal utilities MUx=y and MUy=x. Jim’s utility function is U(x. b) On separate graphs plot Jim’s and Donna’s demand schedule for x for all values of P. Income of Jim of IJ=100 and income of Donna is ID=150. Shape of the demand curve in this case is the same as in part b). Aggregate Demand: Dx(P) = 200 / (3P) + 75 / P = 445 / (3P). When there is one more consumer that has preferences identical to Donna’s then her demand is also 75 / P and hence aggregate demand is Dx(P) = 200 / (3P) + 75 / P + 75 / P = 650 / (3P). 18 . d) Plot aggregate demand when there is one more consumer that has identical utility function and income as Donna.Px x c) Compute and plot aggregate demand when Jim and Donna are the only consumers. or L = Y = w .000 10.000. K is machine‐hours of capital. Chapter 6 Review Questions 6.000.000 15. Then sketch the graphs of the average and marginal product functions. the amount of leisure that w +1 Noah consumes decreases with an increase in the wage rate. So. we see that his supply of labor always increases with an increase in the wage rate. Since PY = 1. w 2 L = (24 − L) w . The 1 MUY = . his labor supply curve is always positively sloped – that is. and this is true no matter what the wage rate is. Also.28 Consider Noah’s preference for leisure (L) and other goods (Y). the number of units of other goods he purchases is Y = (24 – L)w.4 Suppose that the production function for DVDs is given by Q=KL2‐L3. Since the amount of labor that Noah supplies equals (24 – L).000 5.000 0 100 200 300 400 500 600 700 Labor 19 . U(L.000. Find the total product function and graph it over the range L=0 and L=500. and L is man hours of labor. the tangency condition gives us Combining the two conditions.5.000 30. L 24 . it is not backward bending.000 20.000.000.Y ) = associated marginal utilities are MU L = 1 2 L and L + Y . where Q is the number of disks produced per year.000. Suppose that PY=$1. Is Noah’s 2 Y supply of labor backward bending? If Noah’s wage rate is w. then the income he earns from working is (24 – L)w.000. Clearly. At what level of labor L does the average product curve appear to reach its maximum? At what level does the marginal product curve appear to reach its maximum? Total Product Total Product with K=600 35. a) Suppose K=600.000 25. 000.AP and MP with K=600 200000 AP/MP 100000 AP 0 -100000 0 100 200 300 400 500 600 700 -200000 -300000 MP -400000 Labor Based on the figure above it appears that the average product reaches its maximum at L = 300.000 50. The marginal product curve appears to reach its maximum at L = 200 b) Replicate the analysis in (a) for the case in which K=1200 Total Product Total Product with K=1200 300.000.000 200.000 150.000 250.000.000 100.000.000.000 0 500 1000 1500 Labor 20 .000. 9 Suppose the production function is given by the equation Q = L K .0 0 20 40 60 80 Labor 21 . For K = 1200.0 Q=10 4. Q=50. does the total product function have a region of increasing marginal returns? In both instances. The marginal product curve appears to reach its maximum at L = 400. for low values of L the total product curve increases at an increasing rate. it happens for L < 400. For K = 600.0 2. 6.0 Capital 5.0 Q=50 6. Graph the isoquants corresponding to Q=10.0 1. Do these isoquants exhibit diminishing marginal rate of technical substitution? 7. this happens for L < 200.0 0.0 Q=20 3. c) When either K=600 or K=1200.AP and MP with K=1200 1000000 500000 AP AP/MP 0 -500000 0 200 400 600 800 1000 1200 1400 -1000000 -1500000 MP -2000000 Labor Based on the figure above it appears that the average product curve reaches its maximum at L = 600. Q=20. So MRTSL. However. note that MRTSL. K = MPL a = MPK b 6.bL). with the corresponding marginal products MPK=2KL2 and MPL=2K2L. Q = aK + bL = λaK + λbL = λ[ aK + bL = λ[Q] Therefore a linear production function has constant returns to scale. First. What is the marginal rate of technical substitution of labor for capital (MRTSL.18 What can you say about the returns to scale of the Leontief production function Q=min(aK.) Similarly. it cannot have increasing marginal products of capital and labor. the amount of labor is held fixed when we measure MPK. 6. using the production function Q=K2L2. bL ) . 6. 6. where a and b are positive constants? A general fixed proportions production function is of the form Q = min(aK . The MRTS L . Show that this is not true. which diminishes as L increases and K falls as we move along an isoquant. because the amount of capital is held fixed when we measure MPL).Because these isoquants are convex to the origin they do exhibit diminishing marginal rate of technical substitution. K is therefore MRTS L .17 What can you say about returns to scale of the linear production function Q=aK+bL. This exercise demonstrates that it is possible to have a diminishing marginal rate of technical substitution even though both of the marginal products are increasing.K) at any point along an isoquant? For this production function MPL = a and MPK = b . the marginal product of capital MPK is increasing (not diminishing) as K increases (remember. the marginal product of labor is increasing as L grows (again.12 You might think that when a production function has a diminishing marginal rate of technical substitution of labor for capital.K = L/K.11 Suppose the production function is given by the following equation (where a and b are positive constants): Q=aL+bK. 22 .K is diminishing. where a and b are positive constants. for this production function. when? The marginal product of labor is M +1 L Suppose M > 0 . 23 . constant. λbL) = λ min(aK . this production function exhibits constant returns to scale. however. L MPM = 25 +1 M a) Are the returns to scale increasing. will never be MPL = 25 negative since both components of the equation above will always be greater than or equal to zero. In fact. Since output goes up by the same factor as the inputs. increasing L will have the effect of decreasing MPL . increase all inputs by some factor λ and determine if output goes up by a factor more than. or the same as λ Qλ = 50 λ M λ L + λ M + λ L Qλ = 50λ ML + λ M + λ L Qλ = λ ⎡⎣50 ML + M + L ⎤⎦ Qλ = λ Q By increasing the inputs by a factor of λ output goes up by a factor of λ .Q = min(aK . less than.19 A firm produces a quantity Q of breakfast cereal using labor L and material M with the production function Q = 50 ML + M + L . when? Is it ever negative. and if so. The marginal product functions for this production function MPL = 25 are M +1 L . MPL ≥ 1 . or decreasing for this production function? To determine the nature of returns to scale. Holding M fixed. The marginal product of labor is decreasing for all levels of L . b) Is the marginal product of labor ever diminishing for this production function? If so. The MPL . bL) = λ[Q] Therefore the production function has constant returns to scale. 6. bL) = min(λaK . 25 A firm’s production function is initially Q = KL . capital saving. Over time. MRTSL. So there is indeed technological progress.5⎜ ⎟.K is lower with the second production function. or neutral? With the initial production function With the final production function MRTS L . With any positive amounts of K and L. the technological progress is labor-saving. 24 .5⎜ ⎟ and MPL = 0.5 K = . with MPK = L and MPL = 0. Thus. the production function changes to ⎝ K⎠ ⎝ L⎠ ⎛ K ⎞ Q = K L.5⎜ ⎟ . b) Is this change labor saving. with ⎛ L⎞ ⎛ K⎞ MPK = 0. K = MPL 0.6. ⎝ L⎠ a) Verify that this change represents technological progress. KL < K L so more Q can be produced with the final production function. MPK L For any ratio of capital to labor.