Question

firm’s production function is 𝑞(𝐾, 𝐿) = 𝐿

𝛽𝐾

𝛼

, so that the

𝑀𝑃𝐿 = 𝛽𝐿

𝛽−1𝐾

𝛼

and the 𝑀𝑃𝐾 = 𝛼𝐿

𝛽𝐾

𝛼−1

. Let 𝛼 =

2

3

 and 𝛽 =

1

3

. Let the slope of the isocost 

line be −

𝑤

𝑟

, and let 𝑤 = 𝑅4 𝑎𝑛𝑑 𝑟 = 𝑅27.

a) Find the marginal rate of technical substitution. (10)

b) Assume that a firm wants to make 1080 units of a product. What is the lowest cost at which 

it can produce 1080 units? (10)

[20]

QUESTION 4 

4.1 Use graphs to explain utility maximization and choice in the two-goods situation. (10)

4.2 The price of pork falls in the market. Discuss using a diagram the substitution and income 

effects to the purchase of pork given the lower price. How is this related to the law of 

demand? (10)

[20]

QUESTION 5 

5.1 Maximize the function 

 𝑢 = 4𝑥

2 + 3𝑥𝑦 + 6𝑦

Answer 1

(3.1)(a)

$MRT S =- \frac{MP_L}{ MP_K}$

$=$ $=$

$=-\frac{\beta L^{\beta -1}K^{\alpha}}{\alpha L^{\beta}K^{\alpha-1}}$

$=-\frac{\beta L}{\alpha K}$

(b)$MRTS=-\frac{w}{r}$

$q(K,L)=L^{\beta}K^{\alpha}$

$\frac{K}{2L}=\frac{w}{r}\\q=L^{\beta}K^{\alpha}\\K=\frac{2Lw}{r}$

$q=L^{\beta}(\frac{2Lw}{r})^{\alpha}\\q=L^{\beta}(\frac{2Lw}{r})L^{\alpha}\\q=L^{\beta+\alpha}(\frac{2w}{r})^{\alpha}$

$1080=L^{\frac{2}{3}+{1}{3}}(\frac{2\times 4}{27})^{\frac{2}{3}}\\1080=L(\frac{8}{27})^{\frac{2}{3}}=(\sqrt[3]{\frac{8}{27}})^{\frac{2}{3}}$

$1080=L\times \frac{4}{9}\\1080\times \frac{9}{4}=L$

$L=2430$

$K=\frac{2Lw}{r}\\K={2\times 2430\times 4}{27}\\K=720$

$TC=w\times L+r\times K\\=4\times 2430+27\times 720\\=29160$

(4.1)

individuals and firms make choices to seek highest satisfaction from their economic decision. For example in the graph below an individual will spend all the income on t-shirts at point "p" and not watch movies. An individual will also spend all the income on movies at point "T" and not buy t-shirts.

(4.2)

In income effect consumption of a good is based on the income of consumers and in substitution effect consumers can replace the cheaper items with expensive ones. When the price of fork falls it will then be replace with expensive meat like chicken.

(5.1)

$𝑢 = 4𝑥2 + 3𝑥𝑦 + 6𝑦$

subject to

$x+y=56$

Lagrangian Equation:

$L = 4x_2 + 3xy + 6y_2 + λ(56 − x − y)$

first-order partials and set them to zero

$L_x= 8x + 3y − λ = 0$

$L_y= 3x + 12y − λ = 0$

$L_λ= 56 − x − y = 0$

from the first two equations we get

$8x + 3y = 3x + 12y$

$x = 1.8y$

 substitute this result into the third equation

$56 − 1.8y − y = 0\\ y = 20$

therefore

$x = 36\\ λ = 34$