Intermediate Microeconomics

Note: In order to get full marks you need to explain and show all your

working steps

1. Explain following terms: (14 points)

Production function, Marginal product, Isoquant curve,

TRS, Returns to scale, Isoprofit line,

Isocost line,

2. For each of following production functions comment on the ability to

substitute capital for labor. Note that Q, K, and L denote output, capital, and labor

respectively. (6 points)

A:

Q ? K ? L

B:

0.5 0.5 Q ? K L

C:

Q ? min?K, L?

3. Justify whether the following statement is true or false. “If the production

function is

2

2

2

1 2 1

y ? f (x , x ) ? x x

, then it exhibits constant returns to scale.” (5

points)

False

4. Justify whether the following statement is true or false. “If the value of the

marginal product of labor exceeds the wage rate, then a competitive, profit

maximizing firm would want to hire less labor.” (5 points)

False

Problem 1: (10 points)

A profit-maximizing competitive firm uses just one input, x, to produce output y.

Its production function is:

1/ 2

y ? 8x

The price of output is 40$ each unit and the factor price is 8$ each unit.

a. Find profit-maximizing output and the amount of the factor that the firm

demands. (5 points)

b. Find also the maximum profits. (Hint: the marginal product of the factor

is

1/ 2

4

?

x

.) (5 points)

Problem 2: (10 points)

Weeding a vegetable patch only requires labour as an input. The output of such

an activity is the number of square meters weeded per day, defined as Y. The

relationship between Y and the amount of labour L is:

Y = 20L1/2

a. Draw this relationship on a graph and label it. (5 points)

b. Is the marginal product of labour always positive? Does this relationship

obey the “law” of diminishing marginal returns? (5 points)

Problem 3: (26 points)

A farmer uses two inputs (labour:L, and Fertilizer:F) to produce his only output

(corn: C). His production function is :

1/2 1/2 C?f(L,F) ? 2L F .

a- Derive the expressions of marginal product of labour

MPL

and marginal

product of fertilizer

MPF

. (4 points)

b- Use your answers in (a) to check if the decreasing marginal return law is

verified. (4 points)

c- Let

WL

be the wage per hour and

WF

the price per one kg of fertilizer.

What is the long run cost-minimization condition for this farmer given his

technology of production described above? (4 points)

d- What would be the ratio of labour to fertilizer knowing that

W $8

L

?

and

W ? $4

F

? (2 points)

e- What are the optimal amounts of labour and fertilizer that this farmer

should use to produce 1kg of corn? (3 points)

f- What would be the cost of producing 1kg of corn? (2 points)

g- In general, given his production function, if the farmer doubles the

amount of the inputs by how much would the output be multiplied? What

does this tell you about the returns to scale? (3 points)

h- Is the answer in (g) in contradiction with the answers found in (b)?

Clearly explain. (4 points)

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