Consider a Keynesian model with consumption function

C = 100 + c(Y – T), 0

Question

Consider a Keynesian model with consumption function

C = 100 + c(Y – T),              0<c<1

where taxes T are given by

T = 100 + tY,                   0<t<1

with marginal tax rate t. An increase in G will increase equilibrium output by the multiplier

BlackTom AI · Accepted Answer

To analyze the multiplier in this Keynesian setup, I’ll first restate the model components and then derive the equilibrium output in terms of G.

- Consumption: C = 100 + c(Y − T), with 0 < c < 1.
- Taxes: T = 100 + tY, with 0 < t < 1.
- Government spending: G is the exogenous component that changes.
- In a simple closed Keynesian framework with no investment or net exports specified, aggregate output Y equals consumption plus government spending: Y = C + G.

Plug C into the identity Y = C + G and substitute T into the consumption function:
- Y = [100 + c(Y − T)] + G
- T = 100 + tY, so Y = 100 + c(Y − (100 + tY)) + G
- Expand: Y = 100 + cY − 100c − c t Y + G
- Collect Y terms on the left: Y − cY + c t Y = 100 − 100c + G
- Factor Y: Y(1 − c + c t) = 100(1 − c) + G
- Solve for Y: Y = [100(1 − c) + G] / [1 − c + c t]

From this expression, the effect of a change in G on Y—the fiscal multiplier—is the derivative dY/dG, holding c and t constant:
- dY/dG = 1 / [1 − c + c t]

Now evaluate each answer option against this result.

Option 1: 1/(1 − c − c t)
- This denominator uses a subtraction of c t from 1 − c, i.e., 1 − c − c t. That would arise if the tax transmission reduced the denominator more strongly than in the correct model, but the algebra shows the c t term is added in the denominator, not subtracted, so this is inconsistent with the derived expression.
- Misconception addressed: misplacing the effect of the tax channel on the marginal propensity to spend after tax, leading to the wrong sign convention for the tax interaction term.

Option 2: 1/(1 − c + c t)
- This matches the derived multiplier exactly. The term (1 − c + c t) in the denominator reflects two competing effects: the marginal propensity to consume (c) reduces the multiplier, while the tax responsiveness to income (t) increases the multiplier through its interaction with the marginal propensity to consume, c t, in the denominator.
- Correct interpretation: taxes reduce the multiplier by the baseline marginal propensity to consume (via the 1 − c part), but the presence of income taxes that respond to Y (via t) mitigates the reduction, yielding the positive c t term in the denominator.

Option 3: 1/(1 − c t)
- This expression ignores the separate c term that discounts autonomous consumption and the linear tax linkage to income. Dropping the 1 − c portion from the denominator misrepresents how much of Y is affected by autonomous spending versus marginal consumption and tax feedbacks.
- Misconception illustrated: treating the tax interaction as if it only multiplies c by t, without the baseline (1 − c) clearance, which is not what the equilibrium algebra yields.

Option 4: 1/(1 − c)
- This is the familiar multiplier when there is no tax system at all (or when taxes do not respond to income, i.e., t = 0). Here, with T = 100 + tY and 0 < t < 1, taxes matter, and the extra term c t in the denominator cannot be neglected. Hence this option ignores the key dynamic from the tax channel.
- Misconception highlighted: assuming a static fiscal stance identical to the case with no distortion from income-dependent taxes.

Putting it together, the correct structural form of the multiplier in this model, given C = 100 + c(Y − T) and T = 100 + tY, is dY/dG = 1 / (1 − c + c t).

Consider a Keynesian model with consumption function C = 100 + c(Y – T), 0<c<1 where taxes T are given by T = 100 + tY, 0<t<1 with marginal tax rate t. An increase in G will increase equilibrium output by the multiplierSingle choice

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