UNIVERSITY OF TORONTO SCARBOROUGH ...
UNIVERSITY OF TORONTO SCARBOROUGH DEPARTMENT OF MANAGEMENT MGEB02: Price Theory: A Mathematical Approach Sample Test-2 (Solution)
Instructor: A. Mazaheri
Instructions: This is a closed book test, only Calculator is allowed.
You have 2 Hours. Good Luck! Last Name: First Name:
ID
FOR MARKERS ONLY: Q1
Q2
Q3
Q4
Total
30
25
20
20
95
Marks Earned Maximum Marks Possible
Page 1 of 16
Answer all following 4 questions: Question-1 [30 Points] Answer the following Short Questions: a) [5 Points] Refer to the following figure. (a) Find two marginal products for labour (MPL) and comment on them. (b) Comment on returns to scale. Explain how you arrived at your conclusions.
4 90 2
60 40 1
2
4
Solution:
a) MPL is:
∆q 20 = = 20 ∆L 1 ∆q 30 MPL2 = = = 15 ∆L 2 MPL1 =
MPL is declining as expected. b) Doubling the inputs (from (2,2) to (4,4)) will more than double the output (from 40 to 90). Therefore, the production exhibits increasing returns to scale. (or economies of scale)
Page 2 of 16
b) [5 Points] A firmʹs total cost function is given by the equation:
TC = 20q 2 − 4 q + 2 Find the cost-output elasticity of this firm and explain whether the firm’s production is characterized by increasing returns to scale, decreasing returns to scale, or constant returns to scale. Solution:
AC = 20q − 4 + 2 / q MC = 40q − 4 Ec =
MC 40q − 4 = AC 20q − 4 + 2 / q
40q − 4 = 20q − 4 + 2 / q => q = 1 / 10 Therefore : q < 1 / 10 ( IRS ) q = 1 / 10 (CRS ) q > 1 / 10 ( DRS ) We found where AC=MC (CRS) and identified IRS if produced less and DRS if produced more.
Page 3 of 16
c) (6 Points) You own a factory. The wage rate has gone down but the rental price of capital (interest rate) has remained constant. Show graphically, the effect of this change on the expansion path of your firm. Solution: If the price of labor decrease from w1 to w2 (assuming the wages are kept as before) the iso-cost shifts down as a result the new expansion path will lower.
K Old expansion path
New expansion path L C/w1
C/w2
The lower cost of labor capital forces the firm to move to more labor-intensive technology.
Page 4 of 16
d) [8 Points] Suppose a firm‘s technology is given by q = min{L,4 K}. The wage is 4 and the
rental cost of capital is 8. i) In the short run the capital is fixed at K = 10. Find the equation for the short term cost function. Solution: L=4K L = q => SRTC = 4q + 8 * 10 But in the short run the fixed proportion production offers a pre-determined production K = 10, L = 40 => q = 40 => SRTC = 160 + 80 = 240 or infinite There is no real SRTC just a point.
ii) Find the equation for the long run cost function and comment on the returns to scale. Solution: L = 4K L = 4K = q L=q, K=(q/4) LRTC = 4q+8(q/4) =6q Fixed proportion production is characterised by CRS
Page 5 of 16
e) [6 Points] The short run production curve for a firm is shown in the following figure. On the lower diagram, graphically derive the APL and the MPL.
q
q
L
AC MC
Page 6 of 16
Question-2 [25 Points] A consumer has the following utility function U = I 3 / 4 where I is the income. He is currently employed with an annual wage of $80,000. However, he might lose his job and if so he will have to find a second job that pays only $20,000 a year. The probability of him losing his job is is 25%. (p= 0.25)
Note: Round to two digits. a) [8 Points] Graph the utility of this consumer on the following diagram. Calculate the expected utility for this consumer and identify it on the graph. Use the difference between his utility of expected and his expected utility to comment on his risk aversion. U 4756.83 4070.85 3988.07
16811.79
I 20000
65000
80000
E ( I ) = 0.75 × 80000 + 0.25 × 20000 = 65,000 U ( E ( I )) = 65000 3 / 4 = 4070.85 EU = 0.75 × 80000 3 / 4 + 0.25 × 200003 / 4 = 3988.07 His utility of expected is higher than his expected utility (for the same risky proposition), therefore he must dislike risk – i.e. he must be risk averse.
Page 7 of 16
b) [6 Points] An insurance company is offering full insurance in the case of him losing his job. What is the maximum that the consumer will pay for the insurance?
Solution:
EU = 0.75 × 80000 3 / 4 + 0.25 × 200003 / 4 = 3988.07 3988.07 = CE 3 / 4 => CE = 63243.66 RP = 80,000 − 63243.66 = 16756.34 Note: Since all the lost wage is paid by the insurance company the expected income will be 80,000.
c) [6 Points] Now suppose that the insurance company is offering 50% coverage – if the worker loses his job he will get 30,000 from the insurance company - for a premium of $10,000. Would the consumer insure himself?
Solution:
EU with = 0.75 × (80,000 − 10,000) 3 / 4 + 0.25 × ( 20,000 + 30,000 − 10000) 3 / 4 = 3934.75 EU without = 3988.07 EU with < EU without Better off without the insurance. => Will not take the insurance.
Page 8 of 16
d) [5 Points] So far we have assumed that the probability of losing the job is 25%. Now suppose that is not necessarily the case. What would be the maximum probability such that he is indifferent between insuring and not insuring himself if he is offered the insurance as described in part (c).
Solution: EU with = (1 − p ) × (80,000 − 10,000) 3 / 4 + p × (20,000 + 30,000 − 10,000) 3 / 4 = 4303.5 − 4303.5 p + 2828.43 p = 4303.5 − 1475.09 p EU without = (1 − p ) × (80,000) 3 / 4 + p × ( 20,000) 3 / 4 = 4756.83 − 4756.83 p + 1681.79 p = 4756.83 − 3075.04 p EU with = EU withou 4303.5 − 1475.09 p = 4756.83 − 3075.04 p p = 0.2833
Page 9 of 16
Question-3 [20 Points] Suppose that the production function for lava lamps is given by:
q = 0.5KL2 −
L3 3
where q is the number of lamps produced per year, K is the machine-hours of capital, and L is the man-hours of labor. a) [5 Points] Suppose that K = 300: Find average productivity of labor (APL) and the marginal productivity of labor (MPL). At what level of labor does APL reach its maximum? What is the level of output at this point? Solution:
APL = 0.5K L −
L2 3
MPL = K L − L2 APL = MPL L2 0 .5 K L − = K L − L2 3 2L K = = 150 3 2 L = 225 APL is maximized when L=225 L=225, K=300 => q = 3796875 b) [5 Points] Continuing to assume that K = 300: graph the APL and MPL curves on the space below. Make sure to identify the points at which MPL = 0, the point where the output is maximized and the points where APL and MPL are equal?
MPL = KL − L2 = 0 L = 0, L = 300
3796875
Page 10 of 16 225
300
c) [4 Points] Does this production function exhibit constant, increasing, or decreasing returns to scale? Analytically demonstrate your answer.
Solution:
L* = λL K * = λK q
*
( λL )3 L3 3 = 0.5λK (λL ) − = λ (0.5KL − ) = λ3 q > λq 2
3
3
It is characterized by IRS.
Page 11 of 16
d) [6 Points] Find MRTS. Does MRTS exhibit standard diminishing marginal returns to inputs? Graph the isoquant is q = 10. Solution:
q = 0.5KL2 − MRTS =
L3 3
MPL KL − L2 K − L = = MPK 0.5L2 0 .5 L
MRTS exhibits standard characteristics (i.e. declining) only if K-L > 0, K>L, otherwise MPL q = 5.62
q 5.62 K
Expansion Path K=L-10
L 10
Page 16 of 16
Instructor: A. Mazaheri
Instructions: This is a closed book test, only Calculator is allowed.
You have 2 Hours. Good Luck! Last Name: First Name:
ID
FOR MARKERS ONLY: Q1
Q2
Q3
Q4
Total
30
25
20
20
95
Marks Earned Maximum Marks Possible
Page 1 of 16
Answer all following 4 questions: Question-1 [30 Points] Answer the following Short Questions: a) [5 Points] Refer to the following figure. (a) Find two marginal products for labour (MPL) and comment on them. (b) Comment on returns to scale. Explain how you arrived at your conclusions.
4 90 2
60 40 1
2
4
Solution:
a) MPL is:
∆q 20 = = 20 ∆L 1 ∆q 30 MPL2 = = = 15 ∆L 2 MPL1 =
MPL is declining as expected. b) Doubling the inputs (from (2,2) to (4,4)) will more than double the output (from 40 to 90). Therefore, the production exhibits increasing returns to scale. (or economies of scale)
Page 2 of 16
b) [5 Points] A firmʹs total cost function is given by the equation:
TC = 20q 2 − 4 q + 2 Find the cost-output elasticity of this firm and explain whether the firm’s production is characterized by increasing returns to scale, decreasing returns to scale, or constant returns to scale. Solution:
AC = 20q − 4 + 2 / q MC = 40q − 4 Ec =
MC 40q − 4 = AC 20q − 4 + 2 / q
40q − 4 = 20q − 4 + 2 / q => q = 1 / 10 Therefore : q < 1 / 10 ( IRS ) q = 1 / 10 (CRS ) q > 1 / 10 ( DRS ) We found where AC=MC (CRS) and identified IRS if produced less and DRS if produced more.
Page 3 of 16
c) (6 Points) You own a factory. The wage rate has gone down but the rental price of capital (interest rate) has remained constant. Show graphically, the effect of this change on the expansion path of your firm. Solution: If the price of labor decrease from w1 to w2 (assuming the wages are kept as before) the iso-cost shifts down as a result the new expansion path will lower.
K Old expansion path
New expansion path L C/w1
C/w2
The lower cost of labor capital forces the firm to move to more labor-intensive technology.
Page 4 of 16
d) [8 Points] Suppose a firm‘s technology is given by q = min{L,4 K}. The wage is 4 and the
rental cost of capital is 8. i) In the short run the capital is fixed at K = 10. Find the equation for the short term cost function. Solution: L=4K L = q => SRTC = 4q + 8 * 10 But in the short run the fixed proportion production offers a pre-determined production K = 10, L = 40 => q = 40 => SRTC = 160 + 80 = 240 or infinite There is no real SRTC just a point.
ii) Find the equation for the long run cost function and comment on the returns to scale. Solution: L = 4K L = 4K = q L=q, K=(q/4) LRTC = 4q+8(q/4) =6q Fixed proportion production is characterised by CRS
Page 5 of 16
e) [6 Points] The short run production curve for a firm is shown in the following figure. On the lower diagram, graphically derive the APL and the MPL.
q
q
L
AC MC
Page 6 of 16
Question-2 [25 Points] A consumer has the following utility function U = I 3 / 4 where I is the income. He is currently employed with an annual wage of $80,000. However, he might lose his job and if so he will have to find a second job that pays only $20,000 a year. The probability of him losing his job is is 25%. (p= 0.25)
Note: Round to two digits. a) [8 Points] Graph the utility of this consumer on the following diagram. Calculate the expected utility for this consumer and identify it on the graph. Use the difference between his utility of expected and his expected utility to comment on his risk aversion. U 4756.83 4070.85 3988.07
16811.79
I 20000
65000
80000
E ( I ) = 0.75 × 80000 + 0.25 × 20000 = 65,000 U ( E ( I )) = 65000 3 / 4 = 4070.85 EU = 0.75 × 80000 3 / 4 + 0.25 × 200003 / 4 = 3988.07 His utility of expected is higher than his expected utility (for the same risky proposition), therefore he must dislike risk – i.e. he must be risk averse.
Page 7 of 16
b) [6 Points] An insurance company is offering full insurance in the case of him losing his job. What is the maximum that the consumer will pay for the insurance?
Solution:
EU = 0.75 × 80000 3 / 4 + 0.25 × 200003 / 4 = 3988.07 3988.07 = CE 3 / 4 => CE = 63243.66 RP = 80,000 − 63243.66 = 16756.34 Note: Since all the lost wage is paid by the insurance company the expected income will be 80,000.
c) [6 Points] Now suppose that the insurance company is offering 50% coverage – if the worker loses his job he will get 30,000 from the insurance company - for a premium of $10,000. Would the consumer insure himself?
Solution:
EU with = 0.75 × (80,000 − 10,000) 3 / 4 + 0.25 × ( 20,000 + 30,000 − 10000) 3 / 4 = 3934.75 EU without = 3988.07 EU with < EU without Better off without the insurance. => Will not take the insurance.
Page 8 of 16
d) [5 Points] So far we have assumed that the probability of losing the job is 25%. Now suppose that is not necessarily the case. What would be the maximum probability such that he is indifferent between insuring and not insuring himself if he is offered the insurance as described in part (c).
Solution: EU with = (1 − p ) × (80,000 − 10,000) 3 / 4 + p × (20,000 + 30,000 − 10,000) 3 / 4 = 4303.5 − 4303.5 p + 2828.43 p = 4303.5 − 1475.09 p EU without = (1 − p ) × (80,000) 3 / 4 + p × ( 20,000) 3 / 4 = 4756.83 − 4756.83 p + 1681.79 p = 4756.83 − 3075.04 p EU with = EU withou 4303.5 − 1475.09 p = 4756.83 − 3075.04 p p = 0.2833
Page 9 of 16
Question-3 [20 Points] Suppose that the production function for lava lamps is given by:
q = 0.5KL2 −
L3 3
where q is the number of lamps produced per year, K is the machine-hours of capital, and L is the man-hours of labor. a) [5 Points] Suppose that K = 300: Find average productivity of labor (APL) and the marginal productivity of labor (MPL). At what level of labor does APL reach its maximum? What is the level of output at this point? Solution:
APL = 0.5K L −
L2 3
MPL = K L − L2 APL = MPL L2 0 .5 K L − = K L − L2 3 2L K = = 150 3 2 L = 225 APL is maximized when L=225 L=225, K=300 => q = 3796875 b) [5 Points] Continuing to assume that K = 300: graph the APL and MPL curves on the space below. Make sure to identify the points at which MPL = 0, the point where the output is maximized and the points where APL and MPL are equal?
MPL = KL − L2 = 0 L = 0, L = 300
3796875
Page 10 of 16 225
300
c) [4 Points] Does this production function exhibit constant, increasing, or decreasing returns to scale? Analytically demonstrate your answer.
Solution:
L* = λL K * = λK q
*
( λL )3 L3 3 = 0.5λK (λL ) − = λ (0.5KL − ) = λ3 q > λq 2
3
3
It is characterized by IRS.
Page 11 of 16
d) [6 Points] Find MRTS. Does MRTS exhibit standard diminishing marginal returns to inputs? Graph the isoquant is q = 10. Solution:
q = 0.5KL2 − MRTS =
L3 3
MPL KL − L2 K − L = = MPK 0.5L2 0 .5 L
MRTS exhibits standard characteristics (i.e. declining) only if K-L > 0, K>L, otherwise MPL q = 5.62
q 5.62 K
Expansion Path K=L-10
L 10
Page 16 of 16
