Question

Suppose that the amount that people consume is equal to \(80 \%\) of their disposable income of the preceding year, and autonomous consumption is 200 billion. Net investment equals the addition to the capital stock of the preceding period. The capital stock in any year is always equal to \(2.5\) times the level of the same year's consumption. What is the equilibrium income in the economy? How would the economy behave if it is confronted by a sudden drop in aggregate income of 50 billion?

Short answer

The equilibrium income in the economy is 500 billion. If the economy is confronted by a sudden drop in aggregate income of 50 billion, the new disposable income is 450 billion, leading to a consumption level of 560 billion and capital stock of 1400 billion. The economy will adjust by increasing consumption and reducing investments until a new equilibrium is reached.

Step-by-step solution

Consumption Function

Using the provided information, we can write the consumption function as: \(C_t = 0.8Y_{t-1} + 200\), where \(C_t\) is the consumption at time t and \(Y_{t-1}\) is the disposable income of the preceding year.

Capital Stock Function

The capital stock function is given as: \(K_t = 2.5C_t\), where \(K_t\) is the capital stock at time t.

Disposable Income & Capital Stock Relation

Using the capital stock function, we can find a relationship between disposable income and capital stock. From Step 2, we have \(C_t=\frac{1}{2.5}K_t\), substituting into the consumption function from Step 1, we get: \(K_t = 0.8(2.5)Y_{t-1} + 200(2.5)\) Solving for \(Y_{t-1}\), we get: \(Y_{t-1}=\frac{K_t}{0.8(2.5)} - 250\)

Equilibrium Income

In equilibrium, the net investment is equal to the addition to the capital stock of the preceding period. Therefore, the change in capital stock between two consecutive years is zero. So, we have: \(\Delta K_t = K_t - K_{t-1} = 0\) Substituting the relation for disposable income in terms of capital stock from Step 3, we get: \(K_t - \left(\frac{K_t}{0.8(2.5)} - 250\right)(2.5) = 0\) Solving for \(K_t\), we get: \(K_t = 500\) Now substituting the value of \(K_t\) in the relationship between disposable income and capital stock found in Step 3: \(Y_{t-1}= \frac{500}{0.8(2.5)} - 250\) \(Y_{t-1}= 500\) So, the equilibrium income is 500 billion.

Impact of a sudden drop in aggregate income

If the aggregate income suddenly drops by 50 billion, the new disposable income becomes: \(Y'_{t-1} = Y_{t-1} - 50 = 500 - 50 = 450\) Now substituting the new disposable income into the consumption function from Step 1: \(C'_t = 0.8(450) + 200\) \(C'_t = 560\) Now using the capital stock function, we find the new capital stock: \(K'_t = 2.5C'_t = 2.5(560)\) \(K'_t = 1400\) Given that the new capital stock is higher than the initial capital stock, the economy will try to adjust by increasing consumption and reducing investments. This adjustment process will continue until a new equilibrium is reached.

Key concepts

Consumption Function

The consumption function is a fundamental concept in understanding how households decide on spending based on their income. In this exercise, the consumption function is expressed as:\[C_t = 0.8Y_{t-1} + 200\]where:

• \(C_t\) is the consumption at time \(t\).
• \(Y_{t-1}\) represents the disposable income of the previous year.
• Autonomous consumption, represented by 200, is the consumption when income is zero.

This equation reveals that 80% of the previous year's income is spent on consumption in the current year, supplemented by an additional 200 billion from autonomous consumption. This pattern highlights how people's consumption habits can be influenced by their past incomes along with other non-income related factors.

Capital Stock

Capital stock refers to the total value of assets firms use to produce goods and services. In this context, it is related directly to consumption through the relationship:\[K_t = 2.5C_t\]where:

• \(K_t\) is the capital stock at time \(t\).
• \(C_t\) is the consumption at time \(t\).

This formula indicates that the capital stock for any year is 2.5 times the consumption of that year. It implies that as consumption increases, so does the required capital to produce goods and services efficiently. This relationship helps in understanding how capital accumulation and consumption levels are interdependent across economic cycles.

Net Investment

Net investment refers to the actual addition to the capital stock in an economy during a period. It is calculated as the difference between total investment and depreciation of existing capital stock. In equilibrium conditions, where no additional net investment is being made, the capital stock remains constant over time:\[\Delta K_t = K_t - K_{t-1} = 0\]This equation implies that for the economy to be in equilibrium, any new investments exactly replace the depreciated capital, maintaining the capital stock level constant. This balance is crucial as it signifies a stable economic environment where investment meets replacement needs without leading to inflationary pressures.

Aggregate Income

Aggregate income is the total amount of income earned by everyone in an economy. It includes wages, investments, rents, and profits. In this exercise, the equilibrium income acts as a key indicator of the overall economic health and stability:\[Y_{t-1}= \frac{500}{0.8 \times 2.5} - 250\]Achieving equilibrium income at 500 billion illustrates a scenario where the total income is efficiently utilized, balancing consumption and investment to maintain a stable economy. Any fluctuations, such as a sudden drop in aggregate income, can have significant ripple effects, impacting consumption and investment, potentially disrupting the equilibrium state.

Economic Equilibrium

Economic equilibrium occurs when demand equals supply, ensuring that no persuasive forces are acting to change the state of the economy. At this point, production, employment, and income levels are stable. In the given exercise, equilibrium income was determined to be 500 billion. To achieve economic equilibrium, various components such as consumption, investment, and net exports must be aligned. If the aggregate income suddenly drops, it results in shifts in consumption patterns which can initially cause disequilibrium. This new lower disposable income leads to adjustments in consumption functions and capital stock, moving the economy toward a new equilibrium. This process involves both short-term adaptations and long-term structural changes to restore balance.