Question

Suppose that the demand for capital is given by \(K=20-2 \mathrm{r}\), where \(K\) is capital and \(r\) is the rental rate. In a graph with \(r\) on the vertical axis and \(K\) on the horizontal axis, plot the demand for capital. Suppose that in the short run the supply of capital is fixed at 6 units. In a graph show how the rental rate is determined in equilibrium. An earthquake destroys part of the capital available in the economy. The supply of capital shrinks to 4 units in the short run. What happens to the equilibrium rental rate?

Short answer

After the earthquake, the equilibrium rental rate increases to 8.

Step-by-step solution

Understand the Demand Equation

The demand for capital is given by the equation \(K = 20 - 2r\), where \(K\) is the quantity of capital and \(r\) is the rental rate. This linear equation represents a demand curve on a graph.

Plot Demand on the Graph

To plot the demand curve, identify the intercepts. When \(r = 0\), \(K = 20 - 2(0) = 20\). When \(K = 0\), solve for \(r\) giving \(0 = 20 - 2r\), thus \(r = 10\). Plot these points: (0, 20) and (10, 0) and draw a line through them.

Identify Initial Supply of Capital

In the short run, the supply of capital is fixed at 6 units. On the graph, this is a vertical line at \(K = 6\).

Determine Initial Equilibrium

Find the rental rate \(r\) when \(K = 6\) by substituting 6 for \(K\) in the demand equation: \(6 = 20 - 2r\). Solving gives \(2r = 14\), so \(r = 7\). The equilibrium point is (6, 7) on the graph.

Effect of the Earthquake on Supply

After the earthquake, the supply shrinks to 4 units. The vertical supply line now shifts to \(K = 4\).

Determine New Equilibrium

Find the new rental rate for \(K = 4\) by substituting 4 for \(K\) in the demand equation: \(4 = 20 - 2r\). Solving this gives \(2r = 16\), so \(r = 8\). The new equilibrium point is (4, 8) on the graph.

Key concepts

Equilibrium Rental Rate

The equilibrium rental rate represents the rental rate at which the demand for capital perfectly matches the supply of capital. In our given scenario, we started with an equation for capital demand: \(K = 20 - 2r\). This equation helps us understand how changes in the rental rate \(r\) affect the quantity of capital \(K\) demanded. To find the equilibrium rental rate, we set the quantity of capital supplied equal to the quantity demanded. Initially, with a fixed supply of 6 units of capital, we inserted \(K = 6\) into the demand equation. Solving for \(r\) (the rental rate), we found that the equilibrium point is where \(r = 7\). Therefore, the equilibrium rental rate before any change is 7. After the earthquake, which reduces the capital supply to 4 units, we repeat the process with \(K = 4\). Inserting this into the equation, we solve for \(r\) and find the new equilibrium rental rate is 8. This shows how changes in capital supply—due to factors like natural disasters—can affect rental rates.

Demand and Supply Curve

A demand and supply curve showcases the relationship between the rental rate of capital and the quantity of capital demanded and supplied. In our task, the demand equation \(K = 20 - 2r\) represents a linear demand curve. Plotting the Demand Curve

• When \(r = 0\), \(K = 20\). This corresponds to the point (0, 20) on the graph.
• When \(K = 0\), solving gives \(r = 10\). This corresponds to the point (10, 0).

Joining these two points, we get a straight downward-sloping demand curve. The steepness of this curve shows how sensitive the demand for capital is to changes in the rental rate. For the supply of capital, which remains constant at certain levels in the short run, we represent it with a vertical line. Initially, this line is at \(K = 6\), showing that no matter the rental rate, capital remains fixed at 6 units. After the earthquake, this vertical line shifts to \(K = 4\), reflecting the reduced supply. This graph clearly helps to visualize how equilibrium is reached and altered by changing supply conditions.

Short Run Capital Supply

The short run capital supply is characterized by its fixed nature. In economic terms, in the short run, certain inputs like capital cannot be quickly adjusted, causing the supply curve to appear vertical on a graph. In our scenario, the initial short run supply was set at a constant 6 units of capital. Impact of Fixed Supply

• With 6 units of supply, the equilibrium rental rate was determined by the intersection of the demand curve at this level, which initially was 7.

However, external shocks like earthquakes can drastically alter this constant supply. When the earthquake in this hypothetical example destructively reduced capital availability to 4 units, the short run supply curve moved leftward to 4 units. Such shifts highlight the inflexibility in adjusting capital levels quickly, affecting not only the equilibrium rates but also the responsiveness to market dynamics. Students should understand the significance of short run fixed capital supply in influencing rental rates and overall market equilibrium.