Change in Price, Slutzky Equation and Labor Supply application

Change in Price: Revisited

* Price Effect (E → E′)

(E→E′′): Substitution Effect due to price change (E′′→ E′):Income Effect

In this case, good x is a normal good.

cf) Derive substitution as well as income effect if good x is neither a normal nor an inferior good.
cf) Derive substitution and income effect if good x is an inferior good. Maybe you can think of two different cases.

2213_change in price.jpg

Slutzky Equation (Slutzky Decomposition)

x′ − x0 = (x′′ − x0 ) + (x′ − x′′) .............(1)

Dividing both sides by ΔPx , we get the following equation,

1897_slutzky equation.jpg...........(2)

l.h.s. can be explained (Δx/ΔPx)|M And the first term on r.h.s. is equivalent to (Δx/ΔPx)|U

As well as the second term on r.h.s. is now revised as follows:

2498_real income.jpg...........(3)

where R means real income.

ΔR/ΔPx means the change in real income with response to change in price.

So, ΔR/ΔPx = -x (Shepard’s lemma)..........(4)
And Δx/ΔR=Δx/ΔM (why? assume about the definition of income effect!)

Plugging (4) and (5) in (3) and then in (2), we can get: (Slutzky Equation)

1810_slutzky equation.jpg

Labor Supply and Leisure (Application)

H + L = 24, where H is hours worked and L is amount of leisure activities.

And hourly wage is given at w0 as well as non-labor income is given at V0.

The utility function is now U = U(L, M)............(1)

The budget equation is M = w0H + V0 = w0 (24 − L) + V0 = 24w0 − w0L + V0 .............(2)

Rewriting (2), we can get M + w0L = 24w0 + V0 (full-income constraint) .............(3)

If we assume the price level of goods (composite good) is PC = $1, then M on l.h.s would equal total number of goods that this consumer buys.

Can you interpret terms in equation (3)?

1) Now can you depict the budget line?
2) Find the utility-maximizing point.
3) Suppose wage increases. Overlap fresh optimal points on the original budget line.
4) Connect those optimization points to obtain PCC And labour supply curve.
5) Why do you suppose is it a backward bending supply curve?

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