High School

Suppose that an economy has 1500 units of capital and 1000 workers. This economy produces computers and shoes. Computer production requires 4 units of capital per worker, and shoe production requires 1 unit of capital per worker.

A. Solve for the amount of labor and capital used in each industry.

Hint: Note that:
1. [tex]K_c + K_s =[/tex] the total capital stock
2. [tex]L_c + L_s =[/tex] the total labor force
3. [tex]K_c = 4L_c[/tex], and [tex]K_s = 1L_s[/tex]

B. Suppose that the number of workers increases to 1250 due to immigration, keeping total capital fixed at 1500. Solve for the distribution of labor and capital between the two sectors.

Answer :

In the original economy with 1500 units of capital and 1000 workers, the computer industry uses 800 units of capital and 200 workers, while the shoe industry uses 700 units of capital and 800 workers. In the scenario where the number of workers increases to 1250 while keeping the total capital fixed at 1500, the computer industry will use 1000 units of capital and 250 workers, while the shoe industry will use 500 units of capital and 1000 workers.

A. To solve for the amount of labor and capital used in each industry, we'll use the given information and the equations provided.

Let's denote:

Kc = Capital used in the computer industry

Lc = Labor used in the computer industry

Ks = Capital used in the shoe industry

Ls = Labor used in the shoe industry

From the given information, we have:

Total capital stock (Kc + Ks) = 1500 units

Total labor force (Lc + Ls) = 1000 workers

We also have the equations:

Kc = 4Lc (Computer production requires 4 units of capital per worker)

Ks = 1Ls (Shoe production requires 1 unit of capital per worker)

Using the equations, we can substitute the values and solve for the unknowns.

From the equation Kc = 4Lc, we can substitute Kc in the total capital stock equation:

4Lc + Ks = 1500

From the equation Ks = 1Ls, we can substitute Ks in the total capital stock equation:

4Lc + 1Ls = 1500

Now we can use the equation Lc + Ls = 1000 to solve for the labor variables.

We have a system of equations:

4Lc + 1Ls = 1500

Lc + Ls = 1000

Solving this system of equations will give us the values for Lc, Ls, Kc, and Ks.

B. To solve for the distribution of labor and capital between the two sectors after the increase in the number of workers to 1250 while keeping the total capital fixed at 1500, we need to adjust the equations accordingly.

Let's denote the new labor force as Lc' + Ls' = 1250.

Using the same equations as before:

4Lc' + 1Ls' = 1500

Lc' + Ls' = 1250

Solving this new system of equations will give us the updated values for Lc', Ls', Kc', and Ks'.

Note that the total capital remains fixed at 1500, so Kc' + Ks' = 1500.

By solving the system of equations, we can determine the distribution of labor and capital in each industry after the increase in the number of workers.

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The computer industry uses 800 capital units and 200 workers in the original economy, which had 1500 capital units and 1000 workers, whereas the shoe industry uses 700 capital units and 800 workers.

How does the economy benefit from capital?

In a financial sense, capital also refers to funds made available for a business improvement or expansion. The reason why capital is valuable is because it makes it possible for people to consume more and better goods and services than they otherwise could.

From the given information:

Total capital stock = 1500 units of capital

Total labor force = 1000 workers

We can set up the following equations:

(1) Kc + Ks = total capital stock

(2) Lc + Ls = total labor force

(3) Kc = 4Lc (capital per worker in the computer industry)

(4) Ks = 1Ls (capital per worker in the shoe industry)

Substituting equations (3) and (4) into equation (1), we get:

4Lc + Ls = 1500

Substituting equation (2) into the equation above, we have:

4Lc + (1000 - Lc) = 1500

4Lc + 1000 - Lc = 1500

Lc ≈ 166.67

Substituting Lc back into equation (3), we find:

Kc = 4Lc

Kc ≈ 666.67

Substituting Ls into equation (4), we find:

Ks = 1Ls

Ks ≈ 833.33

B. If the number of workers increases to 1250 while keeping the total capital fixed at 1500, we can solve for the new distribution of labor and capital between the two sectors.

Using equation (2):

Lc + Ls = total labor force

166.67 + Ls = 1250

Ls ≈ 1083.33

Using equation (4):

Ks = 1Ls

Ks ≈ 1083.33

Since the total capital stock is fixed at 1500, we can find the capital used in the computer industry:

Kc = total capital stock - Ks

Kc ≈ 416.67

Therefore, The computer industry will use 1000 units of capital and 250 workers in the scenario where the total amount of capital remains constant at 1500, while the shoe industry will use 500 units of capital and 1000 workers.

Learn more about capital from the given link.

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