ECON 323 - Exam 2
Suppose that a firm's cost function is given by: C(q)=4q. If the market price is $10, then the firm's profit from producing and selling 200 units is?
$1200 Profit(q)=pq-C(q)=10(200)-4(200)=1200.
John manages his own company and receives $35,000 a year for it. The best salary that John would be able to find in a different company is $90,000 a year. The economic cost of John's labor is:
$90,000 The economic cost of a resource is value of its best alternative use.
Consider a firm with production function f(L,K)=L^(1/7) * K^(6/7). Assume that the price of capital r=3 and the price of labor w=2. Suppose that a worker strike constrains the firm to a level of labor x units lower than its original level. What is a good approximation of the amount of capital the firm should increase in order to maintain its original level of production?
(2x)/3
Suppose that a firm has a production function f(L,K)=min{2L,K}. From the following combination of labor and capital (L,K), which one belongs to the same iso-quant as (3,90)?
(5,6) (3,90)=min{2*3,90}=6; the only combination (L,K) that produces 6 from the options is (5,6)
The following figure shows the demand and supply in a market. What is the dead weight loss if the government decides to impose a specific tax of $11.9 collected from the producer? (The following is a description of the graph: In a two-axis graph we measure q in the horizontal axis and dollars $ in the vertical axis. Two curves are shown S and Q. Curve S is an increasing function. It is piece-wise linear, i.e., it is formed by two line segments; the first line segment connects the point (0,3) to the point (20,16.7); for q>=20 S continues as an increasing line that starts at (20,16.7) and passes through the point (30,23). Curve Q is also piece-wise linear; it is a decreasing function formed by two line segments; the first segment connects the points (0,40) and (20,28.6); for q>=20 Q continues as a decreasing line that starts at (20,28.8) and passes through (30,23). Six areas in the figure are labeled with lowercase letters: a is the triangle with vertices (0,40), (0,28.6), and (20,28.6); b is the rectangle with extremes (0,28.6) and (20,23); c is the rectangle with extremes (0,23) and (20,16.7); d is the triangle with vertices (0,3), (0,16.7) and (20,16.7); e is the triangle with vertices (20,28.6), (20,23) and (30,23); and f is the triangle with vertices (20,23), (20,16.7) and (30,23).)
-59.5. If the government imposes a specific tax of $11.9 collected from the producer, the market supply will shift up by $11.9. Then, the new equilibrium will be p=28.6 and q=20. Total welfare in equilibrium is 548.5 and total welfare when p=28.6 and q=20 is 489, then DWL=489-548.5=-59.5.
The following figure shows demand and supply in a certain market. (The following is a description of the figure: In a two-axis graph we measure q in the horizontal axis and dollars $ in the vertical axis. Two curves S and Q are shown. S is a line with positive slope whose vertical intercept is positive. Q is a decreasing line whose vertical intercept is above the vertical intercept of S. Denote by (q*,p*) the point at which S and Q intersect; this point is such that q* is positive and p* is in between the vertical intercepts of S and Q. A level of prices p1 is shown. This price is above the vertical intercept of S, and below p*. The level of q associated by S to p1 is labeled q1. This q1 is to the left of q*. Denote by p' the level of price such that Q(p')=q1. This p' is greater than p*. Six labels for areas in the figure are shown. These labels represent the size of the respective area. They are: a is the triangle with vertices given by the vertical intercept of Q, (0,p'), and (q1,p'); b is the rectangle with extremes (0,p') and (q1,p*); c is the rectangle with extremes (0,p*) and (q1,p1); d is the triangle with vertices given by the vertical intercept of S, (0,p1), and (q1,p1); e is the triangle with vertices (q1,p'), (q1,p*), and (q*,p*); and f is the triangle with vertices (q1,p1), (q1,p*), and (q*,p*).) If government imposes a price cap p1 firms will produce their optimal production at price p1, which is q1, and consumers will be willing to buy this production at this price. What is the dead weight loss from this market intervention?
-E-F The loss in producer surplus is -c-f and loss in consumer surplus is c-e. Then the total loss is -e-f.
The following figure shows demand and supply in a certain market. (The following is a description of the figure: In a two-axis graph we measure q in the horizontal axis and dollars $ in the vertical axis. Two curves S and Q are shown. S is a line with positive slope whose vertical intercept is positive. Q is a decreasing line whose vertical intercept is above the vertical intercept of S. Denote by (q*,p*) the point at which S and Q intersect; this point is such that q* is positive and p* is in between the vertical intercepts of S and Q. A level of prices p1 is shown. This price is above the vertical intercept of S, and below p*. The level of q associated by S to p1 is labeled q1. This q1 is to the left of q*. Denote by p' the level of price such that Q(p')=q1. This p' is greater than p*. Six labels for areas in the figure are shown. These labels represent the size of the respective area. They are: a is the triangle with vertices given by the vertical intercept of Q, (0,p'), and (q1,p'); b is the rectangle with extremes (0,p') and (q1,p*); c is the rectangle with extremes (0,p*) and (q1,p1); d is the triangle with vertices given by the vertical intercept of S, (0,p1), and (q1,p1); e is the triangle with vertices (q1,p'), (q1,p*), and (q*,p*); and f is the triangle with vertices (q1,p1), (q1,p*), and (q*,p*).) If government imposes a price cap p1 firms will produce their optimal production at price p1, which is q1, and consumers will be willing to buy this production at this price. What is the loss in producer surplus from this market intervention compared to the market equilibrium?
-c-f Loss in producer surplus from this market intervention is the difference in producer surplus in the market intervention minus the one in market equilibrium = d-(c-f-d)=-c-f.
In a competitive market, the demand and supply curves are Q(p) = 12 - p and S(p) = 5P, respectively. What percentage of the total surplus is lost if the government imposes a sales tax (ad valorem) of 20% collected from the producer?
0.16%. Total welfare here is 60 (50 Consumer surplus and 10 producer surplus). The after tax supply is given by St(p)=S(0.8p)=0.8x5p=4p. The equilibrium in the market with taxes is p=12/5=2.4 and q=48/5=9.6. Notice that if S(p)=9.6, then p=9.6/5=1.92. Thus the DWL is given by the size of the triangle with vertices (9.6,1.92), (9.6,2.4) and (10,2). This triangle has basis 0.48 and height 0.4. Thus, the DWL is 0.096. This is 0.16% of 60.
A call center has a production function: f(L,K)= 30L + 240K. What is the Marginal Rate of Technical Substitution of labor for capital MRTSLK(L,K)?
1/8. MPL(L,K)=df/dL(L,K)= 30 and MPK(L,K)=df/dK(L,K)= 240. Then MRTSLK(L,K)= MPL(L,K)/ MPK(L,K)=1/8
Assume that the economic cost of capital in the market (also called return) is 3% a year. Jane is analyzing the prospect of investing in a restaurant in a segment of the market that is competitive and for which there is free entry. She has the option to invest $100,000.00 in the restaurant, and sign a contract that determines her return as follows: she will be paid 2% of interest on her capital contribution, plus a 50% share of the business profit. What will be the return of Jane's investment if all other resources used by the restaurant are paid its economic cost?
2.5%. Since the market is competitive, and has free entry, then in equilibrium the restaurant will make zero economic profit. Since they pay only 2% for the capital and the economic cost of capital is 3%, then the restaurant will have a $100,000.00(0.01)=$1,000.00 operation profit. She will receive 2% of interest and 0.5($1000)=$500 on profit distribution. She will receive in total 2.5% on her investment.
Consider a firm that has production function f(L,K)=5L1/3K2/3. What is the value of the Marginal Product of labor when L=1 and K=8?
20/3 MPL (L,K) = 5K^(2/3) / 3L^(2/3)
Consider a firm whose production function is f(L,K)=5L^(1/2) * K^(1/2). If K is equal to 1, for what level of labor is the Average Product of Labor equal to 1?
25. APL is f(L,K)/L. If K=1, then APL is 5L^(1/2) / L = 5/L^(1/2). Thus, APL=1 if L^(1/2)=5, or equivalently L=25.
The following figure shows the demand and supply in a market. What is the value of the Consumer Surplus in equilibrium? (The following is a description of the graph: In a two-axis graph we measure q in the horizontal axis and dollars $ in the vertical axis. Two curves are shown S and Q. Curve S is an increasing function. It is piece-wise linear, i.e., it is formed by two line segments; the first line segment connects the point (0,3) to the point (20,16.7); for q>=20 S continues as an increasing line that starts at (20,16.7) and passes through the point (30,23). Curve Q is also piece-wise linear; it is a decreasing function formed by two line segments; the first segment connects the points (0,40) and (20,28.6); for q>=20 Q continues as a decreasing line that starts at (20,28.8) and passes through (30,23). Six areas in the figure are labeled with lowercase letters: a is the triangle with vertices (0,40), (0,28.6), and (20,28.6); b is the rectangle with extremes (0,28.6) and (20,23); c is the rectangle with extremes (0,23) and (20,16.7); d is the triangle with vertices (0,3), (0,16.7) and (20,16.7); e is the triangle with vertices (20,28.6), (20,23) and (30,23); and f is the triangle with vertices (20,23), (20,16.7) and (30,23).)
254. The equilibrium in this market is p=23 and q*=30. CS is the net area below the market demand and the horizontal line for p=23 up to q=30. This area is composed of three sub areas a+b+e: a is a triangle with base length 20 and height 114; thus a=114; b is a rectangle of sides 5.6 and 20, so b=112; e is a triangle with base length 10 and height 5.6, so e=28. Then, a+b+e=254.
Assume that the economic cost of capital in the market (also called return) is 3% a year. Jane is analyzing the prospect of investing in a restaurant in a segment of the market that is competitive and for which there is free entry. She has the option to invest $100,000.00 in the restaurant, and sign a contract that determines her return as follows: she will be paid x% of interest on her capital contribution, plus a 90% share of the business profit. What is the minimum x% for which Jane is willing to invest if all other resources used by the restaurant are paid its economic cost?
3%.
A call center has a production function: f(L,K)= 30L + 240K. What is the Marginal Product of Labor when L=1200 and K=1?
30. MPL(L,K)=df/dL(L,K)=30 independently of where it is measured.
The following figure shows the demand and supply in a market. What is the value of the Producer Surplus in this market if consumers pay a price of $28.6 and consume 20 units? (The following is a description of the graph: In a two-axis graph we measure q in the horizontal axis and dollars $ in the vertical axis. Two curves are shown S and Q. Curve S is an increasing function. It is piece-wise linear, i.e., it is formed by two line segments; the first line segment connects the point (0,3) to the point (20,16.7); for q>=20 S continues as an increasing line that starts at (20,16.7) and passes through the point (30,23). Curve Q is also piece-wise linear; it is a decreasing function formed by two line segments; the first segment connects the points (0,40) and (20,28.6); for q>=20 Q continues as a decreasing line that starts at (20,28.8) and passes through (30,23). Six areas in the figure are labeled with lowercase letters: a is the triangle with vertices (0,40), (0,28.6), and (20,28.6); b is the rectangle with extremes (0,28.6) and (20,23); c is the rectangle with extremes (0,23) and (20,16.7); d is the triangle with vertices (0,3), (0,16.7) and (20,16.7); e is the triangle with vertices (20,28.6), (20,23) and (30,23); and f is the triangle with vertices (20,23), (20,16.7) and (30,23).)
375. PS is the net area below horizontal line for p=28.6 up to q=20 units and above the market supply. This is total area can be decomposed in the subareas b+c+d: b+c is a rectangle with sides 11.9 and 20, so b+c=238; d is a triangle with base length 13.7 and height 20; thus d=137. Thus, PS= 238+137=375.
Suppose a firm has the following cost function when capital is fixed: C(q)=100+4q2. The minimum price necessary for the firm to earn non-negative profit is?
40. The firm has positive profits if p>AC. Then, the minimum price necessary for the firm to earn non-negative profits is that at which p=AC. Since profit maximization implies p=MC, then it must be the case that MC(q)=AC(q). Thus, 8q=(100+4q2)/q. Thus, q2=25, and q=5. Since p=MC(q)=8q, then p=8(5)=40.
A call center has a production function: f(L,K)=40L+200K. The maximal amount of calls that the call center may receive given that L=1 and K=2 is?
440
The following figure shows the demand and supply in a market. What is the total welfare in the market if consumers pay a price of $28.6 and consume 20 units and firms produce this amount and sell at p=$28.6? (The following is a description of the graph: In a two-axis graph we measure q in the horizontal axis and dollars $ in the vertical axis. Two curves are shown S and Q. Curve S is an increasing function. It is piece-wise linear, i.e., it is formed by two line segments; the first line segment connects the point (0,3) to the point (20,16.7); for q>=20 S continues as an increasing line that starts at (20,16.7) and passes through the point (30,23). Curve Q is also piece-wise linear; it is a decreasing function formed by two line segments; the first segment connects the points (0,40) and (20,28.6); for q>=20 Q continues as a decreasing line that starts at (20,28.8) and passes through (30,23). Six areas in the figure are labeled with lowercase letters: a is the triangle with vertices (0,40), (0,28.6), and (20,28.6); b is the rectangle with extremes (0,28.6) and (20,23); c is the rectangle with extremes (0,23) and (20,16.7); d is the triangle with vertices (0,3), (0,16.7) and (20,16.7); e is the triangle with vertices (20,28.6), (20,23) and (30,23); and f is the triangle with vertices (20,23), (20,16.7) and (30,23).)
489 CS=114 and PS=375 so, Total welfare=CS+PS=489.
The following figure shows the demand and supply in a market. What is the total welfare in the market equilibrium? (The following is a description of the graph: In a two-axis graph we measure q in the horizontal axis and dollars $ in the vertical axis. Two curves are shown S and Q. Curve S is an increasing function. It is piece-wise linear, i.e., it is formed by two line segments; the first line segment connects the point (0,3) to the point (20,16.7); for q>=20 S continues as an increasing line that starts at (20,16.7) and passes through the point (30,23). Curve Q is also piece-wise linear; it is a decreasing function formed by two line segments; the first segment connects the points (0,40) and (20,28.6); for q>=20 Q continues as a decreasing line that starts at (20,28.8) and passes through (30,23). Six areas in the figure are labeled with lowercase letters: a is the triangle with vertices (0,40), (0,28.6), and (20,28.6); b is the rectangle with extremes (0,28.6) and (20,23); c is the rectangle with extremes (0,23) and (20,16.7); d is the triangle with vertices (0,3), (0,16.7) and (20,16.7); e is the triangle with vertices (20,28.6), (20,23) and (30,23); and f is the triangle with vertices (20,23), (20,16.7) and (30,23).)
548.5 The equilibrium in this market is p=23 and q*=30. Then CS=254 and PS=294.5. Then Total welfare=CS+PS=548.5.
A firm has supply function S(p)=200p. If consumers pay a price p=50 and the government is collecting an ad-valorem tax of 8.25% from the producers' revenue, then the amount supplied by the firm is?
9175 units. If we have an ad-valorem tax of 8.25% (a sales tax is an ad-valorem tax), then the amount produced by the firm is S(p(1-0.0825))=S(0.9175p). If p=50, then the firm's supply is S(0.9175(50))=S(45.875)=9175.
The following figure shows demand and supply in a certain market. (The following is a description of the figure: In a two-axis graph we measure q in the horizontal axis and dollars $ in the vertical axis. Two curves S and Q are shown. S is a line with positive slope whose vertical intercept is positive. Q is a decreasing line whose vertical intercept is above the vertical intercept of S. Denote by (q*,p*) the point at which S and Q intersect; this point is such that q* is positive and p* is in between the vertical intercepts of S and Q. A level of prices p1 is shown. This price is above the vertical intercept of S, and below p*. The level of q associated by S to p1 is labeled q1. This q1 is to the left of q*. Denote by p' the level of price such that Q(p')=q1. This p' is greater than p*. Six labels for areas in the figure are shown. These labels represent the size of the respective area. They are: a is the triangle with vertices given by the vertical intercept of Q, (0,p'), and (q1,p'); b is the rectangle with extremes (0,p') and (q1,p*); c is the rectangle with extremes (0,p*) and (q1,p1); d is the triangle with vertices given by the vertical intercept of S, (0,p1), and (q1,p1); e is the triangle with vertices (q1,p'), (q1,p*), and (q*,p*); and f is the triangle with vertices (q1,p1), (q1,p*), and (q*,p*).) What is the consumer surplus in this market situation if there is no government intervention?
A+B+E. CS is the net area below the market demand and the horizontal line for equilibrium price and up to the equilibrium quantity. This is a+b+e.
Consider a firm with production function f(L,K)=2L+4K. Assume also that the price of capital r=3 and the price of labor w=2. What is this firm's cost functions when both L and K are variable?
AC(q)=3/4. MC(q)=3/4 C(q)=3q/4
Consider the following figure (The following is a description of the figure: it shows a two-axis graph; the horizontal axis measures labor and the vertical axis measures output; for a level of K fixed, the graph shows that maximal production that the firm can achieve with different levels of labor; the graph starts at zero production for zero labor; then it is increasing in all of its range; five units of labor is shown as reference in the horizontal axis; the corresponding production for this level of labor is 200; the graphs slope is initially increasing, then there is an inflection point to the left of five levels of labor; after this inflection point, the slope of the graph is decreasing; a line that passes through zero and is tangent to the graph is also shown; this line is tangent to the graph for a level of labor that is to the left of 5.). From the graph we learn that for the corresponding K:
APL(5,K)=40
Consider the following graph of a production function when capital is constant. (The following is a description of the figure: it shows a two-axis graph; the horizontal axis measures labor and the vertical axis measures output; for a K fixed, the graph shows that maximal production that the firm can achieve with different levels of labor; the graph starts at zero production for zero labor; then it is increasing in all of its range; three levels of labor are shown as reference; there are L1, L2, and L3; they are related as follows L1<L2<L3; the graph is convex from 0 to L1, that is, its slope is increasing; the graph is concave from L1 on, that is, its slope is decreasing; the line that is tangent to the curve at L2, passes through the origin of the graph.). From the graph we know that for the corresponding K:
APL(L1,K)<MPL(L1,K) APL(L,K) is the slope of the line that interpolates the production function when for K fixed and the origin; for L1, this line is flatter than the tangent to the production function at L1; thus, APL(L1,K)<MPL(L1,K); none of the other relations are true.
Consider a firm with production function f(L,K)=3L+8K. Assume that capital is fixed at K=12. Assume also that the price of capital r=10 and the price of labor w=3. What are the cost functions associated with this?
C(q)=24+q. MC(q)=1. AC(q)=24/(q+1) VC(q)= (q-96) AVC(q)=1-96/q. L(q,12)= (q/3)-32.
Consider a firm with production function f(L,K)=3L^(1/3) * K^(2/3). Assume that capital is fixed at K=1. Assume also that the price of capital r=5 and the price of labor w=3. What are the cost formulas associated with this?
C(q)=5+q^3/9. AC(q)=5/q+ q^2 / 9. MC(q)=q^2 / 3. VC(q) = q^3/9 AVC(q)=q^2 / 9. F = 5. AF(q)=5/q.
The following figure shows demand and supply in a certain market. (The following is a description of the figure: In a two-axis graph we measure q in the horizontal axis and dollars $ in the vertical axis. Two curves S and Q are shown. S is a line with positive slope whose vertical intercept is positive. Q is a decreasing line whose vertical intercept is above the vertical intercept of S. Denote by (q*,p*) the point at which S and Q intersect; this point is such that q* is positive and p* is in between the vertical intercepts of S and Q. A level of prices p1 is shown. This price is above the vertical intercept of S, and below p*. The level of q associated by S to p1 is labeled q1. This q1 is to the left of q*. Denote by p' the level of price such that Q(p')=q1. This p' is greater than p*. Six labels for areas in the figure are shown. These labels represent the size of the respective area. They are: a is the triangle with vertices given by the vertical intercept of Q, (0,p'), and (q1,p'); b is the rectangle with extremes (0,p') and (q1,p*); c is the rectangle with extremes (0,p*) and (q1,p1); d is the triangle with vertices given by the vertical intercept of S, (0,p1), and (q1,p1); e is the triangle with vertices (q1,p'), (q1,p*), and (q*,p*); and f is the triangle with vertices (q1,p1), (q1,p*), and (q*,p*).) If government imposes a price cap p1 firms will produce their optimal production at price p1, which is q1, and consumers will be willing to buy this production at this price. What is the change in consumer surplus from this market intervention compared to the market equilibrium (that is, consumer surplus with government intervention minus consumer surplus in the competitive market without intervention)?
C-E The loss in consumer surplus from this market intervention is the difference in consumer surplus in the market intervention minus the one in market equilibrium =(a+b+c)-(a+b+e)=c-e.
A firm has a production function f. If for each pair (L,K), f(2L, 2K)= 2f(L, K), we say the firm has:
Constant returns to scale
Consumer & Producer Surplus
Consumer Surplus: Above the price level and below the demand curve Producer Surplus: Below the price level and above the supply curve Total Welfare = CS + PS
The following figure shows demand and supply in a certain market. (The following is a description of the figure: In a two-axis graph we measure q in the horizontal axis and dollars $ in the vertical axis. Two curves S and Q are shown. S is a line with positive slope whose vertical intercept is positive. Q is a decreasing line whose vertical intercept is above the vertical intercept of S. Denote by (q*,p*) the point at which S and Q intersect; this point is such that q* is positive and p* is in between the vertical intercepts of S and Q. A level of prices p1 is shown. This price is above the vertical intercept of S, and below p*. The level of q associated by S to p1 is labeled q1. This q1 is to the left of q*. Denote by p' the level of price such that Q(p')=q1. This p' is greater than p*. Six labels for areas in the figure are shown. These labels represent the size of the respective area. They are: a is the triangle with vertices given by the vertical intercept of Q, (0,p'), and (q1,p'); b is the rectangle with extremes (0,p') and (q1,p*); c is the rectangle with extremes (0,p*) and (q1,p1); d is the triangle with vertices given by the vertical intercept of S, (0,p1), and (q1,p1); e is the triangle with vertices (q1,p'), (q1,p*), and (q*,p*); and f is the triangle with vertices (q1,p1), (q1,p*), and (q*,p*).) If government imposes a price cap p1 firms will produce their optimal production at price p1, which is q1, and consumers will be willing to buy this production at this price. What is the producer surplus in this market situation?
D. PS is the net area below horizontal line for p=p1 up to q1 units and above the market supply. This is d.
Does the following figure show the market supply for a firm and the market supply that would be induced by an ad-valorem sales tax of t percent? (This is a description of the figure: In a two-axis graph we measure quantity q in the horizontal axis and dollars $ in the vertical axis. We show two curves: one is labeled S and the other S+t; curve S follows the vertical axis until a positive value, then it is strictly increasing; that is, a higher price has a higher associated quantity; for price p-t, S associates a positive value S(p-t); curve S+t is the vertical translation of curve S by value t; in particular for price p curve S+t associated value S(p-t).)
False. Shows a specific tax: fixed amount of tax/unit.
Does the following production function represents an industry in which there is free entry: f(L,K)=100L^(1/2) * K^(1/3)?
False. The function is a Cobb-Douglas function with alpha=1/2 and beta=1/3. Since alpha+beta=5/6<1 then it satisfies decreasing returns to scale. Free entry would require constant returns to scale. (a+b=1)
Since any market intervention generates a dead weight loss, then there is no reason to impose taxes or regulate markets?
False. We may need to impose taxes to finance the institutions that make possible the operation of the market; we may need to regulate markets because sometimes they are not competitive.
Consider a firm whose cost function when both L and K are variable is shown in the figure below. (The following is a description of the figure. In a two-axis graph we measure q in the horizontal axis and dollars $ in the vertical axis. A single curve labeled C is shown in the figure. It is a linear curve with positive slope).
Flat.
Consider a firm whose cost function when both L and K are variable is shown in the figure below. (The following is a description of the figure. In a two-axis graph we measure q in the horizontal axis and dollars $ in the vertical axis. A single curve labeled C is shown in the figure. It is a linear curve with positive slope). Then, the Average cost function associated with this production is:
Flat. MC = AC. Both are constant
Suppose that the marginal rate of technical substitution of L for K is constant and equal to x. Then?
If the firm substitutes one unit of labor for x units of capital, then production remains constant.
Consider the firm whose MC, AC, AVC, AFC functions are shown in the following graph. (The following is a description of the figure: This figure is a two-axis graph; in the horizontal line we measure output q, and in the vertical line dollars $; there are four curves. The first, MC starts at a positive level when q=0; more precisely, MC(0) is greater than 16 and lower than 30; then MC is decreasing for values of q in between 0 and 50; at 50 MC has a minimum; this minimum is MC(50)=10; after q=50, MC is increasing; in particular MC(100)=16, MC(120)=30. The second curve, AVC, starts at the same level of MC(0); it is decreasing when q is between 0 and 100; in this range AVC is above MC; at q=100, AVC crosses MC; more precisely, AVC(100)=MC(100)=16; for q>100, AVC is increasing and below MC. The third curve, AC, has a positive asymptote at zero, that is, it grows to plus infinity when q is very small; AC is decreasing when q is in between 0 and 120; in this range is above MC; AC(100)=34; AC and MC cross at q=120; more precisely, AC(120)=MC(120)=30; for q>=120, AC is increasing, below MC and above AVC.). Consider the following.
If the output price is equal to $30, then the firm's maximal profits is? $0 If the output price is equal to $16, then firm maximizes profits by producing? 100 units. If the output price is equal to $16, then the maximum profit is equal to? -$1800. If the output price is equal to $21, then the firm maximizes profits by producing? More than 100 but less than 120 units. If the output price is equal to $12, then the firm maximizes profits by producing? 0 units. If the output price is equal to $8, then the firm maximizes profits by producing? 0 units. If the output price is equal to $34, then the firm maximizes profits by producing? More than 120 units. If the output price is equal to $30, then the firm maximizes profit by producing? 120 units.
Consider a firm that has production function f(L,K)=5L^(1/3) * K^(2/3). What is the expression for the marginal rate of technical substitution MRTSLK at (L,K)?
K/2L
Maximizing labor and capital:
Labor = Total Cost/W Capital = Total Cost/R
Consider a firm that has production function f(L,K)=5L^(1/3) * K^(2/3). What is the expression for this firm's Marginal Product of capital?
MPK(L,K)= 10L^(1/3) / 3K^(1/3) MPK(L,K)=derivative of function/derivative of K(L,K)
Consider a firm that has production function f(L,K)=4L^(2/3) * K^(1/3). What is the expression for this firm's Marginal Product of labor?
MPL(L,K)= 8K^(1/3) / 3L^(1/3)
When labor and capital are perfect substitutes:
MRTS are constant ISO-Quants are constant & parallel
Consider a newspaper with production function f(L,K)= 4min{L,K}, where L is the units of labor and K the units of capital they use. Is the Average Product of Labor always equal to the Average Product of Capital for this firm?
No.
The following figure shows the cost function of a firm. It is a two-axis graph in which the horizontal axis measures production output and the vertical axis measures cost in $. The graph shows an increasing function. The slope of the curve is increasing too. Could this be a cost function for a firm participating in a market in which there is free-entry?
No.
The following figure shows the graph of the production function of a firm when capital is fixed at some level K. This is a description of the graph. This is a two axis graph in which the horizontal axis measures L and the vertical axis measures output units. A concave increasing curve is shown. It passes through the points (100,23) and (200,42). Is this the graph of a linear production function when capital is fixed?
No. If f is linear, the graph of the production function when K is fixed is a line.
The following figure shows the graph of the production function of a firm when capital is fixed at some level K. This is a description of the graph. This is a two axis graph in which the horizontal axis measures L and the vertical axis measures output units. A concave increasing curve is shown. It passes through the points (100,23) and (200,42). Is this the graph of a production function f(L,K)=Amin{L,K} for some constant A>0, when capital is fixed?
No. No. If f(L,K)=Amin{L,K} for some constant A>0, the graph of the production function when K is fixed is a line until certain value of L and then is flat.
The MRTSLK(L,K) for a certain firm is constant and equal to 2. Then, if the firm substitutes 2 units of labor FOR one unit of capital?
Production increases.
The MRTSLK(L,K) for a certain firm is constant and equal to 2. Then, if the firm substitutes 2 units of labor for one unit of capital?
Production increases.
Economic Profit =
Revenue - Economic Cost
Suppose that the cost function for an orange juice producer is C(q) = 10 + 0.1q2. If there are 100 identical orange juice producers in the market, the market supply curve is?
S(p)=500p. Then S(p)=5p. Since there are 100 identical firms, then Aggregate-Supply(p)=100(5p)=500p
Suppose that the cost function for an orange juice firm is C(q) = 10 + 0.1q2. What is the supply function of this firm?
S(p)=5p. The optimal quantity for a single firm with cost C(q)=10+0.1q2 is given by q* that satisfies p=MC(q*) and p greater than or equal to AVC(q*). Since MC(q)=0.2q, then p=.2q implies that q=5p. AVC(5p)=.1(5p)=.5p. Since p is always greater than or equal to 0.5p, then p is greater than or equal to AVC(q*).
The marginal rate of technical substitution of L for K at (L,K) is equal to?
The absolute value of the slope of the tangent to the iso-quant through (L,K) at (L,K).
The following figure shows the graph of the production function of a firm when capital is fixed at some level K. This is a description of the graph. This is a two axis graph in which the horizontal axis measures L and the vertical axis measures output units. A concave (concave down & increasing) increasing curve is shown. From the following, which can be the graph of L(q,K), i.e., the labor necessary to produce q units?
The following is a description of the graph that is shown. A two axis graph in which the horizontal axis is labeled q and the vertical axis is labeled Units of Labor. A convex curve is shown. (convex = increasing & concave up) The graph of L(q,K) is obtained by rotating the graph of f when K is fixed and flipping it.
Consider a call center with production function f(L,K)=30L+300K, where L is units of labor and K is units of capital. Suppose that K=2. For which amounts of labor is the Average Product of Labor equal to 10?
There is no level of labor for which APL is equal to 10. APL is f(L,K)/L. In this case (30L+600)/L, which is 30+600/L. No matter what L is, this figure is more than 30. Thus, it is never 10.
A firm has constant MRTSLK. Suppose that when the firm substitutes x units of labor FOR y units of capital production remains constant. Then, if the firm substitutes x units of labor WITH y units of capital production remains constant.
True.
Consider a firm whose cost function when both L and K are variable is shown in the figure below. (The following is a description of the figure. In a two-axis graph we measure q in the horizontal axis and dollars $ in the vertical axis. A single curve labeled C is shown in the figure. It is a linear curve with positive slope). Is it possible that this is the cost function when both L and K are variable of a production function that satisfies the law of decreasing marginal returns of labor?
True.
Consider the following production function when K is fixed. (This is a description of the figure: it shows a two-axis graph; in the horizontal axis we measure labor and in the vertical axis we measure output, in this case units of juice; the graph of the production function is a curve that has decreasing slope in all of its domain.). Can we say that the production function satisfies the law of decreasing marginal returns of labor?
True. Yes, eventually the marginal return of labor decreases (actually MRL is always decreasing for this production function).
Consider the following production function when K is fixed. (This is a description of the figure: it shows a two-axis graph; in the horizontal axis we measure labor and in the vertical axis we measure output, in this case articles as in the production function of a newspaper; the graph of the production function, for the given level of capital K fixed, is composed of two line segments; the first goes from the origin to the point (10,30); the second, starting from 10 on, is a horizontal line; it is also shown that the production for five units of labor is 15). Can we say that the production function satisfies the law of decreasing marginal returns of labor?
True. Yes, eventually the marginal return of labor decreases (becomes zero after L=10).
Does the following production function satisfies increasing returns to scale: f(L,K)=100LK?
True. a+b = 2 > 1
The following figure shows the graph of the production function of a firm when capital is fixed at some level K. This is a description of the graph. This is a two axis graph in which the horizontal axis measures L and the vertical axis measures output units. A concave increasing curve is shown. It passes through the points (100,23) and (200,42). Is the law of diminishing marginal returns of labor satisfied for this production function (if the graph of all production functions when capital is fixed looks like this, i.e., concave)?
Yes. The Marginal Product of Labor is the slope of the production function when K is fixed. When the graph of this functions are concave, the Marginal Product of Labor is always decreasing as L increases. Thus the law of diminishing marginal returns of labor is satisfied for this production function.
A call center has a production function: f(L,K)=40L+200K. If capital is fixed at K=2, what is the expression for the maximal production as a function of labor?
f(L,2)=40L+400
A call center employs workers and automatic answering machines. Each worker is able to answer a maximum of 5 calls per hour (6 hours a day; a total of 30 calls per day); each automatic answering machine is able to answer a maximum of 10 calls per hour (24 hours a day; a total of 240 calls a day). Denote the number of workers employed by the company by L and the number of automatic answering machines employed by the company by K. The firm's daily production function is?
f(L,K) = 30L + 240K
The following figure shows the production function of a restaurant for a fixed level of capital. (This is a description of the figure: it shows a two-axis graph; in the horizontal axis we measure labor and in the vertical axis we measure meals; the graph of the production function is a line that intersects the vertical axis at a positive amount; this graph is a line with positive slope and passes through the point (4,300)). From the following options, which one can be the production function of this restaurant?
f(L,K)=50(L+K). The production function with K fixed is a line. The only functions that generate lines from the options are f(L,K)=50(L+K), f(L,K)=50L, ad f(L,K)=7LK. We know that neither f(L,K)=50L nor f(L,K)=7LK are the choice, for otherwise the graph would go through (0,0). If K is fixed at 2, then the graph of f(L,2) is exactly the one in the problem.
If a competitive firm's marginal profit is positive at an output of 1000 units,
it will not produce 1000 units
Suppose that a competitive firm maximizes profits, when capital is fixed, by producing q>0 units of output. Which of the following must be true?
p ≥ AVC(q)
Total Cost of Labor:
w*L + r*K cost of labor + cost of capital
Cost ratio:
w/r
Consider a Cobb-Douglas production function f(L, K)= ALαKβ, where A, α and β are positive constants. Then, f has increasing returns to scale if:
α+β>1
Consider a Cobb-Douglas production function f(L, K)= AL^(2/3) * K^β, where A and β are positive constants. Then, f has constant returns to scale if and only if:
β=1/3. For a Cobb-Douglas production function f(L, K)= AL^α * K^β, f satisfies constant returns to scale if and only if α+β=1. Then, this is so if β=1/3.
