Question

a. Derive the aggregate supply equation from the sticky price model.

b. Derive the Phillips curve from the aggregate supply equation.

Answer 1

a.

Nominal Wage (W) is set on basis of target w and Price level Pe

W = w*Pe

Dividing both sides by P

W/P = w*(Pe/P)

w = target real wage

Three possibilities:

If P = Pe, then output is at natural level

If P > Pe, then firms higher more workers and output will rise above the natural level

If P < Pe, then output is less than the natural level

Now, a typical firm will set the desired price level as following:'

p = P + (Y - Ybar)

> 0

Y bar is the natural level of output

In the sticky price model, firms must set an expectation of price and output level. So,

p = Pe + (Ye - Ybare)

Now let f = fraction of firms with sticky prices , then above equation can be written as:

P = sPe + (1 - s)[P + (Y - Ybar)]

where Pe = price set by sticky price firms

and P + (Y - Ybar) set by the flexible price firms

Now, we subtract (1-s)P from both LHS and RHS

sP = sPe + (1-s)[(Y - Ybar)]

Divide both sides by s

P = Pe + [(1-s)/s]*(Y - Ybar)

Now, solving for Y

[(1-s)/s]*(Y - Ybar) = P - Pe

(Y - Ybar) = [s /(1-s)] * (P - Pe)

Y = Ybar + * (P - Pe) : AS equation

where = [s /(1-s)]

Y = aggregate output

Ybar= natural rate of output

P = actual price level

Pe = expected price

b.

we know that Aggregate Supply equation is following:

Y = Ybar + * (P - Pe)

P = Pe + (1/)* (Y - Ybar)

subtracting LHS and RHS by P-1

where P-1 is the price in the previous year

So, (P - P-1) = (Pe - P-1) + (1/)* (Y - Ybar)

= e +  (1/)* (Y - Ybar)

We know, (1/)* (Y - Ybar) = -(u - un)

where un = natural unemployment rate

So,

= e - (u - un) : Phillips curve showing inverse relation between unemployment and inflation rate.

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