Chapter 5: Problem 169

Suppose that John's MPC is constant at $3 / 4$. If his breakeven point occurs at $$\$ 7,000$$, how much will John have to borrow when his income is $$\$ 3,000 ?$$

Short Answer

Expert verified

John will have to borrow $1000 when his income is $3,000.

Step by step solution

Understand the MPC formula

The marginal propensity to consume (MPC) is the change in consumption for a change in income. Mathematically, the MPC formula is: $MPC = \frac{dC}{dI}$ where C is consumption, I is income, and d represents the change in the variable. In this exercise, the MPC is constant at $3/4$.

Determine the consumption function

Since MPC is constant in this case, we can integrate the MPC formula with respect to income, I, to find the consumption function: $C = \int MPCdI$ Plug in the value of MPC: $C = \int (3/4)dI$ Now integrate with respect to I: $C = \frac{3}{4}I + k$ where k is the constant of integration.

Find the constant of integration using the breakeven point

At the breakeven point, consumption is equal to income. We know that the breakeven point occurs when income is $7,000. Therefore, we have: $7000 = \frac{3}{4}(7000) + k$ Now solve for k: $k = (1 - \frac{3}{4})(7000)$ $k = \frac{1}{4}(7000)$ $k = 1750$ Now we have the complete consumption function with the constant k: $C = \frac{3}{4}I + 1750$

Determine John's consumption at an income of $3,000

Now we use the consumption function to find John's consumption when his income is $3,000: $C = \frac{3}{4}(3000) + 1750$ $C = 2250 + 1750$ $C = 4000$ John's consumption will be $4,000 when his income is $3,000.

Calculate how much John needs to borrow

Finally, we find the difference between John's consumption and his income to determine how much he needs to borrow: Amount to borrow = Consumption - Income Amount to borrow = $4000 - $3000 Amount to borrow = $1000 When his income is $3,000, John will have to borrow $1,000.