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Econ 360; Fall 2019

Homework 4

Due: November 19, Tuesday (in class)

Problem 1 (Chapt: 19, Production Function) (15)

Consider the following three production functions:

F(K,L)= K1/4 L2 ; F(K, L) = K1/4 L1/4 ; F (K, L)= K2L2

a) In the (K; L) space, sketch the map of isoquants for each of them.

b) Give some economic interpretation for MPK and MPL (for an abstract production function).

c) Find analytically MPk for each production function above and plot it in the graph, with K on

the horizontal axis and MPk on the vertical axis, assuming L = 1. Is MPk increasing, decreasing

or constant?

d) Find MPL for each of the four production functions and plot it on the graph, with L on the

horizontal axis and MPL on the vertical axis, assuming K = 2. Is MPL increasing, decreasing or

constant?

e) Provide some economic intuition behind increasing, constant and decreasing returns to scale.

Give an example of technology from real life that captures each of the three cases (one example

per technology).

f) For each of the three production functions above, show formally whether it exhibits increasing,

decreasing or constant returns to scale.

Problem 2 (Chapt: 20, Profit Maximization- Short run) (25)

Northern DeKalb Auto is producing cars using machines (K) and labor (L). The technology is capital

intensive; the production function is given by

F(K, L) = K1/2 L3/4

The value of Northern DeKalb Auto physical capital (machines, real estate etc.) is equal to

K=$16 billon (in calculations ignore billions). We analyze firm’s behavior in the short run (note

K cannot be changed in the short run). Suppose the price of a car is equal to p and the wage rate

is w (parameters).

a) Write down the profit as a function of L.

b) On a graph with L on the horizontal axis and $ on the vertical axis, plot two components of

profit function: total revenue (pF(K;L)), and labor cost (wL) (when drawing, assume p = 1 and w

= 2). On the graph, mark the profit level as the difference between the two lines (for any given

L).

c) In order to find x that maximizes some function f(x), we take the first derivative of the

function with respect to x and set it equal to zero (we call it a first order condition). Please

explain intuitively why this method allows us to find the optimum.

d) Set the derivative of your profit function with respect to L equal to zero and derive the secret

of producer’s happiness (the equation that tells: MPL = (w/p)). Explain the economic intuition

behind the latter condition.

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e) Find analytically the optimal level of labor L that maximizes the profit as a function of the real

wage w/p. (we call it the firm’s labor demand). Find the values of L for the following values of

parameters

Plot your demand for labor on the graph with the real wage (w/p) on the vertical axis and L on

the horizontal one. Using the table mark three points corresponding to the three values of (w/p).

f) What is the maximal profit for each of the three (w/p) values?

Problem 3 (Chapt: 20, Cost Functions) (15): Consider the following Cobb-Douglas technology:

a) What is the returns to scale for each of these functions (use formal argument with 𝜆 )?

Let r (price of K) = w (price of L) = 1

b) Find the cost functions for each of these production function.

c) Plot the cost function on the same graph with y on the horizontal axis and cost on the vertical one.

d) Find and plot the average and marginal cost functions with y on the horizontal axis and average cost on

the vertical one.

Problem 4 (Chapt: 20, 21, Perfect Complements) (20): Consider the following production function:

a) what are the returns to scale for each of these functions (use formal argument with 𝜆)?

Let r (price of K) = w (price of L) = 1

b) Find the cost function for each of these production function.

c) Plot the cost function on the same graph with y on the horizontal axis and cost on the vertical one.

d) Find and plot the average and marginal cost functions with y on the horizontal axis and average cost on

the vertical one.

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Problem 5 (Chapt: 22, Cost Curves) (25): The Northern DeKalb Auto company is considering building

a new Luxury car factory in Wisconsin. The total (fixed) cost of the investment is F = 8. When built, the

factory will allow to produce q cars at the (variable) cost given by

𝐶(𝑞) = 7𝑞 2

a) Does the technology used in the new factory exhibit increasing, decreasing or constant returns to scale

(ignore the fixed costs in this point)?

b) Find a total costs (TC) of producing 3, 5 and 7 cars. In the graph (q, Cost) plot a TC curve, and

decompose it into a fixed cost curve and a variable cost curve by adding the two curves to your graph.

c) Find the values of the average fixed cost (AFC) for three levels of production q = 3; 5 and 7: Plot an

AFC curve in a separate graph. What happens to the AFC when production becomes very large (close to

infinity) and when it is very small (close to zero). Explain.

d) Find the values of the average variable cost AVC for q = 3; 5 and 7 and mark them in the graph from

question c). Connect the three points to obtain the AVC curve.

e) Find the values of the average total cost ATC for q = 3; 5 and 7 and mark them in your graph from c).

Connect the three points to obtain the ATC curve. What are the values of ATC when the production is

very small and very large? Explain which of the two components of ATC, AFC or AVC- dominates in

each of the two extremes. Why?

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