Production and profit for a one input-one output

Question

Question 1: Using graph, illustrate teh effect of a change in the price of the output on production and profit for a one input-one output firm with decreasing return-to-scale technology.

Question 2: Suppose that a firm uses two inputs.

a) Use isocost curves to find the optimal input bundle to produce the output level y.

b) Show how an increase in the price of the input 1 affects the firm decision.

Question 3: Find the cost function if u₁ = 10 and u₂ = 5 for the following utility functions:

a) f(x₁,x₂) = MlN(x₁,2x₂)

b) f(x₁,x₂) = (x₁^1/2, x₂^1/3)

Question 4: Find the demand functions for good x and good y when the utility function is given by u(x,y) = √x + 4√y

Question 5: Considering that an individual evaluates two goods as equivalents. using graph, explain the effect of an increase in his income on the optimal bundle.

Question 6: Find the production bundle with u₁ = 12 and u₂ = 3 and p = 60 for the following production functions.

a) y = (x₁ + 2x₂)²

b) y = x₁^1/2 x₂^1/2

Question 7:

a) Explain the difference between increasing constant and return-to-scale production functions.

b) explain the difference between the short run and the long. How does it affect the cost function?

Question 8:

Consider a consumer with preferences represented by the utility function u(x,y) = 2√x + 4√y. Assuming that px = 3, py = 3 and m = 100,

a) Draw the budget constraint and rhe budget set.
b) find the optimal bundle.
c) Identify the optimal bundle on your graph
d) Draw the indifference curve passing through the optimal bundle.

Question 9: For each of the following utility functions, find the optimal bundle considering m = 60, p1 = 5 and p2 = 6.

a) u(x₁,x₂) = x₁ + x₂

b) u(x₁,x₂) = (x₁)^1/3 (x₂)^2/3

Question 10: Consider a consumer with preferences represented by the utility function u(x₁,x₂) = (x₁)^α(x₂)^β. Find the optimal bundle as function of α, β, p₁, p₂ and m.

Question 11: Give the definition of the following term.

a) Production set
b) Technical rate of substitution
c) Isoprofit curve
d) Isoquant curve
e) Isocast curve

Question 12: Using graph, illustrate the impact of an increase in the price of good 1 on the optimal bundle for on consumer with convex preferences.

Question 13:

Explain in your words the first and the second fundamental theorems of Welfare economics.

Question 14:

Consider an exchnage economy and suppose the initial endowment is not a pareto optimal allocation.

a) Illustrate graphically trhe competitive market equillibrium in an exchange economy
b) Identify the set of Pareto-improving allocations.
c) Identify the curve
d) Identify the competitive market equilibrium

Question 15:

Find the general equilibrium (L(Labor, l(leisure), x(consumption), y(production) and p/w(price of good/wage)) if

u = 3 * x^1/3 * l^2/3
y = √L
l +  L = 24