Exercise 1
1.1 Number of meals produced:
𝑞 = 5𝐿0.4 𝐾 0.3
𝑞 = 5(3)0.4 (8)0.3
𝑞 = 14.479
So, Jackie can produce 14.479 meals.
1.2 Average number of meals per worker:
𝑞
𝐿
14.479
𝐴𝑃𝐿 =
3
𝐴𝑃𝐿 =
𝐴𝑃𝐿 = 4.826
Average number of meals per kitchen appliance:
𝑞
𝐾
14.479
𝐴𝑃𝐾 =
8
𝐴𝑃𝐾 =
𝐴𝑃𝐾 = 1.81
Since 4.826 >1.81, the average number of meals per worker is higher than the average number of
meals per kitchen appliance. This shows that labor is relatively more productive in this situation.
1.3 Marginal productivity of labor:
𝑀𝑃𝐿 =
𝑑𝑞 𝑑𝐿
×
𝑑𝐿 𝐿
𝑑𝑞
= 0.4 × 5𝐿−0.6 × 𝐾 0.3
𝑑𝐿
𝑑𝑞
= 0.4 × 5 × (3)−0.6 × (8)0.3
𝑑𝐿
𝑀𝑃𝐿 = 1.93
Marginal productivity of Capital:
𝑀𝑃𝐾 =
𝑑𝑞 𝑑𝐾
×
𝑑𝐾 𝐾
𝑑𝑞
= 0.3 × 5𝐿0.4 × 𝐾 −0.7
𝑑𝐾
𝑑𝑞
= 0.3 × 5 × (3)0.4 × (8)−0.7
𝑑𝐾
𝑀𝑃𝐾 = 0.543
Since 1.93 >0.54, the marginal productivity of labor is higher than the marginal productivity of
capital. This indicates that the increasing labor input would be more productive than increasing
capital input.
1.4 Marginal rate of technical substitution:
𝑀𝑅𝑇𝑆𝐿,𝐾 =
𝑀𝑃𝑙
𝑀𝑃𝐾
0.4𝐾
0.3𝐿
1.93
=
0.54
𝑀𝑅𝑇𝑆𝐿,𝐾 =
𝑀𝑅𝑇𝑆𝐿,𝐾
𝑀𝑅𝑇𝑆𝐿,𝐾 = 3.55
The value of 3.55 means that for each unit decrease in the usage of capital, the restaurant would
need to increase 3.55 units of labor to maintain the same level of output.
Exercise 2
2.1
𝑞 = 5𝐿0.4 𝐾 0.3
𝜕𝑞 5 × 0.4 × 𝐾 0.3 2 × 𝐾 0.3
𝑀𝑃𝐿 =
=
=
𝜕𝐿
𝐿0.6
𝐿0.6
𝜕𝑞 5 × 0.3 × 𝐿0.4 1.5 × 𝐿0.4
𝑀𝑃𝐾 =
=
=
𝜕𝐾
𝐾 0.7
𝐾 0.7
Therefore,
𝑀𝑅𝑇𝑆 =
𝑀𝑃𝐿
2
𝐾 4𝐾
=( )× =
𝑀𝑃𝐾
1.5
𝐿 3𝐿
Input mix is optimal when:
𝑀𝑅𝑇𝑆 =
𝑤
𝑟
4𝐾 20
=
=4
3𝐿
5
4𝐾 = 12𝐿
𝐾 = 3𝐿
2.2
𝑞 = 5𝐿0.4 (3𝐿)0.3
5𝐿0.4 (𝐿)0.3 30.3 = 𝑞
5 × 𝐿0.7 × 1.39 = 𝑞
𝐿0.7 = 0.14𝑞
1
𝐿 = (0.14𝑞)0.7 = (0.14𝑞)1.43
𝐿 = (0.14)1.43 (𝑞)1.43
𝐿 = 0.06𝑞1.43
Therefore
𝐾 = 3(0.06𝑞)1.43
𝐾 = (0.18𝑞)1.43
When q is 40
𝐿 = 0.06(40)1.43 = 0.06 × 195.41 = 11.72
𝐾 = 0.18(40)1.43 = 0.18 × 195.41 = 35.17
2.3
𝑞 = 5(11.72)0.4 (35.17)0.3
5(11.72)0.4 (35.17)0.3 = 39
2.4
𝑇𝐶 = 20𝐿 + 5𝐾
𝑇𝐶 = 20 × 0.06𝑞 + 5𝐾 × 0.18𝑞
𝑇𝐶 = 2.1𝑞
𝑞 = 40
𝑇𝐶 = 2.1 + 40
𝑇𝐶 = 84
Exercise 3
3.1 MRTS = (w/r) / (w/K)
MRTS = (2.5/w) / (2.5/K)
MRTS = K/w
Optimal combination of labor and capital:
K = w * MRTS
K = w * 2.5
Therefore, the optimal combination of labor and capital is 2.5 times the cost of labor.
Based on this, it is unlikely that Jackie will fire any of her employees. By halving the cost of
capital, she can maintain the same level of production without having to increase her labor costs.
3.2 Production function: q = f(K,L)
K = w * MRTS
K = w * 2.5
Substituting this into the production function and rearranging: q = f(w * 2.5, L)
K = q / (w * 2.5)
L = (q / w) / 2.5
If Jackie wants to produce 40 meals, the optimal levels of labor and capital are:
K = 40 / (w * 2.5)
L = (40 / w) / 2.5
This result is the same as the previous result. Therefore, Jackie does not need to fire any of her
employees in order to produce 40 meals.
3.3
q = f(K* , L*)
q = f(40 / (w * 2.5), (40 / w) / 2.5)
q = 40
Exercise 4
4.1
Production Function: q(L,K)= 0.6 L +0.4K
r=15 , w=20
𝜕𝑞
𝑀𝑃𝐿 =
= 0.6
𝜕𝐿
𝑀𝑃𝐾 =
New
0.6
20
>
𝑀𝑃𝐿
𝑤
>
𝜕𝑞
= 0.4
𝜕𝐿
𝑀𝑃𝐾
𝑟
0.4
15
So the firm will hire only labor and no capital and the value added to the production by labor is
more than the capital per dollar spent
q=0.6L+0.4(0)
q=0.6L
𝐿=
𝑞
𝑎𝑛𝑑 𝐾 = 0
0.6
Cost Function 𝑐 = 𝑤𝐿 + 𝑟𝐾
𝑞
) + 15(0)
0.6
100𝑞
𝐶=(
)
3
𝐶 = 20 (
4.2
C = ( 100/3) Q
Cost of producing 30 meals
C= ( 100 /3)× 30
C= $1000
Exercise 5
5.1 The input relationship refers to the way in which inputs (in this case, capital (K) and labor
(L)) are used to produce output (q). In this case, the production technology is described as q =
5L^0.4 K^0.3, which means that for every unit of output produced, the production process uses 5
units of K and 4 units of L. However, Jackie has now decided that she will always use exactly 5
units of K and 4 units of L, regardless of the cost or price of the inputs. This means that the input
relationship is now fixed, and the production process will not change the input levels even if the
cost or price of inputs changes.
5.2 The cost function refers to the total cost of producing a given level of output. In this case, the
cost function is given by the equation C(q) = r * K + w * L, where r and w are the prices of
capital and labor, and K and L are the inputs used in production. To compute the cost function,
we simply need to substitute the fixed input values and input prices into the equation:
C(q) = 15 * 5 + 20 * 4 = 75 + 80 = 155
So, the cost function for producing q units of output is C(q) = 155. This means that for any level
of output produced, the total cost will be 155.
5.3 The cost of producing 40 units of output is given by the cost function computed in 5.2, since
the production technology and the input values are fixed. Thus, the cost of producing 40 units of
output is:
C(40) = 155
So, the cost of producing 40 units of output will be 155.
Exercise 6
6.1 This scenario violates the assumption of homogeneous products in perfectly competitive
markets. In a perfectly competitive market, all firms produce identical or nearly identical
products, and there are no differences among the products that can be used to differentiate them
in the market. In this scenario, the unique meals made by each restaurant suggest that the
products are not homogeneous, which would make it difficult for a market to be considered
perfectly competitive.
6.2 This situation violates the assumption of many buyers and sellers in a perfectly competitive
market, as there are only one or two restaurants in the neighborhood. A perfectly competitive
market requires a large number of buyers and sellers, so that no single buyer or seller can
significantly impact market prices.
6.3 This scenario violates the assumption of perfect information in perfectly competitive
markets, as consumers were unable to gather complete information about each restaurant due to
the malfunction of the Yelp app. In a perfectly competitive market, it is assumed that all market
participants have access to complete and accurate information about prices, products, and market
conditions.
6.4 Since Jackie used her political connections to get every other restaurant license denied in her
surrounding area. This violates the assumption of free market entry. Firms are supposed to enter
or exit the market without cost but due to Jackie political influence she is able to limit or control
the market entry and have full control of competitors and therefore the market couldn’t be
consider competitive.